Make Your Move, Kid!

This puzzle is a slight modification of Sam Loyd's Fifteen. The numbers have been replaced with letters that spell a meaningful sentence. As is well known, the solvability of a particular configuration of Fifteen depends on the parity of the number of inversions in the sequence of counters. So that if, for example, we swap the last two counters the parity of the total number of inversions changes which makes the configuration unsolvable.

In the puzzle below, pressing Reset swaps the two last counters. The task is to slide the counters (following the rules of Fifteen) so that at the end the same message is spelled: "Make Your Move, Kid!" Is this possible?

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  1. D.L.Silverman, Your Move, McGraw-Hill, 1971

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Perhaps surprisingly, the modified puzzle is solvable. The key to understanding the reason lies in the observation that now there are two identical counters carrying the letter "O". Swapping these two adds an inversion to the total number. Performing two inversions does not change the parity of the total. Note, however, that after you succeed in spelling the correct message, the total number of inversions will remain odd because the two "O" counters will have to be swapped. With two identical counters we can solve the puzzle without contradicting the theory of Fifteen. Make Your Move, Kid!

(Be cautious, though. There are also two M's. If you swap those two, you may end up with an odd number of inversions.)

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Copyright © 1996-2017 Alexander Bogomolny


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