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March 25, 2017

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how many squares are in an 8x8 checkerboard? (the answer is not 64 because that's too low.)


Thank you for using the Jiskha Homework Help Forum. Unfortunately, you don't state the size of each square. Hopefully the following will answer your question!

Continuing in this way we get squares of size 3 x 3, 4 x 4 and so on.

We can summarize the results as follows:

Size Of square Number of squares
--------------- -----------------
1 x 1 8^2 = 64
2 x 2 7^2 = 49
3 x 3 6^2 = 36
4 x 4 5^2 = 25
5 x 5 4^2 = 16
6 x 6 3^2 = 9
7 x 7 2^2 = 4
8 x 8 1^2 = 1
---------------
Total = 204

There is a formula for the sum of squares of the integers
1^2 + 2^2 + 3^2 + ... + n^2

n(n+1)(2n+1)
Sum = ------------
6

In our case, with n = 8, this formula gives 8 x 9 x 17/6 = 204.

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