Problems

There are three piles of rocks: in the first pile there are 10 rocks, 15 in the second pile and 20 in the third pile. In this game (with two players), in one turn a player is allowed to divide one of the piles into two smaller piles. The loser is the one who cannot make a move. Which player would be the winner?

Louise has a chessboard \(8\times8\) without two opposite corners (see the picture), and 31 dominoes \(2\times1\). Can she tile the crippled chessboard with dominoes she got?

There is a \(3 \times 3\) grid filled with zeros. Louise is allowed to add 1 to each small square inside any \(2\times2\) grid. Can she ever get the following table as a result of her actions?

A rectangular floor is to be covered by \(2 \times 2\) and \(1\times4\) tiles (everything is arranged). Unfortunately one tile got smashed, but we have one more tile of the other kind available. Can we retile the floor perfectly?