James A. Swenson - UW-Plattevillepeople.uwplatt.edu/~swensonj/Surreals.pdf · 4 Surreal numbers...

Hackenbush and the surreal numbers

James A. Swenson

University of Wisconsin–Platteville

[email protected]

September 28, 2017Bi-State Math Colloquium

James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 1 / 37

Thanks for coming!

I hope you’ll enjoy the talk; please feel free to get involved!


Epigraph


Epigraph

Propositiones aliquot, que in Scholis Societatis non sunt docendæ. . .

25 Continuum successiuum & intensio qualitatum solis indiuisibilibusconstant.. . .

30 Infinitum in multitudine, & magnitudine potest claudi inter duasunitates, vel duo puncta.

Ordinatio pro studiis superioribus.. . . A[dmodum]R[everendus] P[ater] N[oster] Francisco Piccolomineoad Prouincias Missa Anno 1651.


Epigraph

Some propositions which must not be taught in the Society’sschools. . .

25 The line of succession and of the intensity of qualities are made up ofindivisible points.. . .

30 Infinity in multitude and infinity in magnitude can be enclosedbetween two units or two points.

Ordinance for higher study. Sent by Our Most Rev-erend Holy Father Francisco Piccolomineo [SuperiorGeneral of the Jesuit Order], year 1651.


Outline

1 Heroes

2 Games

3 Ordering of games

4 Surreal numbers


John H. Conway (1937- )

Conway is a world-famous, award-winning mathematician, who has been aprofessor at Cambridge and (currently) Princeton.


John H. Conway (1937– )

Conway is incredibly untidy. The tables in his room at theDepartment of Pure Mathematics and Mathematical Statisticsin Cambridge are heaped high with papers, books, unansweredletters, notes, models, charts, tables, diagrams, dead cups ofcoffee and an amazing assortment of bric-a-brac, which hasoverflowed most of the floor and all of the chairs, so that it ishard to take more than a pace or two into the room andimpossible to sit down. If you can reach the blackboard thereis a wide range of coloured chalk, but no space to write. Hisroom in College is in a similar state. In spite of his excellentmemory he often fails to find the piece of paper with theimportant result that he discovered some days before, andwhich is recorded nowhere else.

Richard Guy, quoted athttp://www-groups.dcs.st-and.ac.uk/∼history/Biographies/Conway.html


Donald K. Knuth (1938– )


Outline

1 Heroes

2 Games

3 Ordering of games

4 Surreal numbers


The rules of Hackenbush

Hackenbush is a game played by twoplayers, Blue and Red, on a rootedgraph with colored edges.

To move,delete an edge of your color, plus anyedges no longer connected to theground. Blue moves first. If it’s yourturn and you can’t move, you lose.

Red loses!



Hackenbush is a game played by twoplayers, Blue and Red, on a rootedgraph with colored edges. To move,delete an edge of your color, plus anyedges no longer connected to theground.

Blue moves first. If it’s yourturn and you can’t move, you lose.

Red loses!



Hackenbush is a game played by twoplayers, Blue and Red, on a rootedgraph with colored edges. To move,delete an edge of your color, plus anyedges no longer connected to theground. Blue moves first. If it’s yourturn and you can’t move, you lose.

Red loses!


Game notation

To play well, you need to know your options!

=

,

∣∣∣∣∣∣∣ ,


The simplest game

=

∣∣∣∣∣∣∣

Let’s improve our lives by giving this game a name: • = {|}.


The next simplest games

=

∣∣∣∣∣∣∣

In symbols, this game is {•|}. Let’s name it: • = {•|}.

=

∣∣∣∣∣∣∣

In symbols, this game is {| •}. Let’s name it: • = {| •}.


Games with up to two edges

• = {|} • = {•|} • = {•|} ! = {•, •|} ∧• = {•| •}

• = {•| •} • = {| •} • = {| •} ¡ = {| •, •} •∨

= {•| •}


Outline

1 Heroes

2 Games

3 Ordering of games

4 Surreal numbers


Comparing games

Idea

If G and H are games, we want:“G ≤ H” when H is at least as good as G for Blue.

≤ ≤ ≤ ≤

(• ≤ • ≤ • ≤ • ≤ •

)


Order relation on games

Definition

Let G = {GL|GR} and H = {HL|HR} be games.

This means that GL and GR are sets of games smaller than G , etc.,so the following definition is recursive, not circular:

We say G ≤ H provided that:

1 there is no X ∈ GL with H ≤ X ; and

2 there is no Y ∈ HR with Y ≤ G .

(“Blue can’t make G into something as good as H,and Red can’t make H into something as bad as G .”)

Example

Recall • = {|}. Since •L = ∅ = •R , it is vacuously true that • ≤ •.


Comparison: adding a single edge

Theorem

Adding a blue edge makes a game better for Blue; adding a red edgemakes it worse.



Theorem


Corollary

≤ ≤ ≤ ≤

· · · ≤ • ≤ • ≤ • ≤ • ≤ • ≤ · · ·



Theorem


Corollary

≤ and ≤

• ≤ •∨

and∧• ≤ •



Theorem


Proposition

≤ ≤

•∨≤ • ≤ ∧•

Corollary

• ≤ • ≤ •∨≤ • ≤ ∧• ≤ • ≤ •.


Good news: ≤ is reflexive

Theorem

If G is a game, then G ≤ G .

Proof (Induction on number of edges).

We know • ≤ •. Now let G be a game with at least one edge. Suppose for(transfinite) induction that H ≤ H whenever H has fewer edges than G .

Sftsoc: G 6≤ G .

Then either there is some X ∈ GL with G ≤ X or thereis some Y ∈ GR with Y ≤ G .

In the first case, since G ≤ X , there is no Z ∈ GL with X ≤ Z . . . butX ∈ GL and X ≤ X (by induction). �

In the second case, since Y ≤ G , there is no Z ∈ GR with Z ≤ Y . . . butY ∈ GR and Y ≤ Y (by induction). �

In either case,

we reach a contradiction.

Hence G ≤ G .



Theorem



We know • ≤ •. Now let G be a game with at least one edge. Suppose for(transfinite) induction that H ≤ H whenever H has fewer edges than G .Sftsoc: G 6≤ G .

Then either there is some X ∈ GL with G ≤ X or thereis some Y ∈ GR with Y ≤ G .



In either case,

we reach a contradiction. Hence G ≤ G .



Theorem



We know • ≤ •. Now let G be a game with at least one edge. Suppose for(transfinite) induction that H ≤ H whenever H has fewer edges than G .Sftsoc: G 6≤ G . Then either there is some X ∈ GL with G ≤ X or thereis some Y ∈ GR with Y ≤ G .



In either case, we reach a contradiction. Hence G ≤ G .



Theorem



We know • ≤ •. Now let G be a game with at least one edge. Suppose for(transfinite) induction that H ≤ H whenever H has fewer edges than G .Sftsoc: G 6≤ G . Then either there is some X ∈ GL with G ≤ X or thereis some Y ∈ GR with Y ≤ G .In the first case, since G ≤ X , there is no Z ∈ GL with X ≤ Z . . . butX ∈ GL and X ≤ X (by induction). �


In either case, we reach a contradiction. Hence G ≤ G .



Theorem



We know • ≤ •. Now let G be a game with at least one edge. Suppose for(transfinite) induction that H ≤ H whenever H has fewer edges than G .Sftsoc: G 6≤ G . Then either there is some X ∈ GL with G ≤ X or thereis some Y ∈ GR with Y ≤ G .In the first case, since G ≤ X , there is no Z ∈ GL with X ≤ Z . . . butX ∈ GL and X ≤ X (by induction). �In the second case, since Y ≤ G , there is no Z ∈ GR with Z ≤ Y . . . butY ∈ GR and Y ≤ Y (by induction). �In either case, we reach a contradiction. Hence G ≤ G .


More good news: ≤ is transitive

Fact

If G ≤ H and H ≤ K , then G ≤ K .

Proof (Induction on total number of edges in G t H t K ).

Base case: (• ≤ • ≤ •)⇒ (• ≤ •).

Let G ≤ H ≤ K .

1 Sftsoc: X ∈ GL and K ≤ X , so H ≤ K ≤ X .By induction, H ≤ X , so G 6≤ H. �

2 Sftsoc: Y ∈ KR and Y ≤ G , so Y ≤ G ≤ H.By induction, Y ≤ H, so H 6≤ K . �

Hence G ≤ K .


Bad news: ≤ is not antisymmetric

Proposition

1 • ≤ •.2 • ≤ •. • = {|} • = {•|•}

Proof.1 ∗ Let X ∈ •L. Then X = • = {|•}. Now • ∈ XR and • ≤ •, so • 6≤ X .

∗ •R = ∅.

So • ≤ •.2 ∗ •L = ∅.

∗ Let Y ∈ •R . Then Y = • = {•|}. Now • ∈ YL and • ≤ •, so Y 6≤ •.So • ≤ •.



≤ ≤ ≤

≤ ≤ ≤

(•, ¡)≤ • ≤ •

∨≤ (•, •) ≤ ∧• ≤ • ≤

(•, !)


Outline

1 Heroes

2 Games

3 Ordering of games

4 Surreal numbers


Forcing antisymmetry

Definition

Let G and H be games. We say G ∼ H provided that G ≤ H and H ≤ G .

Definition∼ is an equivalence relation; a ∼-equivalence class is called a surreal number.We denote the equivalence class of a game G = {GL|GR} by [G ] = [GL|GR ].

Definition

∗ The number zero is 0 =

[ ]= [•] = [|].

∗ The number one is 1 =

[ ]= [•] = [•|] = [0|] = [[|]|].


Ordering and strategy

Fact

∗ [G ] < 0 ⇐⇒ Red can always win the game G .

∗ [G ] = 0 ⇐⇒ the second player can always win the game G .

∗ 0 < [G ] ⇐⇒ Blue can always win the game G .


Adding games

Definition

If G and H are games, G + H is the game you get by putting G and Hnext to each other.

+ = ∼

1 + [•] = [•] = [•] = 0

Definition

The number negative one is −1 =

[ ]= [•] = [|•] = [|[|]].


The opposite of a game

Definition

∗ −G is the game you get by flipping the color of each edge in G .

∗ G − H is shorthand for G + (−H).

−

=

Proposition

∗ −(−G ) = G .

∗ If G ≤ H, then −H ≤ −G .

∗ G + (−G ) ∼ 0.


Ordering and strategy

Fact

∗ If G ≤ H, then G + K ≤ H + K .

∗ If G1 ∼ G2 and H1 ∼ H2, then G1 ≤ H1 ⇐⇒ G2 ≤ H2.

Definition

We say [G ] ≤ [H] provided that G ≤ H.

Corollary

∗ [G ] < [H] ⇐⇒ Red can always win the game G − H.

∗ [G ] = [H] ⇐⇒ the second player can always win the game G − H.

∗ [H] < [G ] ⇐⇒ Blue can always win the game G − H.


G is between GL and GR

Theorem

Let G be a game.

1 If X ∈ GL, then X ≤ G .

2 If Y ∈ GR , then G ≤ Y .

(“It would always be better to pass.”)

Proof.1 Suppose X ∈ GL is the result when Blue deletes the blue edge e from G .

Now consider the game G − X , and suppose it’s Red’s move. If Red deletesan edge from G that has a mirror image in −X , or an edge from −X thathas a mirror image in G , then Blue responds by deleting that mirror image.Otherwise, Red deletes an edge from G with no mirror image in −X , andBlue responds by deleting e. On Red’s first turn after the deletion of e, theposition has the form H − H ∼ •, so Blue can win (by mirroring Red).This shows that Blue can win G − X , so [G − X ] 6< 0. Thus X ≤ G .

2 (Similar.)


All games are comparable

Theorem

If G 6≤ H, then H ≤ G .

Proof.

Suppose G 6≤ H. We consider two cases.

1 Suppose X ∈ GL and H ≤ X . We know X ≤ G . By transitivity,H ≤ G .

2 Suppose Y ∈ HR and Y ≤ G . We know H ≤ Y . By transitivity,H ≤ G .


What is[∧•]?

Example

The game shown below is∧•+

∧•+ •:

The second player can always win, so[∧•]+

[∧•]+ (−1) = 0.

Definition

The number one half is 12 =

[∧•].


What is[∧•]?

Example

The game shown below is∧•+

∧•+ •:

The second player can always win, so[∧•]+

[∧•]+ (−1) = 0.

Definition

The number one half is 12 =

[∧•].James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 32 / 37


0 1 2 2 12

0 −1 −2 −2 −12


Games with more edges

......

14

18

116

23 π

......

......

......

ε −ω ω ω + 1 2ω


More game arithmetic

Definition

Suppose G and H are games with XG ∈ GL, YG ∈ GR , XH ∈ HL,YH ∈ HR . Then:

∗ {XGH + GXH − XGXH ,YGH + GYH − YGYH} ⊆ (GH)L.

∗ {XGH + GYH − XGYH ,YGH + GXH − YGXH} ⊆ (GH)R .

Definition

Let G and H be games with [G ] > 0 and [H] > 0.1If, for every XG ∈ GL, YG ∈ GR , XH ∈ HL, and YH ∈ HR , we have

∗{

1YG

(1 + (YG − G )XH), 1XG

(1 + (XG − G )YH)}⊆ HL and

∗{

1XG

(1 + (XG − G )XH), 1YG

(1 + (YG − G )YH)}⊆ HR ,

then [G ][H] = 1.


The surreal numbers are universal

Theorem

Every ordered field is isomorphic to a subfield of the surreal numbers.∗

∗The proof requires the axiom of global choice and applies only to sets. . . .James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 36 / 37

References[1] Amir Alexander, Infinitesimal: How a dangerous mathematical theory shaped the modern world, Scientific American/Farrar,

Strauss, and Giroux, New York, 2014.

[2] John H. Conway, On numbers and games, 2nd ed., A K Peters, Ltd., Natick, MA, 2001.

[3] John H. Conway and Richard K. Guy, The book of numbers, Copernicus, New York, 1996.

[4] Tom Davis, Hackenbush (December 15, 2011), http://www.geometer.org/mathcircles/hackenbush.pdf.

[5] Philip Ehrlich, All numbers great and small, Real numbers, generalizations of the reals, and theories of continua, SyntheseLib., vol. 242, Kluwer Acad. Publ., Dordrecht, 1994, pp. 239–258.

[6] Gretchen Grimm, An introduction to surreal numbers (May 8, 2012),https://www.whitman.edu/Documents/Academics/Mathematics/Grimm.pdf.

[7] D. E. Knuth, Surreal numbers: how two ex-students turned on to pure mathematics and found total happiness,Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1974.

[8] Jonas Sjostrand, Combinatorial game theory (March 2015),https://www.math.kth.se/matstat/gru/sf2972/2015/gametheory.pdf.

[9] Wikipedia contributors, Surreal number (July 6, 2017), https://en.wikipedia.org/w/index.php?oldid=789209825.

Image sourcesBook cover: https://www.amazon.com/Surreal-Numbers-Donald-E-Knuth/dp/0201038129/

Conway photo: https://commons.wikimedia.org/wiki/File:John H Conway 2005.jpg

Conway sketch: Fraser, Simon J. https://www.ias.edu/ideas/2015/roberts-john-horton-conway

Knuth photo: Appelbaum, Jacob. https://commons.wikimedia.org/wiki/File:KnuthAtOpenContentAlliance.jpg

Ordinatio: https://books.google.com/books?id=w1O8SZfAioIC

“TikZ Diagrams in Math Mode.” https://tex.stackexchange.com/questions/11105/tikz-diagrams-in-math-mode


http://www.geometer.org/mathcircles/hackenbush.pdf

https://www.whitman.edu/Documents/Academics/Mathematics/Grimm.pdf

https://www.math.kth.se/matstat/gru/sf2972/2015/gametheory.pdf

https://en.wikipedia.org/w/index.php?oldid=789209825

James A. Swenson - UW-Plattevillepeople.uwplatt.edu/~swensonj/Surreals.pdf · 4 Surreal numbers...

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