Chess by the numbers
How many squares are on a chessboard?
A chessboard has 64 squares: eight rows of eight, 32 light and 32 dark. But if you count every square you can draw on it, from a single square up to the whole board, there are 204, and there are 1,296 rectangles.
A square with sides of k fits on the board in 9 − k places along a row and 9 − k places along a column, so there are (9 − k)² of them. There are 36 squares of size 3 × 3, like the one from a1 to c3 here, and just one of size 8 × 8: the whole board.
Squares of every size
Adding up all eight sizes gives the sum of the first eight square numbers, 1 + 4 + 9 + … + 64:
| Size | How many |
|---|---|
| 1 × 1 | 64 |
| 2 × 2 | 49 |
| 3 × 3 | 36 |
| 4 × 4 | 25 |
| 5 × 5 | 16 |
| 6 × 6 | 9 |
| 7 × 7 | 4 |
| 8 × 8 | 1 |
| Total | 204 |
And rectangles?
A rectangle on the board is fixed by choosing two of the nine horizontal lines and two of the nine vertical lines. There are 36 ways to choose two lines out of nine, so there are 36 × 36 = 1,296 rectangles. 204 of them are squares and 1,092 are not.
Light and dark squares
Half the squares are light and half are dark, 32 of each. The board is placed so that each player has a light square in the right-hand corner, h1 for White and a8 for Black, and each queen starts on a square of her own colour: the white queen on d1, a light square, and the black queen on d8, a dark one. The columns are named a to h and the rows 1 to 8, so every square has a name, such as e4.
The wheat and chessboard legend
In an old story, the inventor of chess asks for one grain of wheat on the first square, two on the second, four on the third, doubling on every square. The last square alone would hold 9,223,372,036,854,775,808 grains, and the whole board 18,446,744,073,709,551,615: about 1.2 trillion tonnes of wheat, over 1,400 times what the whole world harvests in a year.
Questions people ask
How many squares are there apart from the 64 small ones?
140: the 204 squares of all sizes minus the 64 single squares.
How many squares can a queen reach?
On an empty board, 27 from one of the four centre squares such as d4, and 21 from a corner. On a real board the other pieces block most of them; how many moves a whole position can have is on the most legal moves in a chess position.
Why does the count matter for chess?
Not much, but the 64 squares do: they keep the number of positions finite, and still vast. With 32 pieces on 64 squares there are about 4.8 × 1044 legal positions; see how many chess positions there are.