# How many squares are there?: visual puzzle | SwaVid

A standard 8×8 chessboard contains how many squares in total, not just the 64 smallest ones? See the complete step-by-step solution.

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# How many squares are there?

## 204 squares.

### Count one size at a time

### Finish the list

### Add the square numbers

SwaVid visual puzzle

A standard 8×8 chessboard contains how many squares in total, not just the 64 smallest ones?

Detailed solution

Count squares of every size from 1×1 up to 8×8 and add them.

A k×k square can start in (9 − k) positions across and (9 − k) down, so there are (9 − k)² of them.

The larger sizes have fewer positions.

The total is the sum of the first eight square numbers.

Answering 64 counts only the smallest squares. Answering 65 adds the board itself but forgets every size in between.

- 1 Count one size at a time A k×k square can start in (9 − k) positions across and (9 − k) down, so there are (9 − k)² of them. 1×1: 64, 2×2: 49, 3×3: 36, 4×4: 25
- 2 Finish the list The larger sizes have fewer positions. 5×5: 16, 6×6: 9, 7×7: 4, 8×8: 1
- 3 Add the square numbers The total is the sum of the first eight square numbers. 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204

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