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How many Squares on a Chessboard?

By Mark Warner
Age Range: 7 to 11

Although this investigation seems quite simple, it requires a methodical approach if the correct answer is to be attained.

As the title suggests, the investigation involves children finding out how squares there are on a chessboard. You might think that there are only 64, but you would be wrong...

The diagram below shows that there are indeed 64 squares, you there are also some more...

Don't forget the one large square (shown in red here)...

And, also the 16 two-by-two squares shown below (although these aren't the only 2x2 squares!)...

There are many more different-sized squares on the chessboard.
The complete list of answers is shown below:

1, 8x8 square
4, 7x7 squares
9, 6x6 squares
16, 5x5 squares
25, 4x4 squares
36, 3x3 squares
49, 2x2 squares
64, 1x1 squares

Therefore, there are actually 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 squares on a chessboard! (in total 204).

A worksheet with a large chessboard which children can use to investigate this problem can be found here.

If the children manage to find all of them, ask them if they can see a pattern in the results (i.e. the square numbers in the table).

Find more challenges like this on our Problem Solving page.
2D Shapes

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