Analysis from the endgame

In this section we describe a general method which will allow us to find a set of winning positions on many games.

We return to Problem 9, the problem about the single king on a chessboard. Let us try to find a set of winning positions. As always, the final position of the game, with the king in square h8 must be a winning one. We therefore place a plus sign in square h8 (see Figure 1). We will place the same sign in every other square at which the king occupies a winning position, and a minus sign in every square which is not a winning position (we call them loosing positions).

Figure1.

Our next investigations (Figure 2) go as follows. The squares from which the king can move to a winning square in a single move are loosing squares (g7, g8 and h7). Next, from squares h6 and f8 we can only go to a loosing squares, so they must be winning ones.These new winning positions lead to new losing positions; h5, g5, g6, f7, e7, and e8.

Figure2.

We continue in analogous fashion see Figure 3 until we finally assign pluses and minuses to all squares on the chess board. This completely describes the winning strategy for the second player.

Figure 3.

The method of finding winning positions just described is called analysis from the endgame. Applying it to the game with the castle (Problem 8) is not hard and and leads to the solution from the previous section (Figure 4).

Figure 4.

Methodological Remark. Students often perform their own ``analysis from the endgame'' intuitively. That is, they can see to the end of the game from a few moves before, and begin to learn which of the possible moves are winning ones, then generalize this to the rest of the game. The best learning will occur if students make this discovery on their own (by playing the game), then are asked to articulate it.