How many squares are on a chessboard?

Question

Everyone says a chessboard has 64 squares, but that only counts the little $1\times 1$ cells. There are also $2\times 2$ squares, $3\times 3$ squares, all the way up to the single $8\times 8$ square that is the whole board.

How many squares of all sizes are there on an 8×8 chessboard in total?

Accepted Answer

The total is 204. Count each size separately. A k 	imes k square can be placed so its top-left corner sits in various spots. On an 8 	imes 8 grid, a k 	imes k square fits in (9 - k) positions horizontally and (9 - k) vertically, so there are (9-k)^2 of them. Size Count ------------- 1	imes1 8^2 = 64 2	imes2 7^2 = 49 3	imes3 6^2 = 36 4	imes4 5^2 = 25 5	imes5 4^2 = 16 6	imes6 3^2 = 9 7	imes7 2^2 = 4 8	imes8 1^2 = 1 Add them up: 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = \mathbf{204}. That's the sum of the first 8 perfect squares, a neat consequence of the counting pattern. Takeaway. "How m…

How many squares are on a chessboard?

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