Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto...
Transcript of Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto...
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Mathematics and Games
Parker Glynn-Adey
University of Toronto
March 23, 2014
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Outline
1 What are games?
2 The 15 Game
3 Nim
4 A Safari in the Land of Games
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This is not a game.(Although, there is a lot of mathematics in Angry Birds.)
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This is not a game.
(Although, there is a lot of mathematics in Angry Birds.)
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This is not a game.(Although, there is a lot of mathematics in Angry Birds.)
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Backgammon (∼1100AD) and Poker (∼1850AD).
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Backgammon (∼1100AD) and Poker (∼1850AD).
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Chess (∼1400AD) and Go (∼2000BC)
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Chess (∼1400AD) and Go (∼2000BC)
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Rules of The 15 Game (A Toy Game)
Players: Two.
Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:2 3 4 5 6 8 9 (L = ,R = )
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Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.
Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:2 3 4 5 6 8 9 (L = ,R = )
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 6 / 28
![Page 12: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/12.jpg)
Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.
Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:2 3 4 5 6 8 9 (L = ,R = )
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 6 / 28
![Page 13: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/13.jpg)
Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:2 3 4 5 6 8 9 (L = ,R = )
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 6 / 28
![Page 14: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/14.jpg)
Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:2 3 4 5 6 8 9 (L = ,R = )
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 6 / 28
![Page 15: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/15.jpg)
Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:2 3 4 5 6 8 9 (L = ,R = )
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 6 / 28
![Page 16: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/16.jpg)
Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:2 3 4 5 6 8 9 (L = ,R = )
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 6 / 28
![Page 17: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/17.jpg)
Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:2 3 4 5 6 8 9 (L = ,R = )
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 6 / 28
![Page 18: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/18.jpg)
Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:1 2 3 4 5 6 7 8 9 (L = 0,R = 0)
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 6 / 28
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Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:1 2 3 4 5 6 7 8 9 (L = 0,R = 0)
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 6 / 28
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Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:1 2 3 4 5 6 7 8 9 (L = 0,R = 0)
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 6 / 28
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Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:
�A1 2 3 4 5 6 7 8 9 (L = 0,R = 1)
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Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:
�A1 2 3 4 5 6 �A7 8 9 (L = 7,R = 1)
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 6 / 28
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Rules of The 15 Game (A Toy Game)
Players: Two.Goal: To collect up numbers adding up to 15.Setup: Write the numbers one through nine on a piece of paper.Play:
Players alternate taking turns.
Both players start with total zero.
On your turn: cross out a number, and add it to your total.
If your total is 15, you win.
Example:
�A1 2 3 4 5 6 �A7 8 9 (L = 7,R = 1)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 6 / 28
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Observations:
Sometimes no one wins.
5 is a good opening move.
The Fifteen Game feels oddly familiar.
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Observations:
Sometimes no one wins.
5 is a good opening move.
The Fifteen Game feels oddly familiar.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 7 / 28
![Page 26: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/26.jpg)
Observations:
Sometimes no one wins.
5 is a good opening move.
The Fifteen Game feels oddly familiar.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 7 / 28
![Page 27: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/27.jpg)
Observations:
Sometimes no one wins.
5 is a good opening move.
The Fifteen Game feels oddly familiar.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 7 / 28
![Page 28: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/28.jpg)
8 1 63 5 74 9 2
The 15 Game is Tic-Tac-Toe.
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8 1
6
3 5 74 9 2
The 15 Game is Tic-Tac-Toe.
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8 1
6
3
5
74 9 2
The 15 Game is Tic-Tac-Toe.
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8 1
6
3
5
74 9
2
The 15 Game is Tic-Tac-Toe.
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8 1
6
3
5 7
4 9
2
The 15 Game is Tic-Tac-Toe.
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8 1
63 5 7
4 9
2
The 15 Game is Tic-Tac-Toe.
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8 1
63 5 7
4
9 2
The 15 Game is Tic-Tac-Toe.
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8
1 63 5 7
4
9 2
The 15 Game is Tic-Tac-Toe.
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8 1 63 5 7
4
9 2
The 15 Game is Tic-Tac-Toe.
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8 1 63 5 74 9 2
The 15 Game is Tic-Tac-Toe.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 8 / 28
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8 1 63 5 74 9 2
The 15 Game is Tic-Tac-Toe.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 8 / 28
![Page 39: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/39.jpg)
Nim
Nim was the first game to be exhaustively analyzed mathematically.
In 1901, Charles Bouton published a solution to the game.This result started the field of mathematical game theory.
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Nim
Nim was the first game to be exhaustively analyzed mathematically.In 1901, Charles Bouton published a solution to the game.
This result started the field of mathematical game theory.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 9 / 28
![Page 41: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/41.jpg)
Nim
Nim was the first game to be exhaustively analyzed mathematically.In 1901, Charles Bouton published a solution to the game.This result started the field of mathematical game theory.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 9 / 28
![Page 42: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/42.jpg)
Rules of Nim
Players: Two.
Goal: To be the last player to pick up a stone.Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)R (1, 1)
L (0, 1)
R (0, 0) (Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 43: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/43.jpg)
Rules of Nim
Players: Two.Goal: To be the last player to pick up a stone.
Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)R (1, 1)
L (0, 1)
R (0, 0) (Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 44: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/44.jpg)
Rules of Nim
Players: Two.Goal: To be the last player to pick up a stone.Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)
Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)R (1, 1)
L (0, 1)
R (0, 0) (Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 45: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/45.jpg)
Rules of Nim
Players: Two.Goal: To be the last player to pick up a stone.Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)R (1, 1)
L (0, 1)
R (0, 0) (Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 46: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/46.jpg)
Rules of Nim
Players: Two.Goal: To be the last player to pick up a stone.Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)R (1, 1)
L (0, 1)
R (0, 0) (Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 47: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/47.jpg)
Rules of Nim
Players: Two.Goal: To be the last player to pick up a stone.Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)R (1, 1)
L (0, 1)
R (0, 0) (Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 48: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/48.jpg)
Rules of Nim
Players: Two.Goal: To be the last player to pick up a stone.Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)R (1, 1)
L (0, 1)
R (0, 0) (Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 49: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/49.jpg)
Rules of Nim
Players: Two.Goal: To be the last player to pick up a stone.Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)
R (1, 1)
L (0, 1)
R (0, 0) (Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 50: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/50.jpg)
Rules of Nim
Players: Two.Goal: To be the last player to pick up a stone.Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)R (1, 1)
L (0, 1)
R (0, 0) (Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 51: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/51.jpg)
Rules of Nim
Players: Two.Goal: To be the last player to pick up a stone.Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)R (1, 1)
L (0, 1)
R (0, 0) (Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 52: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/52.jpg)
Rules of Nim
Players: Two.Goal: To be the last player to pick up a stone.Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)R (1, 1)
L (0, 1)
R (0, 0)
(Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 53: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/53.jpg)
Rules of Nim
Players: Two.Goal: To be the last player to pick up a stone.Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)R (1, 1)
L (0, 1)
R (0, 0) (Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 54: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/54.jpg)
Rules of Nim
Players: Two.Goal: To be the last player to pick up a stone.Setup: Lay out heaps of n1, n2, . . . , nk stones. (ni ≥ 0)Play:
Players alternate taking turns.
A move consists of removing at least one stone from a heap.
If you can’t move, you lose.
Example:
(1, 2)R (1, 1)
L (0, 1)
R (0, 0) (Left loses.)
Let’s play!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 10 / 28
![Page 55: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/55.jpg)
Quiz!
Play and win: (10, 11).
Is (17, 0) a good position?Is (2, 2, 3, 3, 5) a good position?
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 11 / 28
![Page 56: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/56.jpg)
Quiz!
Play and win: (10, 11).Is (17, 0) a good position?
Is (2, 2, 3, 3, 5) a good position?
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 11 / 28
![Page 57: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/57.jpg)
Quiz!
Play and win: (10, 11).Is (17, 0) a good position?Is (2, 2, 3, 3, 5) a good position?
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 11 / 28
![Page 58: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/58.jpg)
Observations:
Nim is tricky!
There are lots of options.
Empty heaps don’t matter.
A single heap is a good position.
Equal heaps ‘cancel out’.
Let’s solve Nim!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 12 / 28
![Page 59: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/59.jpg)
Observations:
Nim is tricky!
There are lots of options.
Empty heaps don’t matter.
A single heap is a good position.
Equal heaps ‘cancel out’.
Let’s solve Nim!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 12 / 28
![Page 60: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/60.jpg)
Observations:
Nim is tricky!
There are lots of options.
Empty heaps don’t matter.
A single heap is a good position.
Equal heaps ‘cancel out’.
Let’s solve Nim!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 12 / 28
![Page 61: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/61.jpg)
Observations:
Nim is tricky!
There are lots of options.
Empty heaps don’t matter.
A single heap is a good position.
Equal heaps ‘cancel out’.
Let’s solve Nim!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 12 / 28
![Page 62: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/62.jpg)
Observations:
Nim is tricky!
There are lots of options.
Empty heaps don’t matter.
A single heap is a good position.
Equal heaps ‘cancel out’.
Let’s solve Nim!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 12 / 28
![Page 63: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/63.jpg)
Observations:
Nim is tricky!
There are lots of options.
Empty heaps don’t matter.
A single heap is a good position.
Equal heaps ‘cancel out’.
Let’s solve Nim!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 12 / 28
![Page 64: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/64.jpg)
Observations:
Nim is tricky!
There are lots of options.
Empty heaps don’t matter.
A single heap is a good position.
Equal heaps ‘cancel out’.
Let’s solve Nim!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 12 / 28
![Page 65: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/65.jpg)
A quick review of numbers
2014 = 2 · 1000 +
0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 66: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/66.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 +
1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 67: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/67.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 +
4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 68: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/68.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1
= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 69: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/69.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 +
0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 70: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/70.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 +
1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 71: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/71.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 +
4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 72: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/72.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 73: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/73.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
=
(2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 74: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/74.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 75: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/75.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
=
1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 76: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/76.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 +
512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 77: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/77.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 +
256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 78: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/78.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 +
128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 79: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/79.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 +
64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 80: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/80.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 +
16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 81: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/81.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 +
8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 82: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/82.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 +
4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 83: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/83.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 +
2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 84: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/84.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 85: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/85.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
=
210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 86: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/86.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 +
29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 87: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/87.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 +
28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 88: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/88.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 +
27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 89: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/89.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 +
26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 90: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/90.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 +
24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 91: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/91.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 +
23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 92: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/92.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 +
22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 93: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/93.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 +
21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 94: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/94.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 95: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/95.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 96: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/96.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 97: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/97.jpg)
A quick review of numbers
2014 = 2 · 1000 + 0 · 100 + 1 · 10 + 4 · 1= 2 · 103 + 0 · 102 + 1 · 101 + 4 · 100
= (2, 0, 1, 4)10
= 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4 + 2
= 210 + 29 + 28 + 27 + 26 + 24 + 23 + 22 + 21
= (1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0)2
Theorem
Every number n can be written as a sum of powers of two in a unique way.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 13 / 28
![Page 98: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/98.jpg)
Nim addition
We define the following operation on binary digits:
⊕ 0 1
0 0 11 1 0
Definition
For a pair of numbers n and m, write:
n = (nk , nk−1, . . . , n0)2 m = (mk ,mk−1, . . . ,m0)2
We define their Nim-sum:
n ⊕m = (nk ⊕mk , nk−1 ⊕mk−1, . . . , n0 ⊕m0)2
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 14 / 28
![Page 99: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/99.jpg)
Nim addition
We define the following operation on binary digits:
⊕ 0 1
0 0 11 1 0
Definition
For a pair of numbers n and m, write:
n = (nk , nk−1, . . . , n0)2 m = (mk ,mk−1, . . . ,m0)2
We define their Nim-sum:
n ⊕m = (nk ⊕mk , nk−1 ⊕mk−1, . . . , n0 ⊕m0)2
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 14 / 28
![Page 100: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/100.jpg)
Nim addition
We define the following operation on binary digits:
⊕ 0 1
0 0 11 1 0
Definition
For a pair of numbers n and m, write:
n = (nk , nk−1, . . . , n0)2 m = (mk ,mk−1, . . . ,m0)2
We define their Nim-sum:
n ⊕m = (nk ⊕mk , nk−1 ⊕mk−1, . . . , n0 ⊕m0)2
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 14 / 28
![Page 101: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/101.jpg)
Nim addition
We define the following operation on binary digits:
⊕ 0 1
0 0 11 1 0
Definition
For a pair of numbers n and m, write:
n = (nk , nk−1, . . . , n0)2 m = (mk ,mk−1, . . . ,m0)2
We define their Nim-sum:
n ⊕m = (nk ⊕mk , nk−1 ⊕mk−1, . . . , n0 ⊕m0)2
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 14 / 28
![Page 102: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/102.jpg)
Nim addition
We define the following operation on binary digits:
⊕ 0 1
0 0 11 1 0
Definition
For a pair of numbers n and m, write:
n = (nk , nk−1, . . . , n0)2 m = (mk ,mk−1, . . . ,m0)2
We define their Nim-sum:
n ⊕m = (nk ⊕mk , nk−1 ⊕mk−1, . . . , n0 ⊕m0)2
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 14 / 28
![Page 103: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/103.jpg)
Nim addition
We define the following operation on binary digits:
⊕ 0 1
0 0 11 1 0
Definition
For a pair of numbers n and m, write:
n = (nk , nk−1, . . . , n0)2 m = (mk ,mk−1, . . . ,m0)2
We define their Nim-sum:
n ⊕m = (nk ⊕mk , nk−1 ⊕mk−1, . . . , n0 ⊕m0)2
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 14 / 28
![Page 104: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/104.jpg)
Examples of Nim Addition
3⊕ 2 =
(2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 105: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/105.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 106: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/106.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 107: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/107.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
=
(1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 108: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/108.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1,
1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 109: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/109.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 110: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/110.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 111: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/111.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 112: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/112.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 =
(4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 113: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/113.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 114: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/114.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 115: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/115.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1,
0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 116: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/116.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0,
1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 117: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/117.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 118: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/118.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 119: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/119.jpg)
Examples of Nim Addition
3⊕ 2 = (2 + 1)⊕ (2)
= (1, 1)2 ⊕ (1, 0)2
= (1⊕ 1, 1⊕ 0)2
= (0, 1)2
= 1
5⊕ 5 = (4 + 1)⊕ (4 + 1)
= (1, 0, 1)2 ⊕ (1, 0, 1)2
= (1⊕ 1, 0⊕ 0, 1⊕ 1)2
= (0, 0, 0)2
= 0
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 15 / 28
![Page 120: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/120.jpg)
5⊕ 7 =
(1 + 4)⊕ (1 + 2 + 4)
= (1⊕ 1)⊕ (0⊕ 2)
⊕(4⊕ 4)
= 0⊕ 2⊕ 0 = 2
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 16 / 28
![Page 121: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/121.jpg)
5⊕ 7 = (1 + 4)⊕ (1 + 2 + 4)
= (1⊕ 1)⊕ (0⊕ 2)
⊕(4⊕ 4)
= 0⊕ 2⊕ 0 = 2
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 16 / 28
![Page 122: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/122.jpg)
5⊕ 7 = (1 + 4)⊕ (1 + 2 + 4)
=
(1⊕ 1)⊕ (0⊕ 2)
⊕(4⊕ 4)
= 0⊕ 2⊕ 0 = 2
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 16 / 28
![Page 123: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/123.jpg)
5⊕ 7 = (1 + 4)⊕ (1 + 2 + 4)
= (1⊕ 1)⊕ (0⊕ 2)
⊕(4⊕ 4)
= 0⊕ 2⊕ 0 = 2
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 16 / 28
![Page 124: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/124.jpg)
5⊕ 7 = (1 + 4)⊕ (1 + 2 + 4)
= (1⊕ 1)⊕ (0⊕ 2)
⊕(4⊕ 4)
=
0⊕ 2⊕ 0 = 2
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 16 / 28
![Page 125: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/125.jpg)
5⊕ 7 = (1 + 4)⊕ (1 + 2 + 4)
= (1⊕ 1)⊕ (0⊕ 2)
⊕(4⊕ 4)
= 0⊕ 2⊕ 0 = 2
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 16 / 28
![Page 126: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/126.jpg)
The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2 = 3.
V (4, 5) = 4⊕ 5 = 1.
V (1, 2, 3) = 1⊕ 2⊕ 3 = 1⊕ 2⊕ (1 + 2) = 0.
Definition
A position is good if V 6= 0. A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
![Page 127: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/127.jpg)
The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2 = 3.
V (4, 5) = 4⊕ 5 = 1.
V (1, 2, 3) = 1⊕ 2⊕ 3 = 1⊕ 2⊕ (1 + 2) = 0.
Definition
A position is good if V 6= 0. A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
![Page 128: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/128.jpg)
The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2 = 3.
V (4, 5) = 4⊕ 5 = 1.
V (1, 2, 3) = 1⊕ 2⊕ 3 = 1⊕ 2⊕ (1 + 2) = 0.
Definition
A position is good if V 6= 0. A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
![Page 129: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/129.jpg)
The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2 = 3.
V (4, 5) = 4⊕ 5 = 1.
V (1, 2, 3) = 1⊕ 2⊕ 3 = 1⊕ 2⊕ (1 + 2) = 0.
Definition
A position is good if V 6= 0. A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
![Page 130: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/130.jpg)
The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2
= 3.
V (4, 5) = 4⊕ 5 = 1.
V (1, 2, 3) = 1⊕ 2⊕ 3 = 1⊕ 2⊕ (1 + 2) = 0.
Definition
A position is good if V 6= 0. A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
![Page 131: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/131.jpg)
The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2 = 3.
V (4, 5) = 4⊕ 5 = 1.
V (1, 2, 3) = 1⊕ 2⊕ 3 = 1⊕ 2⊕ (1 + 2) = 0.
Definition
A position is good if V 6= 0. A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
![Page 132: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/132.jpg)
The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2 = 3.
V (4, 5) = 4⊕ 5
= 1.
V (1, 2, 3) = 1⊕ 2⊕ 3 = 1⊕ 2⊕ (1 + 2) = 0.
Definition
A position is good if V 6= 0. A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
![Page 133: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/133.jpg)
The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2 = 3.
V (4, 5) = 4⊕ 5 = 1.
V (1, 2, 3) = 1⊕ 2⊕ 3 = 1⊕ 2⊕ (1 + 2) = 0.
Definition
A position is good if V 6= 0. A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
![Page 134: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/134.jpg)
The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2 = 3.
V (4, 5) = 4⊕ 5 = 1.
V (1, 2, 3) = 1⊕ 2⊕ 3
= 1⊕ 2⊕ (1 + 2) = 0.
Definition
A position is good if V 6= 0. A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
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The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2 = 3.
V (4, 5) = 4⊕ 5 = 1.
V (1, 2, 3) = 1⊕ 2⊕ 3 = 1⊕ 2⊕ (1 + 2)
= 0.
Definition
A position is good if V 6= 0. A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
![Page 136: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/136.jpg)
The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2 = 3.
V (4, 5) = 4⊕ 5 = 1.
V (1, 2, 3) = 1⊕ 2⊕ 3 = 1⊕ 2⊕ (1 + 2) = 0.
Definition
A position is good if V 6= 0. A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
![Page 137: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/137.jpg)
The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2 = 3.
V (4, 5) = 4⊕ 5 = 1.
V (1, 2, 3) = 1⊕ 2⊕ 3 = 1⊕ 2⊕ (1 + 2) = 0.
Definition
A position is good if V 6= 0.
A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
![Page 138: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/138.jpg)
The Winning Nim Strategy
For a Nim position (n1, . . . , nk) let:
V (n1, . . . , nk) = n1 ⊕ n2 ⊕ · · · ⊕ nk
Examples:
V (1, 1) = 1⊕ 1 = 0.
V (1, 2) = 1⊕ 2 = 3.
V (4, 5) = 4⊕ 5 = 1.
V (1, 2, 3) = 1⊕ 2⊕ 3 = 1⊕ 2⊕ (1 + 2) = 0.
Definition
A position is good if V 6= 0. A position is bad if V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 17 / 28
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The Winning Nim Strategy
If V (n1, . . . , nk) = 0, and it is your turn, you will lose.
If V (n1, . . . , nk) 6= 0 then there is a move, ni n′i , so thatV (n1, . . . , n
′i , . . . , nk) = 0, and your opponent will lose.
That is: Bad positions lose. Good positions win.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 18 / 28
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The Winning Nim Strategy
If V (n1, . . . , nk) = 0, and it is your turn, you will lose.
If V (n1, . . . , nk) 6= 0 then there is a move, ni n′i , so thatV (n1, . . . , n
′i , . . . , nk) = 0, and your opponent will lose.
That is: Bad positions lose. Good positions win.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 18 / 28
![Page 141: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/141.jpg)
The Winning Nim Strategy
If V (n1, . . . , nk) = 0, and it is your turn, you will lose.
If V (n1, . . . , nk) 6= 0 then there is a move, ni n′i , so thatV (n1, . . . , n
′i , . . . , nk) = 0, and your opponent will lose.
That is: Bad positions lose. Good positions win.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 18 / 28
![Page 142: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/142.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 143: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/143.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 144: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/144.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 145: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/145.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 146: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/146.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 147: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/147.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ =
V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 148: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/148.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′
⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 149: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/149.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕
0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 150: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/150.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 151: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/151.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
=
V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 152: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/152.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕
(V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 153: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/153.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 154: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/154.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
=
(V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 155: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/155.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 156: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/156.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
=
(n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 157: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/157.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
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If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 159: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/159.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 160: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/160.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V .
Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 161: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/161.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni .
Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 162: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/162.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni .
V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 163: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/163.jpg)
If V = 0, and there is a legal move, then any ni n′i will makeV ′ 6= 0.
If V 6= 0 then there is a legal move, ni n′i which will make V ′ = 0.
We write V = V (n1, . . . , nk) and V ′ = V (n1, . . . , n′i , . . . , nk).
We compute:
V ′ = V ′ ⊕ 0
= V ′ ⊕ (V ⊕ V )
= (V ′ ⊕ V )⊕ V
= (n′i ⊕ ni )⊕ V
We then get our two claims:
If V = 0 and ni 6= n′i then V ′ 6= 0.
If V 6= 0 then take ni to have the same largest digit as V . Letn′i = V ⊕ ni . Note: n′i ≤ ni . V ′ = ((V ⊕ ni )⊕ ni )⊕ V = 0.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 19 / 28
![Page 164: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/164.jpg)
(1, 2)V=3
R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0 (Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 87 = 1 + 2 + 49 = 1 + 8
⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 165: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/165.jpg)
(1, 2)V=3R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0 (Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 87 = 1 + 2 + 49 = 1 + 8
⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 166: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/166.jpg)
(1, 2)V=3R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0 (Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 87 = 1 + 2 + 49 = 1 + 8
⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 167: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/167.jpg)
(1, 2)V=3R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0
(Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 87 = 1 + 2 + 49 = 1 + 8
⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 168: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/168.jpg)
(1, 2)V=3R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0 (Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 87 = 1 + 2 + 49 = 1 + 8
⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 169: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/169.jpg)
(1, 2)V=3R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0 (Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 87 = 1 + 2 + 49 = 1 + 8
⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 170: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/170.jpg)
(1, 2)V=3R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0 (Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 87 = 1 + 2 + 49 = 1 + 8
⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 171: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/171.jpg)
(1, 2)V=3R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0 (Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 8
7 = 1 + 2 + 49 = 1 + 8
⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 172: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/172.jpg)
(1, 2)V=3R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0 (Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 87 = 1 + 2 + 4
9 = 1 + 8⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 173: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/173.jpg)
(1, 2)V=3R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0 (Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 87 = 1 + 2 + 49 = 1 + 8
⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 174: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/174.jpg)
(1, 2)V=3R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0 (Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 87 = 1 + 2 + 49 = 1 + 8
⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 175: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/175.jpg)
(1, 2)V=3R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0 (Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 87 = 1 + 2 + 49 = 1 + 8
⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 176: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/176.jpg)
(1, 2)V=3R (1, 1)V=0
L (0, 1)V=1
R (0, 0)V=0 (Left loses.)
What about (15, 7, 9, 11) ?
15 = 1 + 2 + 4 + 87 = 1 + 2 + 49 = 1 + 8
⊕ 11 = 1 + 2 + 8
10 = 2 + 8
We must remove: 2 + 8 = 10 from either: 15, 10, 11.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 20 / 28
![Page 177: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/177.jpg)
Winning Ways for Your Mathematical Plays
Winning Ways for Your Mathematical Plays byGuy, Conway, and Berlekamp. (Left to Right)
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 21 / 28
![Page 178: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/178.jpg)
Peg solitaire and the Rubik’s Cube.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 22 / 28
![Page 179: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/179.jpg)
Peg solitaire and the Rubik’s Cube.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 22 / 28
![Page 180: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/180.jpg)
Sprouts and Hackenbush.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 23 / 28
![Page 181: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/181.jpg)
Sprouts and Hackenbush.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 23 / 28
![Page 182: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/182.jpg)
Dots and Boxes
Dots and Boxes by Berlekamp, and a tournament game.
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 24 / 28
![Page 183: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/183.jpg)
Your Move
Nim like games.
Chess problems.
Go problems.
Lateral thinking.
Tic-Tac-Toeproblems !
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 25 / 28
![Page 184: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/184.jpg)
Your Move
Nim like games.
Chess problems.
Go problems.
Lateral thinking.
Tic-Tac-Toeproblems !
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 25 / 28
![Page 185: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/185.jpg)
Your Move
Nim like games.
Chess problems.
Go problems.
Lateral thinking.
Tic-Tac-Toeproblems !
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 25 / 28
![Page 186: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/186.jpg)
Your Move
Nim like games.
Chess problems.
Go problems.
Lateral thinking.
Tic-Tac-Toeproblems !
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 25 / 28
![Page 187: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/187.jpg)
Your Move
Nim like games.
Chess problems.
Go problems.
Lateral thinking.
Tic-Tac-Toeproblems !
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 25 / 28
![Page 188: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/188.jpg)
Your Move
Nim like games.
Chess problems.
Go problems.
Lateral thinking.
Tic-Tac-Toeproblems
!
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 25 / 28
![Page 189: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/189.jpg)
Your Move
Nim like games.
Chess problems.
Go problems.
Lateral thinking.
Tic-Tac-Toeproblems !
Parker Glynn-Adey (UoT) Math & Games March 23, 2014 25 / 28
![Page 190: Mathematics and GamesMathematics and Games Parker Glynn-Adey University of Toronto parker.glynn.adey@utoronto.ca March 23, 2014 Parker Glynn-Adey (UoT) Math & Games March 23, 2014](https://reader033.fdocuments.us/reader033/viewer/2022050107/5f4591bdd1180b7a21550819/html5/thumbnails/190.jpg)
The Golden Age of Board Games
Amazons, Catchup, Slither.
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References
Berlekamp, Elwyn R., John H. Conway, and Richard K. Guy
Winning Ways for Your Mathematical Plays, Volume 1-4.
AMC 10 (2004): 12.
Berlekamp, Elwyn R.
The Dots-and-Boxes Game: Sophisticated Child’s Play.
AK Peters, Ltd., 2000.
Silverman, David L.
Your Move: Logic, Math and Word Puzzles for Enthusiasts.
Dover Publications, 1991.
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Thank you!
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