Answer:
squares: 204
rectangle: 1,296
Step-by-step explanation:
Board Size: 8 x 8
The total number of squares is thus:
[tex]1^{2} + 2^{2} + ... + n^{2}[/tex]
sum = [tex]\frac{n(n + 1)(2n + 1)}{6}[/tex]
For a chess board, with n = 8,
sum = [tex]\frac{(8)((8) + 1)(2(8) + 1)}{6}[/tex]
sum = 204
the number of rectangles on a chess board,
to make a rectangle you need to pick any two of the vertical lines, and any two of the horizontal lines. There are 36 distinct pairs and the same number of vertical pairs, so the answer is 36 × 36 = 1,296.
