Solving the rectangle puzzle

March 15, 2010
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Given 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

k^star = (n+1)(n^2+n+1)

for a Ntimes N=(n^2+n+1)times(n^2+n+1) grid. His theorem is based on the theory of projective planes and n must be such that a projective plane of order n exists, which seems equivalent to impose that n is a prime number. The following graph plots the pairs (N,k^star) when N=1,ldots,13 along with the known solutions, the fit being perfect for the values of N of Mendelsohn’s form (i.e., 3, 7, 13).

Unfortunately, the formula does not extend to other values of N, despite Menselsohn’s comment that using for n the positive root of the equation x^2+x+1=N and then replacing n 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.)

Filed under: Kids, R Tagged: fractal, Le Monde, mathematical puzzle, Mendelsohn, primes, projective planes

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