347.--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 n² squares contains ((n² + n)²)/4 rectangles, of which (2n³ + 3n² + n)/6 are squares and (3n^4 + 2n³ - 3n² - 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.