1. Salvador Badillo-Rios and Verenice Mojica

2. Goal The goal of this research project was to provide an extended analysis of 2-D Chomp using computational and mathematical means in order to provide a pattern that may aid in finding the winning strategy for all board sizes.

3. Game Description Geometric 2-D Chomp: Two-player game Players take turns choosing a square box from an m x n board The pieces below and to the right of the chosen cell disappear after every turn

4. Game Description Geometric 2-D Chomp: Two-player game Players take turns choosing a square box from an m x n board The pieces below and to the right of the chosen cell disappear after every turn Player 1 makes a move

5. Game Description Geometric 2-D Chomp: Two-player game Players take turns choosing a square box from an m x n board The pieces below and to the right of the chosen cell disappear after every turn Player 2 makes a move

6. Game Description Geometric 2-D Chomp: Two-player game Players take turns choosing a square box from an m x n board The pieces below and to the right of the chosen cell disappear after every turn Player 1 makes a move

7. Game Description Geometric 2-D Chomp: Two-player game Players take turns choosing a square box from an m x n board The pieces below and to the right of the chosen cell disappear after every turn Player 2 makes a move

8. Game Description Geometric 2-D Chomp: Two-player game Players take turns choosing a square box from an m x n board The pieces below and to the right of the chosen cell disappear after every turn Player 1 makes a move

9. Game Description Geometric 2-D Chomp: Two-player game Players take turns choosing a square box from an m x n board The pieces below and to the right of the chosen cell disappear after every turn x Player 2 loses!

10. Game Description Numeric 2-D Chomp: Players take turns choosing a divisor of a given natural number, N They are not allowed to choose a multiple of a previously chosen divisor The player to choose 1 loses 2020 21212 2323 2424 3030 124816 3131 36122448 3232 9183672144 3 2754108216432 N = 2 4 * 3 3 = 432

11. Game Description Numeric 2-D Chomp: Players take turns choosing a divisor of a given natural number, N They are not allowed to choose a multiple of a previously chosen divisor The player to choose 1 loses 2020 21212 2323 2424 3030 124816 3131 36122448 3232 9183672144 3 2754108216432 N = 2 4 * 3 3 = 432 Player 1 chooses 12

12. Game Description Numeric 2-D Chomp: Players take turns choosing a divisor of a given natural number, N They are not allowed to choose a multiple of a previously chosen divisor The player to choose 1 loses 2020 21212 2323 2424 3030 124816 3131 36122448 3232 9183672144 3 2754108216432 N = 2 4 * 3 3 = 432 Player 2 chooses 9

13. Game Description Numeric 2-D Chomp: Players take turns choosing a divisor of a given natural number, N They are not allowed to choose a multiple of a previously chosen divisor The player to choose 1 loses 2020 21212 2323 2424 3030 124816 3131 36122448 3232 9183672144 3 2754108216432 N = 2 4 * 3 3 = 432 Player 1 chooses 8

14. Game Description Numeric 2-D Chomp: Players take turns choosing a divisor of a given natural number, N They are not allowed to choose a multiple of a previously chosen divisor The player to choose 1 loses 2020 21212 2323 2424 3030 124816 3131 36122448 3232 9183672144 3 2754108216432 N = 2 4 * 3 3 = 432 Player 2 chooses 2

15. Game Description Numeric 2-D Chomp: Players take turns choosing a divisor of a given natural number, N They are not allowed to choose a multiple of a previously chosen divisor The player to choose 1 loses 2020 21212 2323 2424 3030 124816 3131 36122448 3232 9183672144 3 2754108216432 N = 2 4 * 3 3 = 432 Player 1 chooses 3

16. Game Description Numeric 2-D Chomp: Players take turns choosing a divisor of a given natural number, N They are not allowed to choose a multiple of a previously chosen divisor The player to choose 1 loses 2020 21212 2323 2424 3030 124816 3131 36122448 3232 9183672144 3 2754108216432 N = 2 4 * 3 3 = 432 Player 2 loses!

17. Fair or Unfair? Strategy-Stealing Argument Suppose player one begins by removing the bottom right-most piece

18. Fair or Unfair? Strategy-Stealing Argument Suppose player one begins by removing the bottom right-most piece If that move is a winning move, then player one wins

19. Fair or Unfair? Strategy-Stealing Argument If it is a losing move, player two has a good countermove and player two wins

20. Fair or Unfair? Strategy-Stealing Argument If it is a losing move, player two has a good countermove and player two wins But player one could have gotten to that countermove from the very beginning Therefore, player one has the winning move and can always win, if he/she plays perfectly

21. Known Special Cases m x m Chomp Player one chomps the piece located at (2,2) x 1234 1 2 3 4

22. Known Special Cases m x m Chomp Player one chomps the piece located at (2,2) The board is left as an L- shape, and player one copies player two’s moves symmetrically Player 1 chooses (2,2)

23. Known Special Cases m x m Chomp Player one chomps the piece located at (2,2) The board is left as an L- shape, and player one copies player two’s moves symmetrically Player 2 moves

24. Known Special Cases m x m Chomp Player one chomps the piece located at (2,2) The board is left as an L- shape, and player one copies player two’s moves symmetrically Player 1 moves symmetrically

25. Known Special Cases m x m Chomp Player one chomps the piece located at (2,2) The board is left as an L- shape, and player one copies player two’s moves symmetrically Player 2 moves

26. Known Special Cases m x m Chomp Player one chomps the piece located at (2,2) The board is left as an L- shape, and player one copies player two’s moves symmetrically Player 1 moves symmetrically

27. Known Special Cases m x m Chomp Player one chomps the piece located at (2,2) The board is left as an L- shape, and player one copies player two’s moves symmetrically x Player 2 loses!

28. Known Special Cases Two-Rowed Chomp Proposition 0: (a, a-1) is P-position, where a  1 (a,b) is an N-position ONLY when a  b  0 and a≠ b+1 Winning Moves: (a,a-1) if a=b (b+1,b) if a  b+2

29. Three-Rowed Chomp Zeilberger’s “Chomp3Rows” Doron Zeilberger developed a program that computed P-positions for 3-rowed Chomp for c≤ 115 We will be using this notation throughout [c, b, a] |--------c--------- | |---b---| |--------a--------|

30. Three-Rowed Chomp Proposition 1: The only P-positions [c,b,a], with c = 1, are [1,1,0] and [1,0,2] N-positions with at least 6 pieces and with c = 1: [1,1,1], [1,2,0], [1,0,3+x], and [1,1+y,x] Winning Moves: [1,1,1], [1,2,0],[1,0,3+x] to [1,0,2] [1,1+x,y] to [1,1,0] Proposition 2: [2,b 0,a 0 ] is a P-position iff a 0 = 2 [1,1,4] i.e., [1,0,3+x] where x = 1 Move to: [1,0,2]

31. Our Research Two computers play against each other, both eventually learn to play at their best Displays : Board 1 st computer’s opening winning move P-positions and their total amount Number of games played Adaptive Learning Program

32. Approximation of P-positions

34. Approximation of P-Positions

35. Initial attempt to Analyze P-Positions Initially we decided to look at the sum of the P- positions to note obvious patterns One obvious pattern was found (the one proposed by Zeilberger) Was not much of a success due to the various possible arrangements of pieces

36. Analyzing Opening Winning Moves Computer’s opening winning moves for 3,4, and 5 rows were analyzed One significant pattern was observed for 3-rowed Chomp, and a possible pattern was observed as well No clear patterns were found for 4 and 5-rowed Chomp

37. P-Positions after Computer Learned Opening Move 3-Rows Board Size:Value of NComputer 1's Opening Winning MoveResuting P-Position 3x193[0,0,1] 3x218 [1,1,0] 3x3366[1,0,2] 3x47212[2,0,2] 3x514472[3,2,0] 3x628824[3,0,3] 3x7576144[4,3,0] 3x8115248[4,0,4] 3x92304576[6,3,0] 3x10460896[5,0,5] 3x119216192[6,0,5] 3x12184322304[8,4,0] 3x1336864384[7,0,6] 3x14737289216[10,4,0] 3x15147456768[8,0,7]

38. Type 1: y n = [y 0 [1]+4n,0,y 0 [3]+3n] Opening Winning Move Conjecture for 3-Rowed Chomp Suppose x n is the set of board sizes: 3 x (1+7n), 3 x (3+7n), 3 x (4+7n), 3 x (6+7n), where n≥0. Then the computer’s opening winning moves for x n are to the set of P-positions y n y n has a pattern such that: y n = [y 0 [1]+4n,0,y 0 [3]+3n], where : Board Size (x 0 ): Computer 1's Opening Winning Move: Resulting P-Positions (y 0 ): 3x13[0,0,1] 3x36[1,0,2] 3x412[2,0,2] 3x624[3,0,3] Board Size (x 1 ): Computer 1's Opening Winning Move: Resulting P-Positions (y 1 ): 3x848[4,0,4] 3x1096[5,0,5] 3x11192[6,0,5] 3x13384[7,0,6] y 0 = { [0,0,1] } [1,0,2] [2,0,2] [3,0,3]

39. Type 1: y n = [y 0 [1]+4n,0,y 0 [3]+3n] Board Size (x 0 ): Computer 1's Opening Winning Move: Resulting P-Positions (y 0 ): 3x13[0,0,1] 3x36[1,0,2] 3x412[2,0,2] 3x624[3,0,3] Board Size (x 1 ): Computer 1's Opening Winning Move: Resulting P-Positions (y 1 ): 3x848[4,0,4] 3x1096[5,0,5] 3x11192[6,0,5] 3x13384[7,0,6] Board Size: Computer 1's Opening Winning Move Resuting P-Position 3x218[1,1,0] 3x572[3,2,0] 3x7144[4,3,0] 3x9576[6,3,0] 3x122304[8,4,0] 3x149216[10,4,0] Type 2: In Progress

40. No Patterns Found 4-Rows Board Size Value of N Computer 1's Opening Winning Move Resuting P- Position 4 x1273[0,0,0,1] 4x254 [1,1,0,0] 4x310818[1,0,2,0] 4x42166[1,0,0,3] 4x543236[2,0,3,0] 4x686412[2,0,0,4] 4x7172872[3,0,3,0] 4x8345624[3,0,0,4] 4x969123456[7,2,0,0] 4x1013824288[5,0,5,0] 4x112764848[4,0,0,7] 4x125529696[5,0,0,7] 4x13110592576[6,0,7,0] 5-Rows Board SizeValue of N Computer 1's Opening Winning Move Resuting P- Position 5x1813[0,0,0,0,1] 5x2162 [1,1,0,0,0] 5x3324108[2,0,1,0,0] 5x464836[2,0,0,0,2] 5x512966[1,0,0,0,4] 5x6259272[3,0,0,3,0] 5x7518412[2,0,0,0,5] 5x8103685184[7,1,0,0,0] 5x920736288[5,0,0,4,0] 5x104147224[3,0,0,0,7] 5x118294441472[9,2,0,0,0]

41. Analyzing All P-positions by Grouping 3, 4, and 5-rowed Chomp was analyzed The P-positions within these n-rowed Chomp sets were grouped by the amount of pieces in the bottom row The P-positions for each group were then sorted into their possible permutations 4 Rows: d = 2 [2,0,0,4] [2,1,0,2][2,1,1,5][2,1,2,2][2,1,3,2][2,1,4,3][2,1,5,3][2,1,6,3] [2,2,1,3] [2,3,0,4] [2,4,0,6] [2,5,1,5] [2,6,3,2] [2,7,4,0]

42. Pattern Found After Grouping Constant Row Value Conjecture For n-rowed Chomp, when n≥3, at least one subset of its total P-positions will have a pattern as follows: n-2 columns of the data for the subset will be fixed to distinct constant values In the following column the values will increase by a value of one The values of the remaining columns may vary or have a constant value as well 3-Rows: c = 4 [c,b,a] [4,0,4] [4,1,4] [4,2,4] [4,3,0] 4-Rows: d = 4 [d,c,b,a] [4,0,0,7] [4,0,1,5] [4,0,2,7] [4,0,3,2] [4,0,4,4] 5-Rows: e = 1 [e,d,c,b,a] [1,0,0,0,4] [1,0,0,1,2] [1,0,0,2,3] [1,0,0,3,3] [1,0,0,4,3] [1,0,0,5,3] [1,0,0,6,3] [1,0,0,7,3]

43. Concluding Remarks Developed a learning program to analyze Chomp Approximated amount of P-positions per board size Initially analyzed sum of P-positions to find patterns Analyzed Computer’s opening moves and resulting P-positions Opening Winning Move Conjecture for 3-Rowed Chomp Grouped P-positions of certain board sizes with fixed boards by amount of pieces in bottom row Constant Row Value Conjecture

44. Aknowledgements iCAMP Program Faculty Advisor: Dr. Eichhorn Robert Campbell Game Theory fellow researchers