Chess Squares

How many squares are on a chess board?

Solution:
A 1×1 square can be placed on the chess board in 8 horizontal and 8 vertical positions, thus making a total of 8 x 8 = 64 squares. Let’s consider a 2×2 square. There are 7 horizontal positions and 7 vertical positions in which a 2×2 square can be placed. Why? Because picking 2 adjacent squares from a total of 8 squares on a side can only be done in 7 ways. So we have 7 x 7 = 49 2×2 squares. Similarly, for the 3×3 squares, we have 6 x 6 = 36 possible squares. So here’s a break down.

1×1     8 x 8 = 64 squares
2×2     7 x 7 =  49 squares
3×3     6 x 6 = 36 squares
4×4     5 x 5 = 25 squares
5×5     4 x 4 = 16 squares
6×6     3 x 3 = 9 squares
7×7     2 x 2 = 4 squares

8×8     1×1 = 1 square

Total = 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squares