On Modular Extensions to Nim

Transcription

1 On Modular Extensions to Nim Karan Sarkar Mentor: Dr. Tanya Khovanova Fifth Annual Primes Conference 16 May 2015

2 An Instructive Example: Nim The Rules Take at least one token from some chosen pile. Player who takes last token wins.

3 An Instructive Example: Nim The Rules Take at least one token from some chosen pile. Player who takes last token wins.

4 An Instructive Example: Nim The Rules Take at least one token from some chosen pile. Player who takes last token wins.

5 An Instructive Example: Nim The Rules Take at least one token from some chosen pile. Player who takes last token wins.

6 An Instructive Example: Nim The Rules Take at least one token from some chosen pile. Player who takes last token wins.

7 Nim Positions Position Notation A position with piles of sizes a 1, a 2,..., a n is denoted as the ordered n-tuple: (a 1, a 2,..., a n ). Definition A P-position is a position that guarantees a loss given optimal play Definition An N-position is a position that guarantees a win given optimal play

8 The Winning Strategy for Nim Theorem (Bouton s Theorem) The position (a 1, a 2,..., a n ) is a P-position in Nim if and only if n a i = 0. i=1 Definition (Bitwise XOR) The symbol denotes the bitwise XOR operation. 1 Write both numbers in binary. 2 Add without carrying over.

9 m-modular Nim The Rules Take at least one token from some chosen pile or km tokens total. Player who takes last token wins.

10 An Example: 3-Modular Nim with 2 Piles

11 An Example: 3-Modular Nim with 2 Piles

12 An Example: 3-Modular Nim with 2 Piles

13 An Example: 3-Modular Nim with 2 Piles

14 An Example: 3-Modular Nim with 2 Piles

15 An Example: 3-Modular Nim with 2 Piles (0,0) (1,1) (2,2)

16 An Example: 3-Modular Nim with 2 Piles (0,0) (1,1) (2,2)

17 An Example: 3-Modular Nim with 2 Piles (0,0) (1,1) (2,2)

18 2 Heap m-modular Nim for Odd m Theorem For odd m, a position of m-modular Nim with 2 heaps is a P-position if and only it is of the form (i, i) for integers i where 0 i < m.

19 An Example: 6-Modular Nim with 2 Piles (0,0) (1,1) (2,2)

20 An Example: 6-Modular Nim with 2 Piles (7,8) (8,7) (5,6) (6,5) (3,4) (4,3) (2,2) (1,1) (0,0)

21 An Example: 6-Modular Nim with 2 Piles (7,8) (8,7) (5,6) (6,5) (3,4) (4,3) (2,2) (1,1) (0,0)

22 Another Example: 12-Modular Nim with 2 Piles

23 Another Example: 12-Modular Nim with 2 Piles

24 Another Example: 12-Modular Nim with 2 Piles

25 m-modular Nim for 2 Heaps Theorem Let m = 2 i k where k is odd. A position is a P-position if and only if it is of the form: (2 j 1 b + a, (k + 1)2 j 1 1 a) for all 0 a < 2 j 1, k b < 2k and 0 j < i. Corollary Let m = 2 i k where k is odd. There are ( ) i m P-positions in m-modular Nim with 2 heaps.

26 7-Modular Nim with 3 Heaps

27 A Snapshot of Nim with 3 Heaps

28 A Snapshot of Nim with 3 Heaps

29 A Snapshot of Nim with 3 Heaps

30 7-Modular Nim with 3 Heaps

31 m-modular Nim for odd m Theorem A position (a 1, a 2,..., a n ) is a P-position if and only if: n 1 a i = 0. 2 i=1 n i=1 a i < 2m.

32 14-Modular Nim with 3 Heaps

33 14-Modular Nim with 3 Heaps

34 14-Modular Nim with 3 Heaps

35 7-Modular Nim with 3 Heaps

36 m-modular Nim for Even m: A Partial Result Theorem If a position (a 1, a 2,..., a n ) is a P-position in m-modular Nim for m odd, then it is a P-position in 2m-Modular Nim.

37 Future Research What happens in Miseré Modular Nim? How do the P-positions for even m behave? What happens one can take away km + r tokens for other values of r? What about other polynomials?

38 Acknowledgments I would like to thank My mentor, Dr. Khovanova: for her suggestion of the project and guidance MIT PRIMES: for the opportunity to conduct research My parents: for their encouragement and transportation