The obvious answer: 64

A chessboard is an 8 × 8 grid, so it has 64 small squares — 32 light and 32 dark — and that’s the surface you actually play chess (and draughts) on. The columns are called files (labelled a–h) and the rows are ranks (1–8), and the board is set up “light on right”: each player has a light square in their nearest right-hand corner.

If that were the whole story, this wouldn’t be a famous puzzle. The catch is in the word square — because a 2×2 block of cells is also a square, and so is a 3×3 block, all the way up.

Prefer a picture to a formula? Jump to a completely different way of seeing it ↓

The puzzle answer: 204

Count squares of every size and the total climbs to 204. The trick is to count each size separately. A 1×1 square can sit in 8 positions across and 8 down — that’s 8×8 = 64. A 2×2 square only fits in 7 positions each way (it would hang off the edge otherwise), giving 7×7 = 49. Each larger size loses one position in each direction:

So the answer is 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204.

The real lesson: why it works

Here’s the idea worth taking away. To place an s×s square on an 8×8 board, you just have to choose where its top-left corner goes — and it has (8 − s + 1) choices across and the same number down. So there are (9 − s)² squares of that size. Add those up for s = 1 to 8 and you get 204.

This is a named result: the total is the sum of the first eight square numbers, and it has a tidy formula that works for any n×n board:

Pop in n = 8 and you get 8 × 9 × 17 ⁄ 6 = 204. Try n = 4 and it gives 30; n = 10 gives 385. The interactive board above is really just this formula in disguise — which is why the same logic works on a sudoku grid, a window of panes, or any square lattice you like.

Bonus round: how many rectangles?

If you loosen the rules and count every rectangle (not just squares), the answer leaps to 1,296 — and the reasoning is even neater. A chessboard has 9 lines running each way. To draw any rectangle, you simply pick 2 of the 9 vertical lines and 2 of the 9 horizontal lines. That’s “9 choose 2” each way: 36 × 36 = 1,296.

Since 204 of those rectangles happen to be squares, the remaining 1,092 are the genuinely oblong ones. Two famous chessboard numbers — 204 and 1,296 — and both fall out of one small counting idea.

The sliding frame

Forget the word “square” for a moment. Imagine you cut a small square hole in a piece of card, then slide that card around over the board, peeking through the hole. Every spot where the hole sits neatly on the board — without poking off an edge — counts once.

A tiny hole the size of one cell can rest almost anywhere: eight stops across, eight stops down — sixty-four resting places. Make the hole bigger, two cells wide, and it turns clumsy; push it too far and it hangs over the side, so now it only has seven stops each way. Bigger still, fewer places to sit. The largest hole of all — the size of the whole board — can only rest in one single spot, dead centre on top.

So you were never really counting squares. You were counting parking spots for a frame that keeps getting harder to park. Tot up the spots for every size of frame — 64, then 49, then 36, on down to 1 — and you arrive at 204. Same answer, no formula in sight: just a frame, a board, and a great many places to set it down.

Sources

• Standard combinatorics — squares on an n×n grid = 1² + 2² + … + n² = n(n+1)(2n+1)/6 (204 for n = 8).
• Rectangles on an n×n grid = (n+1 choose 2)² — 36² = 1,296 on a chessboard, of which 204 are squares.

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