Sam Loyd's 14-15 Puzzle: Part III

10/29/2015

SOLUTIONS:

Figure 1: Sam explains that the "real trick" of the puzzle could only be performed by changing the 9 into a 6, and the 6 into a 9, by turning them upside down during the manipulation of the blocks.  And I know this problem can be solved.  The reason the original puzzle with the 14 and 15 reversed cannot be worked is because the reversal changes the parity of the puzzle.  Flipping the 6 and the 9 to make them look like the 9 and 6 is the same as switching the 9 and the 6.  This changes the parity again, returning the puzzle to a solvable state.

Starting from original position figure 2 may be reached in 44 plays as follows: 14, 11, 12, 8, 7, 6, 10, 12, 8, 7, 4, 3, 6, 4,7, 14, 11, 15, 13, 9, 12, 8, 4, 10, 8, 4, 14, 11, 15, 13, 9, 12, 4, 8, 5, 4, 8, 9, 13, 14, 10, 6, 2 and 1.  (Moving the blank space to to top left rather than the bottom right, and then moving all the numbers to show them in the correct order leave every block in a different position - except for the 14 - and ends up changing the parity and makes the parity match the beginning state of thee puzzle).

Figure 3 may be reached in 39 plays: 14, 15, 10, 6, 7, 11, 15, 10, 13, 9, 5, 1, 2, 3, 4, 8, 12 15, 10, 13, , 5, 1, 2, 3, 4, 8, 12, 15, 14, 13, 9, 5, 1, 2, 3, 4, 8, and 12.  (Turning the puzzle a quarter of a turn changes the parity of the puzzle again, and allows a puzzlist to place all of the numbers in the correct order with the blank space back in the bottom right corner).

Figure 4: To produce a magic square adding 30 the following is best: 12, 8, 4, 3, 2, 6, 10, 9, 13, 15, 14, 12, 8, 4, 7, 10, 9, 14, 12, 8, 4, 7, 10, 9, 6, 2, 3, 10, 9, 6, 5, 1, 2, 3, 6, 5, 3, 2, 1, 13, 14, 3, 2, 1, 13, 14, 3, 12, 15, and 3.  (The solution is a magic square with the same parity as the starting position of the puzzle with the 14 and 15 swapped.  The the beginning arrangement and the final arrangement have the same parity this solution is possible.  Now all you have to do is figure out which magic square arrangement to use.  There are 7040 possible magic square arrangement and 3520 have the right parity to make this problem solvable.)

This problem had more meat on it than I thought.

I recommend downloading a copy of Sam Loyd's book from the internet.

I also recommend reading Jerry Slocum's book "The 15 Puzzle".

David