Rick's Tricky Six Puzzle: S 5 Sits Specially in S 6

Many of you will be familiar with the Fifteen Puzzle (FIGURE 1, left). Singmaster [16, §5A, pp. 77–84] gives nearly a hundred references to it. It is often associated with the name of Sam Loyd, but Sam continues to be a controversial figure [9, Chapter 2, pp. 18–30; 17]. In the unlikely event that you’ve never seen the Fifteen Puzzle, you can read about it in the review quoted in the next section. Sliding block puzzles may be represented by graphs in which the vertices represent possible positions of the blocks and the edges represent the permissible moves of a block from one position to another. For example, the Fifteen Puzzle may be thought of as being played on the sixteen vertices of the graph in FIGURE 1. In this graph, don’t think of the numbers as labels for the vertices, but as labeled blocks that can be slid from a vertex to an empty vertex. For example, in the figure, either block 12 or block 15 may be slid onto the vertex where indicates that there is no block.