Joel David Hamkins
Transcript of Joel David Hamkins
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
A mate-in-12 problem
White to move on an infinite, edgeless board.
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0Z0Z0Z0ZKZ0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Can white force mate in 12 moves?
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
The theory of infinite gamesHow to play infinite chess and win!
Joel David Hamkins
The City University of New YorkCollege of Staten Island
The CUNY Graduate Center(& MathOverflow ;-)
Mathematics, Philosophy, Computer Science
Virginia Commonwealth UniversityNovember 2014
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
This talk includes joint work with:
C. D. A. Evans (U.S. national master)
Dan Brumleve
Philip Schlicht
Links to articles available at http://jdh.hamkins.org.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
What is an infinite game?
Consider two-player infinite games of perfect information.
The two players take turns making moves:
White: a0 a2 a4 a6 · · ·
Black: a1 a3 a5 · · ·
Together they build a particular play ~a = 〈a0,a1,a2, . . .〉 of thegame.
White wins this instance of the game if the play ~a ∈ A, a fixedset specifying the winning conditions. Otherwise, Black wins.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
What is an infinite game?
Consider two-player infinite games of perfect information.
The two players take turns making moves:
White: a0 a2 a4 a6 · · ·
Black: a1 a3 a5 · · ·
Together they build a particular play ~a = 〈a0,a1,a2, . . .〉 of thegame.
White wins this instance of the game if the play ~a ∈ A, a fixedset specifying the winning conditions. Otherwise, Black wins.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
What is an infinite game?
Consider two-player infinite games of perfect information.
The two players take turns making moves:
White: a0 a2 a4 a6 · · ·
Black: a1 a3 a5 · · ·
Together they build a particular play ~a = 〈a0,a1,a2, . . .〉 of thegame.
White wins this instance of the game if the play ~a ∈ A, a fixedset specifying the winning conditions. Otherwise, Black wins.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
What is an infinite game?
Consider two-player infinite games of perfect information.
The two players take turns making moves:
White: a0 a2 a4 a6 · · ·
Black: a1 a3 a5 · · ·
Together they build a particular play ~a = 〈a0,a1,a2, . . .〉 of thegame.
White wins this instance of the game if the play ~a ∈ A, a fixedset specifying the winning conditions. Otherwise, Black wins.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
What is an infinite game?
Consider two-player infinite games of perfect information.
The two players take turns making moves:
White: a0 a2 a4 a6 · · ·
Black: a1 a3 a5 · · ·
Together they build a particular play ~a = 〈a0,a1,a2, . . .〉 of thegame.
White wins this instance of the game if the play ~a ∈ A, a fixedset specifying the winning conditions. Otherwise, Black wins.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
What is an infinite game?
Consider two-player infinite games of perfect information.
The two players take turns making moves:
White: a0 a2 a4 a6 · · ·
Black: a1 a3 a5 · · ·
Together they build a particular play ~a = 〈a0,a1,a2, . . .〉 of thegame.
White wins this instance of the game if the play ~a ∈ A, a fixedset specifying the winning conditions. Otherwise, Black wins.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
What is an infinite game?
Consider two-player infinite games of perfect information.
The two players take turns making moves:
White: a0 a2 a4 a6 · · ·
Black: a1 a3 a5 · · ·
Together they build a particular play ~a = 〈a0,a1,a2, . . .〉 of thegame.
White wins this instance of the game if the play ~a ∈ A, a fixedset specifying the winning conditions. Otherwise, Black wins.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
What is an infinite game?
Consider two-player infinite games of perfect information.
The two players take turns making moves:
White: a0 a2 a4 a6 · · ·
Black: a1 a3 a5 · · ·
Together they build a particular play ~a = 〈a0,a1,a2, . . .〉 of thegame.
White wins this instance of the game if the play ~a ∈ A, a fixedset specifying the winning conditions. Otherwise, Black wins.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Winning strategies
A strategy for a player is a function that tells the player whatmove to make next, given the finite sequence of precedingmoves.
σ : 〈a0,a1, . . . ,ak−1〉 7→ ak
A winning strategy for a player is a strategy that, when followed,results in a winning play, no matter how the opponent plays.
Obviously, at most one of the players can have a winningstrategy.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Example infinite games
Consider the following winning conditions for White, where theplayers play 0 or 1 at each step.
A = the set of eventually periodic sequences.B = set of universal sequences (all finite strings occur)C = set of strings containing 00 or 111 but not 0001.D = a countable set.
If white has a winning strategy for X , does black have a winningstrategy for the complement?
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Example infinite games
Consider the following winning conditions for White, where theplayers play 0 or 1 at each step.
A = the set of eventually periodic sequences.B = set of universal sequences (all finite strings occur)C = set of strings containing 00 or 111 but not 0001.D = a countable set.
If white has a winning strategy for X , does black have a winningstrategy for the complement?
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Example infinite games
Consider the following winning conditions for White, where theplayers play 0 or 1 at each step.
A = the set of eventually periodic sequences.B = set of universal sequences (all finite strings occur)C = set of strings containing 00 or 111 but not 0001.D = a countable set.
If white has a winning strategy for X , does black have a winningstrategy for the complement?
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Example infinite games
Consider the following winning conditions for White, where theplayers play 0 or 1 at each step.
A = the set of eventually periodic sequences.B = set of universal sequences (all finite strings occur)C = set of strings containing 00 or 111 but not 0001.D = a countable set.
If white has a winning strategy for X , does black have a winningstrategy for the complement?
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Example infinite games
Consider the following winning conditions for White, where theplayers play 0 or 1 at each step.
A = the set of eventually periodic sequences.B = set of universal sequences (all finite strings occur)C = set of strings containing 00 or 111 but not 0001.D = a countable set.
If white has a winning strategy for X , does black have a winningstrategy for the complement?
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Fundamental question
Question
In any game, must one of the players have a winning strategy?
Definition
A game is determined if one player has a winning strategy.
Question
Is every game determined?
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
DeterminacyAxiom of Determinacy (AD)
The axiom of determinacy is the assertion that every game hasa strategy for one of the players.
Do we have any reason to believe this axiom?
In what sense is this an “axiom”?
Axioms are self-evident principles(Euclidean axioms, Peano axioms, set theory axioms)
Axioms define a collection of structures, a domain of discourse(group axioms)
Axioms express a succinct prescient vision of possibility(CH, MA, V = L, ♦)
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
DeterminacyAxiom of Determinacy (AD)
The axiom of determinacy is the assertion that every game hasa strategy for one of the players.
Do we have any reason to believe this axiom?
In what sense is this an “axiom”?
Axioms are self-evident principles(Euclidean axioms, Peano axioms, set theory axioms)
Axioms define a collection of structures, a domain of discourse(group axioms)
Axioms express a succinct prescient vision of possibility(CH, MA, V = L, ♦)
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Evidence for AD
White has a winning strategy: ∃a0 ∀a1 ∃a2 · · · A(~a)
Black has a winning strategy: ∀a0 ∃a1 ∀a2 · · · ¬A(~a)
AD asserts: White has no ws ⇐⇒ Black has a ws.
In other words, AD is the assertion:
¬(∃a0 ∀a1 ∃a2 · · · A(~a)
)⇐⇒
(∀a0 ∃a1 ∀a2 · · · ¬A(~a)
).
Thus, AD is an infinitary de Morgan law.
¬∃x A(x) ⇐⇒ ∀x ¬A(x)
¬∃x ∀y A(x , y) ⇐⇒ ∀x ∃y ¬A(x , y)
Conclusion: all finite-length games are determined.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Evidence for AD
White has a winning strategy: ∃a0 ∀a1 ∃a2 · · · A(~a)
Black has a winning strategy: ∀a0 ∃a1 ∀a2 · · · ¬A(~a)
AD asserts: White has no ws ⇐⇒ Black has a ws.
In other words, AD is the assertion:
¬(∃a0 ∀a1 ∃a2 · · · A(~a)
)⇐⇒
(∀a0 ∃a1 ∀a2 · · · ¬A(~a)
).
Thus, AD is an infinitary de Morgan law.
¬∃x A(x) ⇐⇒ ∀x ¬A(x)
¬∃x ∀y A(x , y) ⇐⇒ ∀x ∃y ¬A(x , y)
Conclusion: all finite-length games are determined.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Evidence for AD
White has a winning strategy: ∃a0 ∀a1 ∃a2 · · · A(~a)
Black has a winning strategy: ∀a0 ∃a1 ∀a2 · · · ¬A(~a)
AD asserts: White has no ws ⇐⇒ Black has a ws.
In other words, AD is the assertion:
¬(∃a0 ∀a1 ∃a2 · · · A(~a)
)⇐⇒
(∀a0 ∃a1 ∀a2 · · · ¬A(~a)
).
Thus, AD is an infinitary de Morgan law.
¬∃x A(x) ⇐⇒ ∀x ¬A(x)
¬∃x ∀y A(x , y) ⇐⇒ ∀x ∃y ¬A(x , y)
Conclusion: all finite-length games are determined.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Evidence for AD
White has a winning strategy: ∃a0 ∀a1 ∃a2 · · · A(~a)
Black has a winning strategy: ∀a0 ∃a1 ∀a2 · · · ¬A(~a)
AD asserts: White has no ws ⇐⇒ Black has a ws.
In other words, AD is the assertion:
¬(∃a0 ∀a1 ∃a2 · · · A(~a)
)⇐⇒
(∀a0 ∃a1 ∀a2 · · · ¬A(~a)
).
Thus, AD is an infinitary de Morgan law.
¬∃x A(x) ⇐⇒ ∀x ¬A(x)
¬∃x ∀y A(x , y) ⇐⇒ ∀x ∃y ¬A(x , y)
Conclusion: all finite-length games are determined.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Further evidence: AD has nice consequences
The axiom of determinacy (without using AC) has amazinglyattractive consequences:
Every set of reals is Lebesgue measurable.Every set of reals differs from an open set by a meager set.Every uncountable set of reals contains a perfect set.
On consequentialist grounds, this is strong evidence for AD.
AD leverages innate human capacity for strategic reasoning.
AD is deeply connected with large cardinals, equiconsistentwith infinitely many Woodin cardinals.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Evidence against ADNevertheless,
Theorem
AD contradicts the axiom of choice.
Proof idea
To prove this, construct the counterexample set in a transfiniterecursion of length continuum. Well-order the binary-playstrategies in order type continuum, and at stage α, we’ve madefewer than continuum many promises, and so we can add aplay to the set and to the complement so as to prevent a givenstrategy from winning for either side.
Set theorists often study fragments of AD in ZFC.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Open gamesOpen games generalize the concept of finite games.
A game is open for a player, if every winning instance of thegame for that player is won already at a finite stage. That is, if ~ais a winning play, then there is some finite play s = ~a � n suchthat all extensions of s are winning, no matter how playcontinues from s.
This is equivalent to saying that A is an open set in the producttopology on the space of all plays.
Similarly, a game is closed for a player, if it is open for theopposing player.
To win an open game, you must win at some finite stage. Towin a closed game, it suffices never to lose at any finite stage.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Open Determinacy
Theorem (Gale,Stewart 1953)
Every open game is determined. One of the players has awinning strategy.
We give a proof using the concept of ordinal game values.
Recall the ordinals...
0 1 2 · · · ω ω+1 · · · ω·2 ω·2+1 · · · ω·n+k · · · ω2
ω2+1 · · · ωn · · · ωω · · · ε0 · · · ωCh1
?= ωCK
1 · · · ω1 · · ·
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Open Determinacy
Theorem (Gale,Stewart 1953)
Every open game is determined. One of the players has awinning strategy.
We give a proof using the concept of ordinal game values.
Recall the ordinals...
0 1 2 · · · ω ω+1 · · · ω·2 ω·2+1 · · · ω·n+k · · · ω2
ω2+1 · · · ωn · · · ωω · · · ε0 · · · ωCh1
?= ωCK
1 · · · ω1 · · ·
How to win at infinite chess Joel David Hamkins, New York
![Page 30: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/30.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Open Determinacy
Theorem (Gale,Stewart 1953)
Every open game is determined. One of the players has awinning strategy.
We give a proof using the concept of ordinal game values.
Recall the ordinals...
0 1 2 · · · ω ω+1 · · · ω·2 ω·2+1 · · · ω·n+k · · · ω2
ω2+1 · · · ωn · · · ωω · · · ε0 · · · ωCh1
?= ωCK
1 · · · ω1 · · ·
How to win at infinite chess Joel David Hamkins, New York
![Page 31: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/31.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Open Determinacy
Theorem (Gale,Stewart 1953)
Every open game is determined. One of the players has awinning strategy.
We give a proof using the concept of ordinal game values.
Recall the ordinals...
0 1 2 · · · ω ω+1 · · · ω·2 ω·2+1 · · · ω·n+k · · · ω2
ω2+1 · · · ωn · · · ωω · · · ε0 · · · ωCh1
?= ωCK
1 · · · ω1 · · ·
How to win at infinite chess Joel David Hamkins, New York
![Page 32: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/32.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Open Determinacy
Theorem (Gale,Stewart 1953)
Every open game is determined. One of the players has awinning strategy.
We give a proof using the concept of ordinal game values.
Recall the ordinals...
0 1 2 · · · ω ω+1 · · · ω·2 ω·2+1 · · · ω·n+k · · · ω2
ω2+1 · · · ωn · · · ωω · · · ε0 · · · ωCh1
?= ωCK
1 · · · ω1 · · ·
How to win at infinite chess Joel David Hamkins, New York
![Page 33: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/33.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Open Determinacy
Theorem (Gale,Stewart 1953)
Every open game is determined. One of the players has awinning strategy.
We give a proof using the concept of ordinal game values.
Recall the ordinals...
0 1 2 · · · ω ω+1 · · · ω·2 ω·2+1 · · · ω·n+k · · · ω2
ω2+1 · · · ωn · · · ωω · · · ε0 · · · ωCh1
?= ωCK
1 · · · ω1 · · ·
How to win at infinite chess Joel David Hamkins, New York
![Page 34: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/34.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Open Determinacy
Theorem (Gale,Stewart 1953)
Every open game is determined. One of the players has awinning strategy.
We give a proof using the concept of ordinal game values.
Recall the ordinals...
0 1 2 · · · ω ω+1 · · · ω·2 ω·2+1 · · · ω·n+k · · · ω2
ω2+1 · · · ωn · · · ωω · · · ε0 · · · ωCh1
?= ωCK
1 · · · ω1 · · ·
How to win at infinite chess Joel David Hamkins, New York
![Page 35: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/35.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Open Determinacy
Theorem (Gale,Stewart 1953)
Every open game is determined. One of the players has awinning strategy.
We give a proof using the concept of ordinal game values.
Recall the ordinals...
0 1 2 · · · ω ω+1 · · · ω·2 ω·2+1 · · · ω·n+k · · · ω2
ω2+1 · · · ωn · · · ωω · · · ε0 · · · ωCh1
?= ωCK
1 · · · ω1 · · ·
How to win at infinite chess Joel David Hamkins, New York
![Page 36: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/36.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Open Determinacy
Theorem (Gale,Stewart 1953)
Every open game is determined. One of the players has awinning strategy.
We give a proof using the concept of ordinal game values.
Recall the ordinals...
0 1 2 · · · ω ω+1 · · · ω·2 ω·2+1 · · · ω·n+k · · · ω2
ω2+1 · · · ωn · · · ωω · · · ε0 · · · ωCh1
?= ωCK
1 · · · ω1 · · ·
How to win at infinite chess Joel David Hamkins, New York
![Page 37: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/37.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Open Determinacy
Theorem (Gale,Stewart 1953)
Every open game is determined. One of the players has awinning strategy.
We give a proof using the concept of ordinal game values.
Recall the ordinals...
0 1 2 · · · ω ω+1 · · · ω·2 ω·2+1 · · · ω·n+k · · · ω2
ω2+1 · · · ωn · · · ωω · · · ε0 · · · ωCh1
?= ωCK
1 · · · ω1 · · ·
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Game value
In any open game, we define the ordinal game value ofpositions:
A position that is already won has value 0.A position one move away from being won has value 1.
The values continue transfinitely:value is α+ 1, if white may play to a position with value α.if black to play, value is supremum of the values of thepositions to which black may play.
Some positions may be left without value.
White can play to strictly decrease value. Black can maintainvalue, but not increase it.
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Game values provide strategies
White wants to reduce value. From a valued position, white cando so.
Black wants to maintain value. He cannot increase it.
Conclusion
A position is winning for white if and only if it has an ordinalvalue.
White wins via the value-reducing strategy.
Black wins via the value-maintaining strategy.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Illustrating game value ω2
ω2 The initial value, with black to play.
ω · 3 + 17 Black must play to some ordinal below ω2.
ω · 3 + 16 White plays to reduce 1 each step, leading in 17 moves to
ω · 3 Black to play.
ω · 2 + 2014 Black must play below ω · 3
... Black survives another 2014 moves until...
ω · 2 Black to play.
ω + 10100 Black can now survive another googol many moves...
...ω Black to play.
G With Graham’s number, Black can survive a bit longer.
...0 Eventually, the value hits 0, at which time white wins.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Concrete meaning of the game values
Finite value n. . . . . . white can force a win in n moves, but not faster.
Value ω. . . . . . black to move, but every move leads to some (arbitrarilylarge) finite value n. Black will lose, but he can cause an arbitrarilylong delay.
Value ω + 10. . . . . . white can force value ω in 10 moves.
Value ω2. . . . . . black to play, but every move creates value ω · n + k .Think: n is the number of future announcements he will make, foreach of which play will take that long before the next announcement.
Value ω3. . . . . . black can announce an arbitrary number, which will bethe number of large announcements. Each large announcement isthe number of small announcements to be made; each smallannouncement is the number of moves before the nextannouncement.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite chess
I’d like to illustrate various transfinite game values as they arisein the case of infinite chess.
However much I’d like to sit down in a cafe for a game of infinitechess, over espresso, nevertheless it remains a game purely ofthe mind.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Infinite chess
Infinite chess is chess played on an infinite edgeless chessboard, arranged like the integer lattice Z× Z.
The familiar chess pieces—kings, queens, bishops, knights,rooks and pawns—move about according to their usual chessrules, with bishops on diagonals, rooks on ranks and files andso on, with each player striving to place the opposing king intocheckmate.
There is no standard starting configuration in infinite chess, butrather a game proceeds by setting up a particular position onthe board and then playing from that position.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Clarifying the rules
Let me clarify a few of the rules as they relate to infinite chess.
At most one king of each colorThere is no boundary, hence no pawn promotionThere is no castling and no en passantAbandon the 50 move rule as limitingInfinite play is a drawWe may abandon the three-fold repetition ruleFor arbitrary starting positions, one must clarify someweird boundary cases, e.g. White to move, black in check,white in checkmate
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Infinite chess is an open game
Checkmate, when it occurs, does so after finitely many moves.Infinite chess is therefore an open game, and thus subject tothe theory of transfinite game values.
So we can imagine chess positions with various large ordinalgame values.
(The possibility of draw does not significantly upset thetransfinite game value analysis.)
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZrZrZrZ0Z0Z0j0Z0Z0Z0Z0S0Z0Z0ZQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Black to move.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZrZ0ZrZ0Z0Z0j0Z0Z0Z0Z0S0Z0Z0ZQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Black moves up arbitrary height
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZrZ0ZrZ0Z0Z0jRZ0Z0Z0Z0Z0Z0Z0ZQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Check
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZrZkZrZ0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0ZQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZrZkZrZ0Z0ZQZRZ0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Check
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0Z0ZrZ0ZrZ0Z0ZQZRZ0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0jRZ0Z0ZrZ0ZrZ0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Check
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0Z0ZRZ0Z0ZrZ0ZrZ0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0ZQZRZ0Z0ZrZ0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Check
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0Z0Z0Z0Z0Z0Z0ZQZRZ0Z0ZrZ0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0jRZ0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0ZrZ0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Check
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0ZrZ0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0ZQZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZrZ0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Check
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0Z0Z0Z0Z0Z0Z0ZQZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZrZ0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0jRZ0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZrZ0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Check
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZrZ0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0ZQZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZrZ0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Check
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0j0Z0Z0Z0Z0Z0Z0Z0ZQZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZrZ0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
A finite position with value ω
Z0Z0Z0Z0Z0Z0ZrZ0Z0Z0Z0jRZ0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZrZ0ZrZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Checkmate. Black can cause arbitrary delay, but the onlychoice is on first move, so the initial value is ω.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Several infinite positions with value ω
OPZ0ZPOPOPO0Z0OPOPOPZ0ZPOPOPO0Z0OPOPOPZ0ZPAPOPO0jrOPOPOPo0oPSROPZPZPZPOPOBOBOBOPOPOPOPOPOPOPOPOPOPO
OPZ0ZPOPPO0Z0OPOOPZ0ZPOPPO0Z0OPOOPZ0ZPAPPO0jrOPOOPZpZPOPPO0O0OPOOPAPAPOPPOPOPLPOOPOPOPOP
Z0ZPO0OPO0Z0OPZPOPZ0ZPO0OPO0Z0OPZPOPZpZPO0OPO0Z0OPsPOPZ0ZPOBOPO0Z0OPjPOPZ0ZPSPSPO0Z0OPOPOPZ0ZPOPOPO
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Positions with value ω2
Z0ZPO0OPO0Z0Z0Z0OPZPOPZ0Z0Z0ZPO0OPO0Z0Z0o0OPZPOPZ0J0Z0ZPO0OPO0Z0Z0Z0OPsPOPZ0Z0Z0ZPOBOPO0Z0Z0Z0OPjPOPZ0Z0Z0ZPSPSPO0s0Z0Z0OPOPOPZ0Z0Z0ZPOPOPO0Z0Z
Z0ZPO0OPOPZ0Z0Z0Z0OPZPOPO0Z0Z0ZpZPO0OPOPZ0Z0Z0Z0OPZPOPO0Z0Z0Z0ZPO0jPOPZKZ0Z0Z0OPsPZPO0Z0Z0Z0ZPOBOBOPZ0s0Z0Z0OPOPOPO0Z0Z0Z0ZPOPOPOPZ0Z0Z0Z0OPOPOPO0Z0Z0Z0ZPOPOPOPZ0Z0Z
Z0ZPO0OPOPOPZ0Z0Z0Z0OPZPOPOPO0Z0Z0Z0ZPO0OPOPOPZ0Z0Z0Z0OPsPOPOPO0Z0Z0ZpZPO0OPOPOPZ0J0Z0Z0OPZPOPOPO0Z0Z0Z0ZPONOPjPOPZ0Z0Z0Z0OPOPMPMPO0s0Z0Z0ZPOPOPOPOPZ0Z0Z0Z0OPOPOPOPO0Z0Z0Z0ZPOPOPOPOPZ0Z0Z
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0ZrZ0Z0Z0Z0Z0Z0Z00Z0ZPZPZKZ0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OrO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
Black to move.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OrO0Z0ZrZ0Z0Z0Z0Z0Z0Z00Z0ZPZPZKZ0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
He moves trapped rook up arbitrary height.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0ZrZ0Z0Z0Z0Z0Z0Z00Z0Z0ZPZKZ0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
White should capture from left side.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZKZ0s0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
Now black begins to harass white king.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0ZKZ0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0s0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
White must chase down the rook to avoid perpetual check.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0ZKZrZ0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0ZrZ0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0ZKZ0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0ZrZ0Z0Z0Z0Z00Z0Z0ZPZ0ZKZ0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
Black must move away to save rook.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0ZrZ0Z0Z0Z0Z00Z0ZPZPZ0ZKZ0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
Now is white’s chance to advance a pawn.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0ZKZ0Z0s0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0ZKZ0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0s0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0ZrZ0Z0Z0Z0Z00Z0ZPZPZ0Z0ZKZ0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0ZKZ0s0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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Black moves arbitrary distance out.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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Another chance to advance a pawn.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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Black harasses the white king.
How to win at infinite chess Joel David Hamkins, New York
![Page 99: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/99.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0ZKZ0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0s0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
White must chase him down.
How to win at infinite chess Joel David Hamkins, New York
![Page 100: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/100.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0ZKZ0Z0Z0ZrZ00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0ZrZ00Z0ZPZPZ0Z0Z0Z0ZKZ0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0ZKZ0Z0Z0s0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0ZKZ0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0s0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0ZKZ0Z0ZrZ00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0ZrZ00Z0ZPZPZ0Z0Z0Z0Z0ZKZ0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0ZKZ0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0s0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0ZKZ0ZrZ00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0ZrZ00Z0ZPZPZ0Z0Z0Z0Z0Z0ZKZ0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0ZKZ0s0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0ZKZ0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0s0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0ZKZrZ00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0ZrZ00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0ZKZ0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
![Page 114: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/114.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0ZrZ0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0ZKZ0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
(Black should actually move arbitrary distance to the right.)
How to win at infinite chess Joel David Hamkins, New York
![Page 115: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/115.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0ZrZ0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0ZKZ0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
![Page 116: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/116.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0s0Z0Z0Z0Z0Z0ZKZ0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0ZKZ0Z00Z0ZPZPZ0s0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0ZrZ0Z0Z0Z0Z0ZKZ0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0ZrZ0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0ZKZ0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0s0Z0Z0Z0Z0ZKZ0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0ZKZ0Z0Z00Z0ZPZPZ0s0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPAPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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The bishop unlocks the door.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0ZKZ0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0s0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0ZrZ0Z0Z0Z0Z00Z0ZPZPZ0Z0ZKZ0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0ZKZ0s0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0ZKZ0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0s0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0ZKZrZ0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0ZrZ00Z0ZPZPZ0Z0Z0ZKZ0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
Black can move rook arbitrary distance.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0ZKZ0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0s0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0ZKZrZ00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0ZrZ0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0ZKZ0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
The portcullis opens...
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
![Page 180: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/180.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0ZKZ0Z0Z0Z0Z0Z00Z0ZPZPZ0s0Z0Z0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0ZrZ0ZKZ0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0ZrZ0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0ZKZ0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0s0ZKZ0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0ZKZ0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0s0Z0Z0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0ZrZKZ0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0ZrZ0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0ZKZ0Z0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0s00Z0ZPZPZ0ZKZ0Z0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QLQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0s00Z0ZPZPZ0ZKZ0Z0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QL0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZkZ0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
Queens enter the mating chamber.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0s00Z0ZPZPZ0ZKZ0Z0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QL0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0j0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Releasing the Hordes, with value ω2
Z0Z0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0OPO0Z0Z0Z0Z0Z0Z0Z0Z0s00Z0ZPZPZ0ZKZ0Z0Z0Z0Z0Z0Z0ZZ0A0O0O0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZPZPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0ZPOPZ0Z0Z0Z0Z0Z0Z0Z0Z0ZLQZ0OPO0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0Z0LQZ0Z0Z0Z0Z0Z0Z0Z0ZLQZ0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0QZ0Z0O0j0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
Checkmate
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPZPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPsPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZKZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
a b c d e f g h i j k l m n o p q r s t u v w x y
Black to move.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPsPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZKZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
a b c d e f g h i j k l m n o p q r s t u v w x y
The first black tower ascends.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZKZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
a b c d e f g h i j k l m n o p q r s t u v w x y
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZKZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
a b c d e f g h i j k l m n o p q r s t u v w x y
Black harasses white king.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
a b c d e f g h i j k l m n o p q r s t u v w x y
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0ZPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0ZPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black should actually move arbitrary distance right.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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White advances a pawn.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZ0O0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black should actually move arbitrary distance right.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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White advances pawn.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0ZPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black should actually move arbitrary distance right.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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White advances pawn.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZ0O0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black should actually move arbitrary distance right.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0ZPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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White advances pawn.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0ZPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0ZPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0ZPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0ZPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0ZPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0ZPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0ZPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0ZPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZPoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black should actually move arbitrary distance right.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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White advances pawn.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZ0oBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black should actually move arbitrary distance right.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0ZKZ05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0ZKZ05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPO0ZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOBOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black should actually move arbitrary distance right.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0ZKZ03 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black should actually move arbitrary distance right.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0ZKZ03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPOPOBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black should actually move arbitrary distance right.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0ZKZ0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0ZKZ05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0ZKZ0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0ZKZ0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0ZKZ03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBoBZPZBOBOPOPZ0Z0Z05 Z0ZPOBOPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0ZKZ03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black should actually move arbitrary distance right.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPZPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPsPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0ZKZ03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black key pawn attacks second tower.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPsPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0ZKZ03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Second tower ascends.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0ZKZ03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0ZKZ03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0ZKZ0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0ZKZ0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0s0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0ZPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZPO0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0ZPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0ZPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black should actually move arbitrary distance right.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0ZKZ011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0ZKZ0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0ZKZ09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0ZKZ0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0ZKZ07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0ZKZ0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0ZKZ0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0ZKZ05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0Z0s0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0ZKZ0Z4 0Z0OPO0OBOPOPOPOPOPZ0Z0Z03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0Z0Z0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0ZKZ03 Z0ZPOPZPOPOPOPOPOPO0ZrZ0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
a b c d e f g h i j k l m n o p q r s t u v w x y
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
Iterated lock and key, value ω2 · 4
15 Z0ZPO0OPZPO0OPZPOPO0Z0Z0Z14 0Z0OPZPO0OPZPO0OPOPZ0Z0Z013 Z0ZPO0OPOPO0OPZPOPO0ZrZ0Z12 0Z0OPOPO0OPZPO0OPOPZ0Z0Z011 Z0ZPO0OPZ0O0OPZPOPO0Z0Z0Z10 0Z0OPZPO0OPZPO0jPOPZ0Z0Z09 ZpZPO0OPZPO0OPsPZPO0Z0Z0Z8 0Z0OPZPO0OPsPoBZBOPZ0Z0Z07 Z0ZPO0OPZPoBZPZBOPO0Z0Z0Z6 0Z0OPZBOBZPZBOBOPOPZ0Z0Z05 Z0ZPOBZPZBOBOPOPOPO0Z0Z0Z4 0Z0OPO0OBOPOPOPOPOPZ0ZKZ03 Z0ZPOPZPOPOPOPOPOPO0Z0Z0Z2 0Z0OPOPOPOPOPOPOPOPZ0Z0Z01 Z0ZPOPOPOPOPOPOPOPO0Z0Z0Z
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Black should actually move arbitrary distance right.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
State of the art: value ω3
Z0Z0Z0OPOPO0OPZPO0OPZPO0OB0Z0Z0ZPOPOPZPO0OPZPO0OPsPZZ0Z0Z0OPOPO0OPZPO0OPZPZBoB0Z0Z0ZPOPOPZPO0OPZPO0OBZPZZ0Z0Z0OPOPO0OPZPO0OPsPZBOB0Z0Z0ZPOPOPZPO0OPZPZBoBZPOZ0Z0Z0OPOPO0OPZPO0OBZPZBOB0Z0Z0ZPOPOPZPO0OPsPZBOBZPoZ0Z0Z0OPOPO0OPZPZBoBZPOBoP0Z0Z0ZPOPOPZPO0OBZPZBOBoPZZ0J0Z0OPOPO0OPsPZBOBZPoPZ00Z0Z0ZPOPOPZPZBoBZPOBoPZ0oZ0Z0Z0OPOPO0OBZPZBOBoPZ0oP0Z0Z0ZPOPOPsPZBOBZPoPZ0oPZZ0Z0Z0OPOPZBoBZPOBoPZ0oPZ00Z0Z0ZPOPOBZPZBOBoPZ0oPZ0ZZ0ZrZ0OPOPZBOBZPoPZ0oPZ0Z00Z0Z0ZPOPjBZPOBoPZ0oPZ0Z0ZZ0Z0Z0OPZPZBOBoPZ0oPZ0Z0Z00Z0Z0ZPOBOBZPoPZ0oPZ0Z0Z0ZZ0Z0Z0OPOPOBoPZ0oPZ0Z0Z0Z00Z0Z0ZPOPOBoPZ0oPZ0Z0Z0Z0ZZ0Z0Z0OPOPZPZ0oPZ0Z0Z0Z0Z00Z0Z0ZPOPOBOboPZ0Z0Z0Z0Z0ZZ0Z0Z0OPOPOPZpZ0Z0Z0Z0Z0Z00Z0Z0ZPOPOPO0O0Z0Z0Z0Z0Z0ZZ0Z0Z0OPOPOPZ0Z0Z0Z0Z0Z0Z00Z0Z0ZPOPOPO0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0OPOPOPZ0Z0Z0Z0Z0Z0Z0
ZPZBOBoPZ0oPZBOBZPoPZ0oPZ0ZPOBoPZ0oPZ0ZBOBoPZ0oPZ0Z0ZPoPZ0oPZ0Z0ZBoPZ0oPZ0Z0Z0oPZ0oPZ0Z0Z0ZPZ0oPZ0Z0Z0Z0Z0oPZ0Z0Z0Z0Z0oPZ0Z0Z0Z0Z0oPZ0Z0Z0Z0Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
The omega one of chess
ωCh1 = supremum of values of finite positions in infinite chess
ωCh∼1 = supremum of values of all valued positions
Question
How big is the omega one of chess?
ωCh1 ≤ ω
CK1
ωCh∼1 ≤ ω1
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Transfinite game values in infinite chess
The omega one of chessConjecture (Evans, Hamkins)
The omega one of chess is as large as it could be.
ωCh1 = ωCK
1 ωCh∼1 = ω1.
The best lower bounds are still very small...
ω · n + k < ωCh1 ≤ ω
CK1
ω3 · 4 < ωCh∼1 ≤ ω1
But we conjecture that the upper bounds are met.
We haven’t been able yet to prove this in two-dimensionalinfinite chess (but meanwhile, in three dimensions...)
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Computability
Let’s explore a few computability issues that arise in connectionwith infinite chess.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Computable strategies
Deterministic computable strategies
A position is computable if there is a computable function givingthe locations and number of the pieces. (Complications...)
Consider that the players will both play according todeterministic computable strategies.
Question
Will that requirement affect our judgement of whether a positionis winning?
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Computable strategies
Requiring computable play matters
Yes. Requiring players to play according to a deterministiccomputable procedure can affect our judgement of whether agiven computable position is drawn or a win.
Theorem
There is a position in infinite chess such that:1 The position is computable.2 The position is drawn. Neither player has a winning
strategy, and either player can force a draw.3 But among computable strategies, the position is a win for
white. White has a computable strategy defeating everycomputable strategy for black.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Computable strategies
A position where computable play mattersFix a computable tree T ⊆ 2<ω with no computable branch.
ZPSRSRSRA0ARSRSRS0ORSRSRA0ARSRSRSRZPSRSRA0ARSRSRARS0ORSRSRA0ARSRA0ARZPSRSRSRA0ARAPARS0ORSRSRSRA0A0ORSRZPSRARSRSRA0ARSRS0ORA0ARSRA0ARSRSRZPSRAPARA0ARSRSRA0ORSRO0A0ARSRSRA0ZPSRSRA0ARSRSRA0A0ORSRSRA0ARSRA0ARZPSRSRSRA0ARA0ARS0ORSRSRSRA0A0ARSRZPSRSRSRSRA0ARSRS0ORSRSRSRAkARSRSRZPSRSRSRA0ARSRSRS0ORSRSRAKARSRSRSRZPSRSRA0ARSRSRSRS0ORSRSRARSRSRSRSRZPSRSRSRSRSRSRSRS0OPOPOPOPOPOPOPOPZ0Z0Z0Z0Z0Z0Z0Z0Z White forces black to climb the
tree; black hopes to avoid the dead-end traps.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Computable strategies
The mate-in-n problem
There is a genre of chess problems, the mate-in-n problems
White to mate in 2Black to mate in 3
Consider the corresponding class of problems in infinite chess.
The mate-in-n problem
Given a finite position in infinite chess, can a designated playerforce a win in at most n moves?
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Computable strategies
A mate-in-12 problem
White to move on an infinite, edgeless board.
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0Z0Z0Z0ZKZ0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Can white force mate in 12 moves?
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Computable strategies
The mate-in-n problem
Question
Is the mate-in-n problem computably decidable?
A naive formulation of white to mate-in-3, with white to move:There is a white move, such that for every black reply,there is a white move, such that for every black reply,there is a white move that delivers checkmate.
∃w1∀b1∃w2∀b2∃w3 · · ·
Very high arithmetical complexity, 2n alternating quantifiers.
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Computable strategies
Infinite game tree
In finite n × n chess, one can search the entire game tree,which is finite.
In infinite chess, the game tree is not only infinite, but infinitelybranching. The unbounded pieces (queens, rooks, bishops)may have infinitely many possible moves.
Thus, one cannot expect computably to search the entire gametree, even to finite depth.
The naive, brute-force-search manner of deciding the mate-in-nproblem is simply inadequate.
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Computable strategies
The infinitary mate-in-n problem is decidable
Theorem (Brumleve, Hamkins, Schlicht)
The mate-in-n problem of infinite chess is decidable.
Further, there is a computable strategy for optimal play from anymate-in-n position.
Two proof methods:
The structure of chess Ch, the space of all chess positions, withchess concept relations, is an automatic structure, whose theoryis therefore decidable.
The structure of chess is encodable in Presburger arithmetic〈Z,+, <〉, whose theory is decidable.
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Computable strategies
The winning-position problem still open
Our theorem does not show that the winning-position problemis decidable.
Question (Richard Stanley)
Is the winning-position problem for infinite chess decidable?
This question remains open, and is MathOverflow-hard.
The point is that a player may have a winning strategy from aposition, but it is not mate-in-n for any n. So it is not clear thatthe winning positions are even c.e., or even arithmetic orhyperarithmetic.
These positions are precisely the positions with transfinitegame value.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
3D chess
Let us turn to three-dimensional chess.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
3D chessThe history of 3D chess spans more centuries than you might expect:
19th century: Kieseritzkys Kubikschack in 1851.
early 20th century: Maacks raumschach chess clubs, Hamburgfrom 1919 (first 8× 8× 8, then 5× 5× 5)
late 20th century: I played 8× 8× 3 chess as a child.
23rd century: Spock and Kirk play on several Star Trek episodes.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
3D chessThe history of 3D chess spans more centuries than you might expect:
19th century: Kieseritzkys Kubikschack in 1851.
early 20th century: Maacks raumschach chess clubs, Hamburgfrom 1919 (first 8× 8× 8, then 5× 5× 5)
late 20th century: I played 8× 8× 3 chess as a child.
23rd century: Spock and Kirk play on several Star Trek episodes.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
3D chessThe history of 3D chess spans more centuries than you might expect:
19th century: Kieseritzkys Kubikschack in 1851.
early 20th century: Maacks raumschach chess clubs, Hamburgfrom 1919 (first 8× 8× 8, then 5× 5× 5)
late 20th century: I played 8× 8× 3 chess as a child.
23rd century: Spock and Kirk play on several Star Trek episodes.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
3D chessThe history of 3D chess spans more centuries than you might expect:
19th century: Kieseritzkys Kubikschack in 1851.
early 20th century: Maacks raumschach chess clubs, Hamburgfrom 1919 (first 8× 8× 8, then 5× 5× 5)
late 20th century: I played 8× 8× 3 chess as a child.
23rd century: Spock and Kirk play on several Star Trek episodes.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Infinite 3D chess
We consider infinite three-dimensional chess, 3D chess oninfinite boards with no boundary.
One must specify the piece movement; there is room forreasonable disagreement.
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Infinite 3D chess
The omega one of 3D infinite chess
Theorem (Evans, Hamkins)
The ω1 of infinite 3D chess with infinite positions is true ω1, aslarge as it could possibly be.
ωCh3∼1 = ω1
Every countable ordinal therefore arises as the game value of aposition in three-dimensional infinite chess.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
The ω1 of 3D infinite chess
The basic idea is to code an arbitrary well-founded tree T ⊆ ωωas a position in infinite chess.
Every such tree admits an ordinal ranking function.
We create a chess position that is like climbing through such awell-founded tree.
The difficulty is that the tree must be infinite-branching toachieve large ranks. How to force black to make a choice?
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Embedding a tree into 3 space
We embed the tree, with the infinite-branching nodes of Tsimulated individually on separate layers
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Ascending stairs in 3D chess
9 Z0Z0Z0Z0Z8 0Z0Z0Z0Z07 Z0Z0Z0Z0Z6 0Z0Z0Z0Z05 Z0ZPZPZ0Z4 0Z0O0O0Z03 Z0ZpOpZ0Z2 0Z0OpO0Z01 Z0Z0O0Z0Z
a b c d e f g h i
α
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZPZPZ0Z0Z0OkO0Z0Z0ZpOpZ0Z0Z0OpO0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0a b c d e f g h i
β
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZPZPZ0Z0Z0O0O0Z0Z0ZpOpZ0Z0Z0OpO0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Za b c d e f g h i
γ
0Z0Z0Z0Z0Z0ZPZPZ0Z0Z0O0O0Z0Z0ZpOpZ0Z0Z0OpO0Z0Z0Z0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0a b c d e f g h i
δ
Z0ZPZPZ0Z 9
0Z0O0O0Z0 8
Z0ZpOpZ0Z 7
0Z0OpO0Z0 6
Z0Z0O0Z0Z 5
0Z0Z0Z0Z0 4
Z0Z0Z0Z0Z 3
0Z0Z0Z0Z0 2
Z0Z0Z0Z0Z 1
a b c d e f g h i
ε
The black king is forced to ascend the stairs via
1.αe4+ Kγe6 2.βe5+ Kδe7 3.γe6+ Kεe8 4.δe7+.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Branching node layer in 3D infinite chess
6 0Z0Z05 Z0Z0Z4 0O0O03 ZPZPZ2 0oPo01 ZPoPZ0 0ZPZ0
c d e f g
ω
6 0Z0Z05 ZPZ0Z4 0O0Z03 ZpOpZ2 0OpO01 Z0O0Z
c d e f g
α
Z0Z0ZPOp0Z0ZPOpOZ0ZPOpO00ZPOpO0ZZPO0OpZ00O0ZpO0ZZPZpO0Z00Z0O0Z0ZZ0Z0Z0Z0b c d e f g h i
β
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZPO0O25
0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZPO0Op24
Z0Z0Z0Z0Z0Z0Z0Z0Z0ZPO0OpO23
0Z0Z0Z0Z0Z0Z0Z0Z0ZPO0OpO022
Z0Z0Z0Z0Z0Z0Z0Z0ZPO0OpO0Z21
0Z0Z0ZPZPZ0Z0Z0ZPO0OpO0Z020
Z0Z0Z0O0OPZ0Z0ZPO0OpO0Z0Z19
0Z0Z0ZpO0OPZ0ZPO0OpO0Z0Z018
Z0Z0Z0OpO0OPZPO0OpO0Z0Z0Z17
0Z0Z0Z0OpO0OPO0OpO0Z0Z0Z016
Z0Z0Z0Z0OpO0O0OpO0Z0Z0Z0Z15
0Z0Z0Z0Z0O0O0OpO0Z0Z0Z0Z014
Z0Z0Z0Z0ZPO0OpO0Z0Z0Z0Z0Z13
0Z0Z0Z0ZPO0OpO0Z0Z0Z0Z0Z012
Z0Z0O0ZPO0OpO0Z0Z0Z0Z0Z0Z11
0Z0ZPZPO0OpO0Z0Z0Z0Z0Z0Z010
Z0ZBOPO0OpO0Z0Z0Z0Z0Z0Z0Z 9
0Z0ZPO0OpO0Z0Z0Z0Z0Z0Z0Z0 8
Z0ZPObOpO0Z0Z0Z0Z0Z0Z0Z0Z 7
0ZPO0OpO0Z0Z0Z0Z0Z0Z0Z0Z0 6
Z0O0OpO0Z0Z0Z0Z0Z0Z0Z0Z0Z 5
0ZPZ0O0Z0Z0Z0Z0Z0Z0Z0Z0Z0 4
Z0ZPZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z 3
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0 2
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z 1
a b c d e f g h i j k l m n o p q r s t u v w x y
γ
Black has mate-in-2 elsewhere. Play is tightly forced. Transitionfrom stairwell to branching layer γ; additional pawns in layerabove and below will confine black king to the channel.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
The omega one of infinite 3D chessSummary of the argument:
Every well-founded tree T can be embedded into aposition of infinite 3D chess.Chess play from those positions corresponds toclimbing-through-T .Thus, the rank of T is a lower bound for the game value ofthe position.Consequently, the values of positions in infinite 3D chessare unbounded in ω1.
The ω1 of infinite 3D chess, therefore, is true ω1, as large as itcould possibly be.
ωCh3∼1 = ω1
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0Z0Z0Z0ZKZ0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White to move.
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0Z0Z0Z0ZKZ0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0Z0Z0Z0Z0ZKZ0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0Z0Z0Z0Z0ZKZ0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0Z0Z0ZKZ0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0Z0Z0ZKZ0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0Z0Z0Z0ZKZ0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0Z0Z0Z0ZKZ0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0Z0ZKZ0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0Z0ZKZ0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0Z0Z0ZKZ0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0Z0Z0ZKZ0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0ZKZ0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0ZKZ0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0Z0ZKZ0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0Z0ZKZ0Z0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0ZKZ0Z0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0ZKZ0Z0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0ZKZ0Z0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0ZKZ0Z0Z0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0ZKZ0Z0Z0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0ZKZ0Z0Z0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0ZKZ0Z0Z0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0ZKZ0Z0Z0Z0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZKZ0Z0Z0Z0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
White mates in 13, but not 12. The kings must come together.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Infinite 3D chess
Solution to the mate-in-13 problemLastly, let’s give the solution of the mate-in-13 problem.
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZKZ0Z0Z0Z0Z0Z0Z0Z0Z0ZRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Checkmate
White mates in 13, but not 12. The kings must come together.
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Infinite 3D chess
Thank you.
C. D. A. Evans, J. D. Hamkins, “Transfinite game values in infinitechess,” to appear in Integers Journal.
D. Brumleve, J. D. Hamkins, and P. Schlicht, “The mate-in-nproblem of infinite chess is decidable,” in How the worldcomputes, S. Cooper, A. Dawar, and B. Lowe, Ed., SpringerBerlin Heidelberg, 2012, vol. 7318, pp. 78-88.
Slides and articles available on http://jdh.hamkins.org.
Joel David HamkinsThe City University of New York
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
New starting procedureFrom the December 2011 IBM Ponder This problem.
White places any number of queens on the board. Black thenplaces his king anywhere, and play commences.
Question
How many queens suffice to ensure checkmate?
Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZQLQZQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0LQL0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0LQZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZkZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Two queens suffice
First, trap the king in a column. Then move a knight’s moveaway, then again a knight’s move away.
0Z0Z0ZZ0L0Z00Z0Z0ZZ0Z0Z00Z0j0ZZ0Z0Z00Z0Z0ZZ0Z0Z00Z0ZQZZ0Z0Z0
Black to move.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Two queens suffice
First, trap the king in a column. Then move a knight’s moveaway, then again a knight’s move away.
0Z0Z0ZZ0L0Z00Z0Z0ZZ0Z0Z00Z0j0ZZ0Z0Z00Z0ZQZZ0Z0Z00Z0Z0ZZ0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Two queens suffice
First, trap the king in a column. Then move a knight’s moveaway, then again a knight’s move away.
0Z0Z0ZZ0L0Z00Z0Z0ZZ0ZkZ00Z0Z0ZZ0Z0Z00Z0ZQZZ0Z0Z00Z0Z0ZZ0Z0Z0
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Two queens suffice
First, trap the king in a column. Then move a knight’s moveaway, then again a knight’s move away.
0Z0Z0ZZ0L0Z00Z0Z0ZZ0ZkZ00ZQZ0ZZ0Z0Z00Z0Z0ZZ0Z0Z00Z0Z0ZZ0Z0Z0
Checkmate
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many rooks do you need?
Consider the similar problem, where white places rooks, andthen black places a king.
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0SRS0S0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZRSRZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZRS0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0j0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
How many rooks suffice?
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Three rooks are necessary
Clearly, with three rooks, we can do it.
0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0S0Z0Z0Z0Z0Z0Z0Z0Z00Z0Z0Z0Z0ZkZ0Z0Z0Z0ZZ0Z0Z0Z0Z0Z0Z0ZRZ0Z00Z0ZRZ0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
Black to move.
Two rooks are insufficient, since there is no checkmate positionwith two rooks.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Three rooks are necessary
Clearly, with three rooks, we can do it.
0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0S0Z0Z0Z0Z0Z0Z0Z0Z00Z0ZRZ0Z0ZkZ0Z0Z0Z0ZZ0Z0Z0Z0Z0Z0Z0ZRZ0Z00Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
Checkmate
Two rooks are insufficient, since there is no checkmate positionwith two rooks.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many bishops
0Z0Z0Z0Z0Z0Z0Z0Z0Z0ZZ0ABZ0Z0Z0Z0Z0Z0Z0Z00Z0ABA0Z0Z0Z0j0Z0Z0ZZ0ABZ0Z0Z0Z0Z0Z0Z0Z00Z0ABA0Z0Z0Z0Z0Z0Z0ZZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
How many bishops do you need?
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
Six bishops required
Six suffice: use two pairs of two to close two walls. Extra bishop(of correct color) delivers checkmate
0Z0Z0Z0Z0Z0ZBZ0Z0Z0ZZ0Z0Z0Z0Z0Z0A0Z0Z0Z00Z0Z0Z0Z0j0Z0Z0Z0Z0ZZ0A0Z0Z0Z0Z0Z0Z0Z0Z00ZBZ0Z0Z0Z0Z0ABZ0Z0ZZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0
Black to move.
Five do not suffice: the black king will stay on the color havingat most two bishops of that color. There is always a free move(or stalemate), since double check cannot arise.
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0Z0Z0Z0Z0Z0Z0Z0Z0ZNMNMNMNZkZ0Z0Z0Z0Z0Z0MNMNZNM0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Black to move.
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0M0Z0Z0Z0Z0Z0Z0Z0ZNMNM0MNZkZ0Z0Z0Z0Z0Z0MNMNZNM0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Check
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0M0Z0Z0Z0Z0Z0Z0Z0ZNMNM0MNZ0j0Z0Z0Z0Z0Z0MNMNZNM0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0M0Z0Z0Z0Z0Z0Z0Z0ZNMNM0ZNZ0j0Z0Z0Z0Z0Z0MNMNZNM0ZNZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Check
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0M0Z0Z0Z0Z0Z0Z0Z0ZNMNM0ZNZ0ZkZ0Z0Z0Z0Z0MNMNZNM0ZNZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0M0M0Z0Z0Z0Z0Z0Z0ZNMNM0Z0Z0ZkZ0Z0Z0Z0Z0MNMNZNM0ZNZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Check
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0M0M0Z0Z0Z0Z0Z0Z0ZNMNM0Z0Z0Z0j0Z0Z0Z0Z0MNMNZNM0ZNZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0Z0M0Z0Z0Z0Z0Z0Z0ZNMNM0Z0ZNZ0j0Z0Z0Z0Z0MNMNZNM0ZNZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0Z0M0Z0Z0Z0Z0Z0Z0ZNMNM0Z0ZNZ0ZkZ0Z0Z0Z0MNMNZNM0ZNZ0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
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Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0Z0M0Z0Z0Z0Z0Z0Z0ZNMNM0Z0ZNM0ZkZ0Z0Z0Z0MNMNZNM0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
![Page 498: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/498.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0Z0M0Z0Z0Z0Z0Z0Z0ZNMNM0Z0ZNM0Z0j0Z0Z0Z0MNMNZNM0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
![Page 499: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/499.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0Z0M0Z0Z0Z0Z0Z0Z0ZNMNM0Z0Z0M0Z0j0Z0Z0Z0MNMNZNM0Z0Z0M0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
![Page 500: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/500.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0Z0M0Z0Z0Z0Z0Z0Z0ZNMNM0Z0Z0M0Z0ZkZ0Z0Z0MNMNZNM0Z0Z0M0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
![Page 501: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/501.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0Z0Z0Z0Z0Z0Z0Z0Z0ZNMNM0Z0Z0MNZ0ZkZ0Z0Z0MNMNZNM0Z0Z0M0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
![Page 502: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/502.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0Z0Z0Z0Z0Z0Z0Z0Z0ZNMNM0Z0Z0MNZ0Z0j0Z0Z0MNMNZNM0Z0Z0M0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
![Page 503: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/503.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0Z0Z0ZNZ0Z0Z0Z0Z0ZNMNM0Z0Z0ZNZ0Z0j0Z0Z0MNMNZNM0Z0Z0M0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York
![Page 504: Joel David Hamkins](https://reader031.fdocuments.us/reader031/viewer/2022012506/61816343c20d0b72ff603bb4/html5/thumbnails/504.jpg)
Infinite games and transfinite game values Infinite chess Computability Higher dimensions Extras
How many knights?
0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0MNMNZNMNZ0Z0Z0Z0Z0Z0Z0Z0ZNMNMNM0Z0Z0ZNZ0Z0Z0Z0Z0ZNMNM0Z0Z0ZNZ0Z0ZkZ0Z0MNMNZNM0Z0Z0M0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z0Z
Is this enough to ensure checkmate?
No, black can outrun the knights, since two knights areinsufficient to checkmate.
How to win at infinite chess Joel David Hamkins, New York