(related to Problem: Counting The Rectangles)

There are $1,296$ different rectangles in all, $204$ of which are squares, counting the square board itself as one, and $1,092$ rectangles that are not squares. The general formula is that a board of $n^2$ squares contains $$\frac{(n^2+n)^2 }{4}$$ rectangles, of which $$\frac{2n^3 + 3n^2 + n}6$$ are squares and $$\frac{3n^4 + 2n^3 - 3n^2 - 2n}{12}$$ are rectangles that are not squares. It is curious and interesting that the total number of rectangles is always the square of the triangular number whose side is $n.$

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Project Gutenberg

  1. Dudeney, H. E.: "Amusements in Mathematics", The Authors' Club, 1917

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