What's the smallest army that can form into two square blocks 12 different ways?
A sultan wants an army that can be formed into two perfect squares — one large block and one small block standing together — in exactly twelve different ways, with every soldier used each time. What is the smallest possible size of that army?
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160,225 men. The number factors as 5 × 5 × 13 × 17 × 29, and each of the primes 5, 13, 17 and 29 can be written as a sum of two squares; combining them this way yields exactly twelve distinct pairs of squares that sum to 160,225 — such as 400²+15² and 265²+300², among ten others. Four distinct such prime factors would normally give sixteen ways, but because the 5 is repeated, two of those sixteen collapse together, leaving exactly twelve.
— Henry Dudeney, Amusements in Mathematics — Thomas Nelson & Sons, 1917, Puzzle No. 136
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