COUNTING THE RECTANGLES.
(
The Guarded Chessboard)
Can you say correctly just how many squares and other rectangles the
chessboard contains? In other words, in how great a number of different
ways is it possible to indicate a square or other rectangle enclosed by
lines that separate the squares of the board?
Answer:
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 squared squares
contains ((n squared + n) squared)/4 rectangles, of which (2n cubed + 3n squared + n)/6 are
squares and (3n^4 + 2n cubed - 3n squared - 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.
