Answer
-Proving that the first player has a winning strategy for the
game of Chomp,
-if the initial board is square.
Work Step by Step
--Let the Chomp board have n rows and n columns.
-We claim that the first player can win the game by making the first move to leave just the top row and leftmost column.
-Let P(n) be the statement that if a player has presented his opponent with a Chomp configuration consisting of just n cookies in the top row and n cookies in the leftmost column, then he can win the game.
-We will prove ∀ n P (n) by strong induction.
-We know that P(1) is true, because the opponent is forced to take the poisoned cookie at his first turn.
- Fix k ≥ 1 and assume that P(j) is true for all j ≤ k.
- We claim that P(k + 1) is true. It is the opponent’s turn to move.
If she picks the poisoned cookie, then the game
is over and she loses.
-Otherwise, assume she picks the cookie in the top row in column j , or the cookie in the left column in row j ,
-for some j with 2 ≤ j ≤ k + 1.
-The first player now picks the cookie in the left column in row j , or the cookie in the top row in column j , respectively.
-This leaves the position covered by P(j −1) for his opponent, so by the inductive hypothesis,
he can win.
