Enigma 1444: Backslide
6 April 2013
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From New Scientist #2605, 26th May 2007
You may be familiar with the 15-tile puzzle that starts as shown, with one gap, after which you can slide into the gap any one of the tiles adjacent to it.
Continuing in this way you can rearrange the tiles into all sorts of orders.
My nephew is so good at his 15-tile puzzle that I have given him a larger version. It is again a rectangle, with more than two tiles along each side, the tiles are numbered 1, 2, 3 …, and it starts with them in the natural order and with the gap in the bottom right-hand corner.
My nephew has mastered the new version. For example, in less than quarter of an hour he can rearrange the tiles into their reverse order, ending up with … 3, 2, 1 and gap in the bottom row. He has also looked into the properties of the numbers on the tiles and has noted, for example, that their sum is divisible neither by 2 nor by 3.
How many tiles are there in his new version of the puzzle?