2. Crystal Bennett Joshua Chukwuka Advisor: Dr. K. Berg

3. Chomp.  Game Setup  Rules of gameplay  Positions  Theorem 1  Proof of Theorem 1  Approach to 3x5  Approach to 3xn  Theorem 1  Theorem 2  Theorem 3

4. Game Setup  Standard game consists of a grid of size mxn.  m = # of rows, n = # of columns.  Position on grid denoted by (i, j)

5. Game Setup  The (1, 1) square marked by “P” for poison is the least desired square.  You lose the game if you take the (1, 1) piece.

6. Rules of Gameplay

7. Rules of Game play. P P P

8. Positions

9. Plan  We will analyze Chomp positions for w = 0, 1, 2.

10. Theorem 1

11. Proof

35. Theorem 2  For any complete chomp grid of size m by n, there is a winning strategy. In other words, any one player before the starting of the game can always be sure of winning. PROOF: Because it is a complete m by n grid, it will have a square at its upper-right hand corner. Either taking that square is a winning move or it is not. Case 1: If it is a winning move, player 1 makes it and wins the game. Case 2: If it is a losing move then player 2 has a winning move in response to the first move which he makes and then wins the game. However, player 1 could have made player 2 move because it would require the removal of the square at the upper right hand corner and he would win the game.

36. Theorem 2. P

37. Theorem 3  For any chomp grid of size m by n (where m=n), that is a m by m grid, there is a winning strategy and the winning move is to take the (2,2) square. If the first player take the (2,2) square we would have 1 row and 1 column. Excluding the poisoned square,, there is (m-1) columns and rows. Then, the second player can only chomp from the bottom squares or the left squares. Since both are equal, it does not matter which squares player 2 takes from. His move always makes them unequal and the winning strategy for player 1 is to make his next move such that the 1nth row and column(without poison) have equal number of squares. This way, player 1 takes the last square from either the row or the column and then player 2 is left with the poisoned square and then loses.

38. Theorem 3 P

39. Theorem 4  For all stair case kind of chomp grid of step size 1, that is any row has one square more than the row above it there is a winning strategy.(except the first staircase) Case 1: For the staircase with three rows, the winning move is to chomp the (2,2) square. Case 2: For any staircase, the winning move is to chomp any square that makes the staircase a 2 by n with a winning move already made. PROOF: Case 1 is true by theorem 3. Case 2 is true by theorem 1.  Therefore, there is always a winning strategy for all staircase(excluding the (1,1))

40. Theorem 4 contd.

41. Approach to 3x5  Start at the 2x2 Square finding the winning positions.  Add one square at a time to the grid solving for the winning moves up until the 3x5 grid.  Analyze different types of staircase grids where the bottom, middle, and top row all have different lengths. P

42. Approach to 3x5  The position (3,4) is a winning position for the 3x5 grid.  How do I prove this. Analyze the winning moves for all grids up to the 3x5. Identify any patterns, or relations that may exist. P

43. Approach to 3x5 **** P

44. *** P

45. ** P

46. * P

47. 3x6 Chomp Grid. P W 1 Losing Position.

48. 3x6 Chomp Grid. CASE 10CASE 11CASE 12 CASE 7CASE 8CASE 9 CASE 1CASE 2CASE 3 Possible Cases. CASE 4CASE 5CASE 6

49. 3x6 Chomp Grid. Case 1 Case 1 is true by the Rules of Game play

50. 3x6 Chomp Grid. Case 2 P Case 2 is true by Theorem 2

51. 3x6 Chomp Grid. P Case 3 is true by Theorem 2. Case 3

52. 3x6 Chomp Grid. Case 4 P Case 4 is true by Theorem 2.

53. 3x6 Chomp Grid. Case 5 P Case 5 is true by Theorem 1.

54. 3x6 Chomp Grid. Case 6 P Case 6 is true by Theorem 1.

55. 3x6 Chomp Grid. Case 7 P Case 7 is true by Theorem 1.

56. 3x6 Chomp Grid. Case 8. P Case 8 is true by the square chomp Theorem 3.

57. 3x6 Chomp grid. Case 9 P

58. 3x6 Chomp Grid. Case 10 P Case 10 is true by Theorem 3.(2xn)

59. 3x6 Chomp Grid. Case 11 P Case 11 is true by Theorem C.

60. 3x6 Chomp Grid Case 12 P Case 12 is true by Theorem c

61. Approach to 3xn  Use 2x2 up to 3x5 findings to solve larger grids at finding the winning positions.  Add one square at a time to the grid solving for the winning moves.  Analyze different types of staircase grids where the bottom, middle, and top row all have different lengths.  Record winning positions to see if we can find a relationship between the winning positions.

62. 3x7 Chomp Grid.

63. CASE 15CASE 16CASE 17CASE 18 CASE 8CASE 9CASE 10CASE 11 CASE 1CASE 2CASE 3CASE 4 CASE 12CASE 13CASE 14 CASE 5CASE 6CASE 7

65. Q U E S T I O N S COMMENTS CONCERNS