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He showed that 2-connected graphs without odd cycles generate precisely the even permutations of their counters (eg. He considered labeling all but one vertex of a graph with counters and allowing a counter to move along an edge to the unlabelled vertex. Richard Wilson, in 1974, gave a complete treatment of puzzles on graphs of which the Fifteen puzzle is a particular case. A mathematical explanation that the puzzle could not be solved (known to Loyd much earlier) surfaced in the early 1880's and the excitement petered out. It is a single transposition from the desired configuration.
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This does not include the one which Sam Loyd was willing to pay money for, since So reachable permutations of the counters that fix the blank position are an even number of transpositions away from the desired configuration. Then you would represent a sequence of moves, that start and end with the blank in the lower right hand corner, as (n 1,b), (n 2,b).
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Suppose you record your moves using (n,b), to indicate that you slid the counter n into the blank position. Why? The answer to this question is not difficult (1). Despite this no one ever collected the prize. Historical accounts of the puzzle indicate that it generated an excitement similar to the Rubic's cube in the 1970's. A $1,000 prize was offered by the inventor to anyone who could solve the puzzle. The original puzzle was constructed with the 14 and 15 counters permuted. The puzzle above, contains 15 counters which can be moved using a vacant position in an attempt to reach this desired configuration. This paper focuses on a variation of an old puzzle, called the Fifteen puzzle, invented by Sam Loyd in the early 1870's. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet. This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Cut The Knot! An interactive column using Java applets
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