A puzzle introduced by Sam Loyd in 1878. It consists of 15 squares numbered from 1 to 15 which are placed in a box leaving one position out of the 16 empty. The goal is to rearrange the squares from a given arbitrary starting arrangement by sliding them one at a time into the configuration shown above. For some initial arrangements, this rearrangement is possible, but for others, it is not.

To address the solubility of a given initial arrangement, proceed as follows. If the Square containing the number
appears ``before'' (reading the squares in the box from left to right and top to bottom) numbers which are less
than , then call it an inversion of order , and denote it . Then define

where the sum need run only from 2 to 15 rather than 1 to 15 since there are no numbers less than 1 (so must equal 0). If is Even, the position is possible, otherwise it is not. This can be formally proved using Alternating Groups. For example, in the following arrangement

(2 precedes 1) and all other , so and the puzzle cannot be solved.

**References**

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations and Essays, 13th ed.* New York:
Dover, pp. 312-316, 1987.

Bogomolny, A. ``Sam Loyd's Fifteen.'' http://www.cut-the-knot.com/pythagoras/fifteen.html.

Bogomolny, A. ``Sam Loyd's Fifteen [History].'' http://www.cut-the-knot.com/pythagoras/history15.html.

Johnson, W. W. ``Notes on the `15 Puzzle. I.''' *Amer. J. Math.* **2**, 397-399, 1879.

Kasner, E. and Newman, J. R. *Mathematics and the Imagination.* Redmond, WA: Tempus Books, pp. 177-180, 1989.

Kraitchik, M. ``The 15 Puzzle.'' §12.2.1 in *Mathematical Recreations.* New York: W. W. Norton, pp. 302-308, 1942.

Story, W. E. ``Notes on the `15 Puzzle. II.''' *Amer. J. Math.* **2**, 399-404, 1879.

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1999-05-25