The answer is 204 squares.
This is the hardest count we’ve ever posted, because a chessboard hides squares inside squares. The board is a perfect 8×8 grid — and the trick is that a square doesn’t have to be one little cell. Any square whose corners sit on grid points counts, of EVERY size. So count them size by size:
- 1×1 squares: 8 × 8 = 64
- 2×2 squares: 7 × 7 = 49
- 3×3 squares: 6 × 6 = 36
- 4×4 squares: 5 × 5 = 25
- 5×5 squares: 4 × 4 = 16
- 6×6 squares: 3 × 3 = 9
- 7×7 squares: 2 × 2 = 4
- 8×8 squares: 1 × 1 = 1 (the whole board itself!)
Now add them up: 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204.
See the pattern that makes it provable? Every time the square grows by one, the count shrinks to the next square number down — because an n×n square can start in (9 − n) columns and (9 − n) rows. The two traps are forgetting the medium sizes (most people stop after 64 and maybe 49) and forgetting the biggest one of all: the whole board is a square too, and it’s worth exactly 1.
Did you land on 204, or did the board’s own outline escape your count? Be honest in the comments — and watch our page for tomorrow’s counting game.
