How many squares of any size are there on an 8x8 chessboard?

.1..2...3...4..5...6..7...8
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The question is asking how many different squares of differing sizes you can make out of a chessboard.
For example, you can make 64 different 1x1 size squares, as there are 64 different squares.
Alternatively, you can make 1 single 8x8 size squares, as that is as large as the chessboard is.

e.g. For a 5x5 size square, how many different ways can you position it?

.1..2...3...4..5...6..7...8
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[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 2
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[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 8

In this example for a 5x5 square, you can see that there are 4 positions it can take vertically and horizontally.
The current position, and 3 others vertically and horizontally.
So, there are a total of 4x4 ways to position a 5x5 square.

Now to find how many 2x2, 3x3, 4x4, 5x5, 6x6, and 7x7 squares with different positioning on the chessboard.

Extrapolating this...
Size ..... Possible Squares
1............64 (8x8)
2............49 (7x7)
3............36 (6x6)
4............25 (5x5)
5............16 (4x4)
6............9 (4x4)
7............4 (4x4)
8............1 (1x1)

Sum of which is 204.

You need squares of any size.

Imagine an 8 inches by 8 inches chessboard with 64 squares of 1 inch by 1 inch. Now does the chessboard have only 64 squares? It has 64 squares of size 1x1 but what about squares of size 2x2? or 3x3 etc?
If you take 4 small squares together (like the orange square at the top left corner), you get squares of size 2x2. Move one step to the right and you get another such square and keep going till you get the 7th square at the end. Similarly if you go down one step, you get another square till you get 7 squares. So overall, you will get 7 * 7 = 49 squares of size 2x2.



Similarly you will get squares of size 3x3, size 4x4 and so on till you get one big square of size 8x8.

Total number of squares = 8*8 + 7*7 + 6*6 + 5*5 ... 1*1 = 8^2 + 7^2 + 6^2 + ... + 2^2 + 1^2

If you know the formula, use it:

1^2 + 2^2 + 3^2 + ... + n^2 = n(n+1)(2n+1)/6
When n = 8, Sum = 8*9*17/6 = 204

Else you can manually calculate.

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