Subject: Re: Sliding puzzles
Date: Wed, 01 Jan 1997 14:42:43 -0500
From: Alex Bogomolny

Dear Vivienne:

I'd like to help you very much. However, the slider puzzles are not all the same. Given that your grandson is five years old, I assume you are talking of the Fifteen puzzle which is the only one I am aware of that has been implemented as a physical gadget. If your grandson plays a computer then the possibilities are too numerous for me to discuss in a short letter. Please be more specific.

As to the Fifteen, it's a simple matter to compose the first two rows just by sliding the counters into their positions. Sometimes, you'll have to shift the counters from their rightful position. For example, assume the first row is 1,2,3,10 while the second row is 12,5,8,4. The task is now to swap 4 and 10. Give room for 1 immediately beneath it. Slide 1,2,3,10 counterclockwise by one square. After which the upper right corner must be empty. Slide 4 into there and 8 to the right into the freed position. Now the empty square will be just below 10 so make good use of this opportunity and remove 10 from the first row. Slide 3,2,1 back. The first row will thus be finished. The second row is tackled in exactly same manner.

The worst thing that may happen with the last two rows is to get

9,10,11,12, and
14,15,13,_

Somehow you must manage to rotate 13 to the first position in the last row. Rotate the two rows clockwise 2 squares:

14,_,9,10
15,13,12,11

Slide 13 up and rotate the two rows back two squares counterclockwise. This must solve the puzzle.

I would guess it matters very little to your grandson that there exists a theory that specifies which starting configurations are solvable and which are not. However, make sure that the starting configuration is obtained backwards from the original 1,2,3... position by sliding counters randomly. For, otherwise, it's a 50/50 chance you'll get an unsolvable position.

Hope it's what you needed.

Sincerely,
Alexander Bogomolny

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