How Many Squares Are on a Chessboard?
The obvious answer is 64. But if you count squares of every possible size, the real answer is 204. This is a classic maths puzzle with a surprisingly elegant solution.
The breakdown
A standard 8×8 chessboard contains squares of 8 different sizes:
| Size | Count | How? |
|---|---|---|
| 1×1 | 64 | 8 × 8 |
| 2×2 | 49 | 7 × 7 |
| 3×3 | 36 | 6 × 6 |
| 4×4 | 25 | 5 × 5 |
| 5×5 | 16 | 4 × 4 |
| 6×6 | 9 | 3 × 3 |
| 7×7 | 4 | 2 × 2 |
| 8×8 | 1 | 1 × 1 |
Total: 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204
The formula
For an n×n chessboard, the total number of squares is the sum of squares from 1 to n: n(n+1)(2n+1)/6. For n=8: 8 × 9 × 17 / 6 = 204.
Bonus: rectangles
The total number of rectangles (including squares) on a chessboard is 1,296. The formula is (n(n+1)/2)² = (8×9/2)² = 36² = 1,296.