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Outline
1. Games
2. Nim
3. Strategies
4. Jim
5. The Winning Strategy for Nim
6. Addition of Games
7. Equivalence of Games
8. The Sprague-Grundy Theorem
Outline
1. Games
2. Nim
3. Strategies
4. Jim
5. The Winning Strategy for Nim
6. Addition of Games
7. Equivalence of Games
8. The Sprague-Grundy Theorem
Outline
1. Games
2. Nim
3. Strategies
4. Jim
5. The Winning Strategy for Nim
6. Addition of Games
7. Equivalence of Games
8. The Sprague-Grundy Theorem
Outline
1. Games
2. Nim
3. Strategies
4. Jim
5. The Winning Strategy for Nim
6. Addition of Games
7. Equivalence of Games
8. The Sprague-Grundy Theorem
Outline
1. Games
2. Nim
3. Strategies
4. Jim
5. The Winning Strategy for Nim
6. Addition of Games
7. Equivalence of Games
8. The Sprague-Grundy Theorem
Outline
1. Games
2. Nim
3. Strategies
4. Jim
5. The Winning Strategy for Nim
6. Addition of Games
7. Equivalence of Games
8. The Sprague-Grundy Theorem
Outline
1. Games
2. Nim
3. Strategies
4. Jim
5. The Winning Strategy for Nim
6. Addition of Games
7. Equivalence of Games
8. The Sprague-Grundy Theorem
Outline
1. Games
2. Nim
3. Strategies
4. Jim
5. The Winning Strategy for Nim
6. Addition of Games
7. Equivalence of Games
8. The Sprague-Grundy Theorem
1 3 5 79 11 13 1517 19 21 2325 27 29 31
2 3 6 710 11 14 1518 19 22 2326 27 30 31
4 5 6 712 13 14 1520 21 22 2328 29 30 31
8 9 10 1112 13 14 1524 25 26 2728 29 30 31
16 17 18 1920 21 22 2324 25 26 2728 29 30 31
13 =
8 + 4 + 1 = 011012
0 1 1 0 116 8 4 2 124 23 22 21 20
8 9 10 1112 13 14 1524 25 26 2728 29 30 31
01000 01001 01010 0101101100 01101 01110 0111111000 11001 11010 1101111100 11101 11110 11111
13 = 8 + 4 + 1
= 011012
0 1 1 0 116 8 4 2 124 23 22 21 20
8 9 10 1112 13 14 1524 25 26 2728 29 30 31
01000 01001 01010 0101101100 01101 01110 0111111000 11001 11010 1101111100 11101 11110 11111
13 = 8 + 4 + 1 = 011012
0 1 1 0 116 8 4 2 124 23 22 21 20
8 9 10 1112 13 14 1524 25 26 2728 29 30 31
01000 01001 01010 0101101100 01101 01110 0111111000 11001 11010 1101111100 11101 11110 11111
13 = 8 + 4 + 1 = 011012
0 1 1 0 116 8 4 2 1
24 23 22 21 20
8 9 10 1112 13 14 1524 25 26 2728 29 30 31
01000 01001 01010 0101101100 01101 01110 0111111000 11001 11010 1101111100 11101 11110 11111
13 = 8 + 4 + 1 = 011012
0 1 1 0 116 8 4 2 124 23 22 21 20
8 9 10 1112 13 14 1524 25 26 2728 29 30 31
01000 01001 01010 0101101100 01101 01110 0111111000 11001 11010 1101111100 11101 11110 11111
13 = 8 + 4 + 1 = 011012
0 1 1 0 116 8 4 2 124 23 22 21 20
8 9 10 1112 13 14 1524 25 26 2728 29 30 31
01000 01001 01010 0101101100 01101 01110 0111111000 11001 11010 1101111100 11101 11110 11111
Express in Binary
6 =
4 + 2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 + 1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 + 1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 +
2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 + 1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 + 1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 + 2
= 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 + 1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 + 1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 + 2 = 1 · 22 + 1 · 21 + 0 · 20
= 1102
5 = 4 + 1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 + 1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 + 2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 + 1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 + 1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 + 2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 =
4 + 1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 + 1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 + 2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 +
1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 + 1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 + 2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 + 1
= 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 + 1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 + 2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 + 1 = 1 · 22 + 0 · 21 + 1 · 20
= 1012
3 = 2 + 1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 + 2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 + 1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 + 1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 + 2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 + 1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 =
2 + 1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 + 2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 + 1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 +
1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 + 2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 + 1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 + 1
= 0 · 22 + 1 · 21 + 1 · 20 = 0112
Express in Binary
6 = 4 + 2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 + 1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 + 1 = 0 · 22 + 1 · 21 + 1 · 20
= 0112
Express in Binary
6 = 4 + 2 = 1 · 22 + 1 · 21 + 0 · 20 = 1102
5 = 4 + 1 = 1 · 22 + 0 · 21 + 1 · 20 = 1012
3 = 2 + 1 = 0 · 22 + 1 · 21 + 1 · 20 = 0112
Some Games Collected by David Hankin
There are 5 checkers on a table. A move consists of taking one ortwo checkers from the table. The winner is the one who takes thelast checker.
Some Games Collected by David Hankin
There are 100 checkers on a table. A move consists of taking mcheckers from the table, where m is a positive integer power of 2.The winner is the one who takes the last checker. Find the set L oflosing positions.
Some Games Collected by David Hankin
There are 100 checkers on a table. A move consists of taking mcheckers from the table, where m is a prime or m = 1. The winneris the one who takes the last checker. Find the set L of losingpositions.
Some Games Collected by David Hankin
There are 100 checkers on a table. A move consists of taking 1, 3,or 8 checkers from the table. The winner is the one who takes thelast checker. Find the set L of losing positions.
Some Games Collected by David Hankin
There are two piles of checkers on a table. A move consists oftaking any number of checkers from one pile or the same numberof checkers from each. The winner is the one who takes the lastchecker. Find the set L of losing positions.
Some Games Collected by David Hankin
Given an initial integer n0 > 1, two players, A and B, chooseintegers n1, n2, n3, . . . alternately according to the following rules.Knowing n2k , A chooses any integer n2k+1 such thatn2k ≤ n2k+1 ≤ n2
2k . Knowing n2k+1, B chooses any integer n2k+2
such that n2k+1/n2k+2 is a positive power of a prime. Player Awins by choosing the number 1990, player B wins by choosing thenumber 1. For which n0 does
A have a winning strategy,
B have a winning strategy,
neither player have a winning strategy?