Pseudo-games

This class of games we first examine are games that turn out to be jokes. The outcomes of these pseudo games do not depend on the play proceeds. For this reason, the solution of such a pseudo-game does not consist of a winning strategy, but of a proof that one or the other of the two players will always win.

Problem1. Two children take turns breaking up a rectangular chocolate bar 6 squares wide by 8 squares long. They may break the bar only along the divisions between the squares. If the bar breaks into several peaces, they keep breaking the pieces up until only the individual squares remain. The player who cannot make a break loses the game. Who will win?

Solution. After each move, the number of the pieces increases by one. At first, there is only one piece. At the end of the game, when no more moves are possible, the chocolate is divided into small squares ant there are 48 of these. So there must have been 47 moves of which the last, as well as every other odd-numbered move, was made by the first player. Therefore, the first player will win, no matter how the play proceeds.

Problem2. There are three piles of stones: one with 10 stones, one with 15 stones and one with 20. At each turn a player can choose one of the piles and divide it into two smallest piles. The looser is the player who cannot do this. Who will win and how?

Problem3. Two players take turns placing rooks on a chessboard so that they cannot capture each other. The loser is the player who cannot place a rook. Who will win?