## Wednesday, October 8, 2014

### Sam Loyd and the 15 Puzzle

The 15-puzzle was a version of a sliding puzzle (or sliding piece puzzle) invented by Noyes Palmer Chapman, possibly as early as 1874.  The puzzle is played by mixing up the number arrangement and then try to move the blocks (without removing them from the box – only by sliding them around) to rearrange the numbers in order from 1 to 15, as shown below.

The 15-puzzle was a version of a sliding puzzle (or sliding piece puzzle) invented by Noyes Palmer Chapman, possibly as early as 1874.  The puzzle is played by mixing up the number arrangement and then try to move the blocks (without removing them from the box – only by sliding them around) to rearrange the numbers in order from 1 to 15, as shown below.
 A properly solved 15-Puzzle.
Sam Loyd (a chess player and puzzlist) claimed from 1891 until his death in 1911 that he had invented the puzzle.  This dispute, however, is not the big story behind this puzzle.
Sam Loyd offered a new challenge.  He offered a version of this puzzle with the 14 and 15 switched (see below), and he offered a \$1000 prize to anybody who could provide a written solution for this puzzle.  Many people accepted the challenge, and a few did submit “solutions” trying to obtain the \$1000 prize, but no one was able to submit a correct solution.  This is because this puzzle, in this configuration, is impossible to solve.

 The impossible version of the 15-Puzzle by Sam Loyd.  This puzzle cannot be rearranged with all of the number in the proper order.
The solution to this problem depends on “parity” – in short there is an even and an odd version of this puzzle.  Swapping the 14 and 15 changes the parity and makes the problem unsolvable.
Modern versions of this puzzle are usually construct of plastic pieces that have slots and grooves along the sides of the pieces that make in impossible (or almost impossible) to remove the pieces.  It is impossible to change the parity of these puzzles without removing the pieces.  In other words you can’t arrange the pieces to swap the 14 and 15 blocks, and leave the rest of the blocks in the correct order.

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David