# Puzzle #7: Squares on a Chess Board | Amazon

### How many squares are on a chess board?

### If you thought the answer is 64, think again!

### How about all the squares that are formed by combining smaller squares on the chess board (2×2, 3×3, 4×4 squares and so on)?

### Puzzle 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

x

1^2 + 2^2 + 3^2 + . . . n ^2

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