Alice and Bob play the following game. They start with an empty 4x4 grid: Alice

Question

Alice and Bob play the following game. They start with an empty 4x4 grid: Alice makes the first move by writing either 0 or 1 in a slot of the grid. Then Bob writes either 0 or 1 in an empty slot, and they continue like this until all slots are filled. At each turn they can choose any empty slot and they can put either 0 or 1 there. At the end they calculate the determinant of the resulting 4x4 matrix If the determinant is non-zero Alice wins, otherwise Bob is the winner. Show that there is a strategy that Bob can follow to win regardless of what Alice does. Alice and Bob play the same game as above but the grid is 5 x 5. Alice still makes the first move but now she wins if the determinant is zero, and Bob wins il it is non-zero. Show that in this case there is a strategy that Alice can follow to win no matter what Bob does.

Explanation / Answer

a) bob should put the opposite of what alice put in any of the slot. if alice put a zero, bob should put a 1 and vice versa. this will always give value of determinant 0.

b) Alice should put either zero or 1 at any place, then just put the opposite of whatever bob put, the determinant will be zero if this strategy is foolowed

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