how many squares are in an 8x8 checkerboard? (the answer is not 64 because that's too low.)
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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.
18
But wouldn't the answer be 64 though???? 8x8=64???
To find the number of squares in an 8x8 checkerboard, we can calculate the sum of squares of the side lengths of all possible squares.
In this case, we start with squares of size 1x1, which are simply the individual cells of the checkerboard. Since there are 8x8 = 64 cells, there are 64 squares of size 1x1.
Next, we move on to squares of size 2x2. On an 8x8 checkerboard, there are only 7x7 = 49 possible 2x2 squares.
Continuing this process, we can find the number of squares of each size until we reach the largest square of size 8x8, which is the entire checkerboard itself.
By summing up the number of squares of each size, we get a total of 204 squares in an 8x8 checkerboard.
Alternatively, we can use the formula for the sum of squares of the first n natural numbers:
Sum = n(n+1)(2n+1)/6
In this case, n = 8, so the formula becomes:
Sum = 8(8+1)(2(8)+1)/6 = 204
Therefore, there are 204 squares in an 8x8 checkerboard.