Crystal Bennett Joshua Chukwuka Advisor: Dr. K. Berg

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

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

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.

Rules of Gameplay

Rules of Game play. P P P

Positions

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

Theorem 1

Proof

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.

Theorem 2. P

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.

Theorem 3 P

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))

Theorem 4 contd.

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

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

Approach to 3x5 **** P

*** P

** P

* P

3x6 Chomp Grid. P W 1 Losing Position.

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

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

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

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

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

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

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

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

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

3x6 Chomp grid. Case 9 P

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

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

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

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.

3x7 Chomp Grid.

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

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