Quick, how many squares are in an 8x8 chessboard?!

Did you answer 64? If so, it's okay. That's technically correct, yet wrong at the same time. If you count including the suares made by squares, however, the number is much bigger.

Let's say this is a chessboard:

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We can see immediately that there are 64 squares, and more. So how do we calculate for this? The answer is quite simple: 204 or 64+49+36+25+16+9+4+1 squares.

How about a 6x6 square? Well, it's equal to 36+25+16+9+4+1 or 91 squares. Do you see a pattern?

Basically, the sum of the squares on an x-by-x grid is equal to:

\(\sum_{i=1}^{x}i^2\)

Wherein x is the side of the grid. For example: we have a 12x12 grid, so the number of squares is:

\(\sum_{i=1}^{12}i^2=12^2+11^2+10^2...+2^2+1\)

\(=144+121+100+81+64+49+36+25+16+9+4+1\)

\(=650\)

A 12x12 grid has 650 squares.

Mathhemathh
Aug 17, 2017