Question

Total number of squares and rectangles in a chess set.

Answer 1

Consider the lefthand vertical edge of a square of size 1 x 1. This edge can be in any one of 8 positions. Similarly, the top edge can occupy any one of 8 positions for a 1 x 1 square. So the total number of 1 x 1 squares = 8 x 8 = 64. For a 2 x 2 square the lefthand edge can occupy 7 positions and the top edge 7 positions, giving 7 x 7 = 49 squares of size 2 x 2. Continuing in this way we get squares of size 3 x 3, 4 x 4 and so on. We can summarize the results as follows: Size Of square Number of squares --------------- ----------------- 1 x 1 8^2 = 64 2 x 2 7^2 = 49 3 x 3 6^2 = 36 4 x 4 5^2 = 25 5 x 5 4^2 = 16 6 x 6 3^2 = 9 7 x 7 2^2 = 4 8 x 8 1^2 = 1 --------------- Total = 204 There is a formula for the sum of squares of the integers 1^2 + 2^2 + 3^2 + ... + n^2 n(n+1)(2n+1) Sum = ------------ 6 In our case, with n = 8, this formula gives 8 x 9 x 17/6 = 204 number of rectangles on a chessboard. The answer is "Many, in fact 1296." There are 64 one-by-one squares, 49 two-by-two squares, ... (8-n)^2 n-by-n squares, ... 1 eight-by-eight square; 2 x (7x8) one-by-two rectangles, 2 x (6x8) one-by-three rectangles, ... 2 x (1x8) one-by-eight rectangles; 2 x (6x7) two-by-three rectangles, ... 2 x (1x7) two-by-eight rectangles; ...; 2 x (1x2) seven-by-eight rectangles. This can all be simplified to find the sum total as the sum of the cubes of integers 1 to 8, which is 8^2 x 9^2 / 4 or 36^2 = 1296