Enigma 1383: Unique line-up
14 October 2013
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From New Scientist #2543, 18th March 2006
I have a set of cards numbered consecutively from 1 up to and including my nephew’s age. I gave him the cards and asked him to arrange them in a row so that the sum of any adjacent pair was a perfect square. He managed it and reported that there were actually only two ways of doing it, namely the line-up which he had displayed and the complete reverse of that order.
I asked him if the same would have been true had he been one year younger and I had correspondingly given him one card fewer. He said that it would, and he produced the second line-up by simply removing the left hand card from his line-up. I asked him if the same would have been true if he had been two years younger and I had originally given him two fewer cards. He said that it would, and he produced the line-up by simply removing the right hand card from his line-up.
How old is my nephew, and what was the number on the middle card in the original line-up?