How Many Squares Are on a Chessboard? (It Is Not 64)

Short answer: a chessboard has 204 squares, not 64. The 64 small squares are only the $1\times 1$ ones; squares of every size from $1\times 1$ up to the whole $8\times 8$ board count too.

Contents5 sections

Counting by size

A $k\times k$ square fits on an $8\times 8$ board in $9-k$ positions across and $9-k$ positions down, so there are $(9-k)^2$ of them:

Size	Positions	Squares
1 × 1	8 × 8	64
2 × 2	7 × 7	49
3 × 3	6 × 6	36
4 × 4	5 × 5	25
5 × 5	4 × 4	16
6 × 6	3 × 3	9
7 × 7	2 × 2	4
8 × 8	1 × 1	1
Total		204

The total is the sum of the first 8 square numbers, and there is a formula for it:

$$1^2+2^2+\dots+n^2=\frac{n(n+1)(2n+1)}{6},$$

and for $n=8$:

$$\frac{8\cdot 9\cdot 17}{6}=204.$$

And how many rectangles?

A rectangle on the board is fixed by choosing 2 of the 9 vertical grid lines and 2 of the 9 horizontal grid lines:

$$\binom{9}{2}\times\binom{9}{2}=36\times 36=1296.$$

So the board has 1296 rectangles, of which 204 are squares.

Try these

Puzzle 1

How many squares are in a $5\times 5$ grid?

Show Answer

$25+16+9+4+1=\mathbf{55}$, or $\frac{5\cdot 6\cdot 11}{6}=55$.

Puzzle 2

How many squares are in a $3\times 5$ rectangle of unit squares?

Show Answer

$1\times 1$: $3\cdot 5=15$. $2\times 2$: $2\cdot 4=8$. $3\times 3$: $1\cdot 3=3$. Total $\mathbf{26}$.

Puzzle 3

How many rectangles that are not squares are on a chessboard?

Show Answer

$1296-204=\mathbf{1092}$.

Related

• Stars and bars practice problems
• Circular permutations practice problems
• The square in a semicircle puzzle

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