Chapter 6 - Questions Concerning Games and Puzzles

Summary

Puzzles and Directed Graphs

Previously (Section 2.6) we explained how puzzle problems can be formulated in terms of graphs. The vertices of the graph correspond to the positions in a puzzle; the edges of the graph correspond to the possible moves from one position to another. The solution of the puzzle consists in finding a path from a given initial position to one (or possibly more) terminal or winning positions.

In dealing with these puzzles, we used undirected graphs. This was based upon the tacit assumption that the moves can be made both ways from one position to another. Such a procedure is permissible for the puzzles of the ferryman, the three jealous husbands, and the moves of the knight on the chessboard.

But for many puzzles the moves can only be made in one direction, and in this case we are compelled to use directed graphs in the representation. If some moves can be made in both directions, we can include an edge for each direction, or we can use a mixed graph in which these edges are undirected. To solve the puzzle, we must find a directed path from the initial position in the graph to the desired terminal position.

We shall illustrate these remarks by considering an ancient and familiar puzzle.