# Hint #27: Number of Squares on a Chessboard

27. How many squares are there on a chessboard or checkerboard?? (the answer is not 64)

Can you extend your technique to calculate the number of rectangles on a chessboard?

Can you extend your technique to calculate the number of rectangles on a chessboard?

Lets first establish what we mean by a square to be clear why the answer is not 64. Below are some examples of valid squares.

The method I used was to think about a square of a specific size say a 2x2. And consider the number of locations that square can go. There is a simple relationship between the size of a square and the number of locations it can be placed.

Moving on to rectangles the same technique can still be used. Although it's a lot more fiddly. A better solution involves counting the number of available diagonals.

Where next?

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