**From New Scientist #2366, 26th October 2002** [link]

You are probably familiar with the puzzle consisting of 15 sliding tiles as shown.

By sliding the tiles around you can make various arrangements and then read off each row as one long number. For example, the first row might consist of 3, 1, 8 and 13, which could be read as the palindromic number 31813.

I formed one arrangement recently in which the numbers formed by three of the four rows were palindromic and the number formed by the remaining row was exactly six times a palindrome.

What was that non-palindromic row?

[enigma1210]

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As discussed in

Enigma 1444, an arrangement of the 15-puzzle is only possible if the parity (i.e. odd or even) of the number of “inversions” is the opposite of the parity of the row number of the empty space.This Python program runs in 367ms.

Solution:The non-palindromic row is: 7, 9, 8, 6.When read as a number 7986 = 6 × 1331.

There are many ways to arrange the puzzle. My program finds 1296 solutions, but in all cases the non-palindromic row is (7, 9, 8, 6). (Thanks to Brian Gladman for pointing out a flaw in my original program that cause it to miss some of the solutions).

Here is one arrangement which I made using the

sliding-puzzle.pyprogram I wrote forEnigma 1444.See also

Enigma 106,Enigma 1275,Enigma 1444.