Enigma 43: Scalenian squares
29 January 2012
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From New Scientist #1185, 13th December 1979 [link]
Scalenia is a country bounded by three straight frontiers – called A, B and C, of course – each of a different length but each an exact number of kilometres long. The curious thing is that the length of A+B+C, and the lengths of A+B−C, A+C−B and B+C−A are all precisely square numbers of kilometres.
If Scalenia has the smallest perimeter consistent with this curious fact, what is the length of each of its frontiers?