Gale's Problem (1974). The game is played between two players, I and II, on a board of n m squared. Each square is identified by a pair (i, j) where i and j are numbers between 1 and n and between 1 and m respectively. The coordinate i is the horizontal coordinate and the coordinate j is the vertical coordinate. Players I and II play alternately according to the following rules:

-Player I starts the game. -The player who selects the square (1, 1) loses the game. -In each turn, the player chooses a square (i, j) that has not been chosen or disabled in previous moves. After each turn, the squares (k, l) such that k i and l j are disabled.

Answer the following questions:

a) In the game of Gale played on a 2 board, is the statement that player I has a winning strategy true? Justify your answer. If the statement is true, describe a winning strategy for I.

b) Which player has a winning strategy when Gale's game is played on a n board? Justify your answer. Describe the winning strategy.

c) Find two winning strategies for player I when the board is n .