## Solving the rectangle puzzle

**G**iven the wrong solution provided in Le Monde and comments from readers, I went to look a bit further on the Web for generic solutions to the rectangle problem. The most satisfactory version I have found so far is Mendelsohn’s in * Mathematics Magazine*, which gives as the maximal number

for a grid. His theorem is based on the theory of projective planes and must be such that a projective plane of order exists, which seems equivalent to impose that is a prime number. The following graph plots the pairs when along with the known solutions, the fit being perfect for the values of of Mendelsohn’s form (i.e., 3, 7, 13).

**U**nfortunately, the formula does not extend to other values of , despite Menselsohn’s comment that *using for the positive root of the equation and then replacing by nearby integers* (in the maximal number) should work. (The first occurrence I found of a solution for a square-free set did not provide a generic solution, but only algorithmic directions. While it is restricted to squares. the link with fractal theory is nonetheless interesting.)

April 5, 2010 at 12:12 am

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