Winning positions

Problem8.   On a chessboard a rook stand on square a1. Players take turns moving the rook as many squares as they want, either horizontally to the right or vertically upward. The player who can place the rook on square h8 wins.

Solution. In this game, the second player will win. The strategy is quite simple: at each turn, place the rook on the diagonal from a1 to h8. The reason this works is that the first player is forced to move the rook of the diagonal at each turn, while the second player can always put the rook back on this diagonal. Since the winning square belongs to the diagonal, the second player will eventually be able to place the rook on it.

Let us analyze this solution a little more deeply. We have been able here to define a class of winning positions (in which the rook is on the diagonal from a1 to h8) which enjoys the following properties:

1.

The final position of the game is a winning one;

2.

A player can never move from one winning position to another an a single turn;

3.

A player can always move from a non-winning position to a winning one in a single move.

The discovery of such a class of winning positions for a given game is equivalent to solving the game. Indeed, moving to a winning position at each move constitutes a winning strategy. If the initial position of the game is a winning one, then the second player will win (as in the game described above). Otherwise, the first player will win.

Methodological Remark. As the concept of winning positions generalizes a set of strategies it can only be understood after solving several games as the presented in this section. As always, it is important to have students play each game before solving it.

Problem9.   A king is placed on a1 square of a chessboard. Players take turns moving the king either upwards, to the right, or along a diagonal going upwards and to the right. the player who places the king on square h8 is the winner.