Jim and Nim. Japheth Wood New York Math Circle. August 6, 2011
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1 Jim and Nim Japheth Wood New York Math Circle August 6, 2011
2 Outline 1. Games
3 Outline 1. Games 2. Nim
4 Outline 1. Games 2. Nim 3. Strategies
5 Outline 1. Games 2. Nim 3. Strategies 4. Jim
6 Outline 1. Games 2. Nim 3. Strategies 4. Jim 5. The Winning Strategy for Nim
7 Outline 1. Games 2. Nim 3. Strategies 4. Jim 5. The Winning Strategy for Nim 6. Addition of Games
8 Outline 1. Games 2. Nim 3. Strategies 4. Jim 5. The Winning Strategy for Nim 6. Addition of Games 7. Equivalence of Games
9 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
10 But First...
11
12 13 =
13 13 =
14 13 = =
15 13 = =
16 13 = =
17 13 = =
18 Express in Binary 6 =
19 Express in Binary 6 = 4 +
20 Express in Binary 6 = 4 + 2
21 Express in Binary 6 = =
22 Express in Binary 6 = = = 110 2
23 Express in Binary 6 = = = =
24 Express in Binary 6 = = = = 4 +
25 Express in Binary 6 = = = = 4 + 1
26 Express in Binary 6 = = = = =
27 Express in Binary 6 = = = = = = 101 2
28 Express in Binary 6 = = = = = = =
29 Express in Binary 6 = = = = = = = 2 +
30 Express in Binary 6 = = = = = = = 2 + 1
31 Express in Binary 6 = = = = = = = =
32 Express in Binary 6 = = = = = = = = = 011 2
33 Which is Bigger?
34 Which is Bigger? Base 10:
35 Which is Bigger? Base 10: 1894 or 2011
36 Which is Bigger? Base 10: 1894 or or 1993
37 Which is Bigger? Base 10: 1894 or or 1993 Base 2:
38 Which is Bigger? Base 10: 1894 or or 1993 Base 2: or 10101
39 Which is Bigger? Base 10: 1894 or or 1993 Base 2: or or 10011
40 And now back to our talk...
41
42
43
44
45
46 Nim
47 Nim Challenges Who has the winning strategy?
48 Nim Challenges Who has the winning strategy?
49 Nim Challenges Who has the winning strategy?
50 Nim Challenges Who has the winning strategy?
51 Nim Challenges Who has the winning strategy?
52 Nim Challenges Who has the winning strategy?
53 Nim Challenges Who has the winning strategy?
54 Nim Challenges Who has the winning strategy?
55 Nim Challenges Who has the winning strategy?
56 Nim Challenges Which two pile Nim games are L positions?
57 W and L games L W W W W
58 W and L games L W W W W W W L W W
59 W and L games L W
60 W and L games L W L W
61 W and L games L W
62 Jim
63 Jim Challenges Who has the winning strategy?
64 Jim Challenges Who has the winning strategy?
65 Jim Challenges Who has the winning strategy?
66 Jim Challenges Who has the winning strategy?
67 Jim Challenges Who has the winning strategy?
68 Jim Challenges Who has the winning strategy?
69 Jim Challenges Who has the winning strategy?
70 Jim Challenges Who has the winning strategy?
71 Jim Challenges Who has the winning strategy?
72 Jim Challenges Which two row Jim games are L positions?
73 3-Row Jim Show this is an L position.
74 3-Row Jim Goal: Describe all 3-Row Jim L positions.
75 Jim and Nim Can a Strategy for Jim help solve Nim?
76 Jim and Nim Can a Strategy for Jim help solve Nim?
77 Thank You! Nim and Jim Japheth Wood, New York Math Circle
78 Some Games Collected by David Hankin There are 5 checkers on a table. A move consists of taking one or two checkers from the table. The winner is the one who takes the last checker.
79 Some Games Collected by David Hankin There are 100 checkers on a table. A move consists of taking m checkers 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 of losing positions.
80 Some Games Collected by David Hankin There are 100 checkers on a table. A move consists of taking m checkers from the table, where m is a prime or m = 1. The winner is the one who takes the last checker. Find the set L of losing positions.
81 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 the last checker. Find the set L of losing positions.
82 Some Games Collected by David Hankin There are two piles of checkers on a table. A move consists of taking any number of checkers from one pile or the same number of checkers from each. The winner is the one who takes the last checker. Find the set L of losing positions.
83 Some Games Collected by David Hankin Given an initial integer n 0 > 1, two players, A and B, choose integers n 1, n 2, n 3,... alternately according to the following rules. Knowing n 2k, A chooses any integer n 2k+1 such that n 2k n 2k+1 n 2 2k. Knowing n 2k+1, B chooses any integer n 2k+2 such that n 2k+1 /n 2k+2 is a positive power of a prime. Player A wins by choosing the number 1990, player B wins by choosing the number 1. For which n 0 does A have a winning strategy, B have a winning strategy, neither player have a winning strategy?
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