17 The Monk’s Puzzle
The Monk might have placed dogs in the kennels in two thousand nine hundred and twenty-six different ways, so that there should be ten dogs on every side. The number of dogs might vary from twenty to forty, and as long as the Monk kept his animals within these limits the thing was always possible.
The general solution to this puzzle is difficult. I find that for n dogs on every side of the square, the number of different ways is (n4 + 10n3 + 38n2 + 62n + 33) / 48, where n is odd, and ((n4 + 10n3 + 38n2 + 68n) / 48) + 1, where n is even, if we count only those arrangements that are fundamentally different. But if we count all reversals and reflections as different, as the Monk himself did, then n dogs (odd or even) may be placed in ((n4 + 6n3 + 14n2 + 15n) / 6) + 1 ways. In order that there may be n dogs on every side, the number must not be less than 2n nor greater than 4n, but it may be any number within these limits.
An extension of the principle involved in this puzzle is given in No. 42, “The Riddle of the Pilgrims.” See also “The Eight Villas” and “A Dormitory Puzzle” in A. in M.