Chess Squares

Question:

How many squares are on a chess board?

.

.

C

A

P

T

A

I

N

I

N

T

E

R

V

I

E

W

.

.

Solution: If you thought the answer was 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)?

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 breakdown.

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

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