Solving the rectangle puzzle

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 $N\times N=(n^2+n+1)\times(n^2+n+1)$ 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.)

One Response to “Solving the rectangle puzzle”

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