How many squares are there on a checkerboard?

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- Apr 6th 2009, 05:28 AMphillyfan09Checkerboard
How many squares are there on a checkerboard?

- Apr 6th 2009, 07:32 AMstapel
- Apr 6th 2009, 09:46 AMGrandadSquares on a Checkerboard
Hello phillyfan09Of course, there are $\displaystyle 8^2=64$ little squares, but we suspect that this isn't the answer that we're supposed to come up with!

There's one very big square $\displaystyle (8\times 8)$, of course, and there will be some number between $\displaystyle 1$ and $\displaystyle 64$ squares with a size between these two. So how do we find out how many there are in an organised way? Well, look first at the next size square down: $\displaystyle (7 \times 7)$. You'll see that you can find $\displaystyle 4$ of these.

Then look at a size $\displaystyle (6 \times 6)$ square. There's one up in the top left-hand corner, and two more along the top row of the board. If we move down a square there are another $\displaystyle 3$; and a final row of $\displaystyle 3$ along the bottom of the board. That's $\displaystyle 3^2 = 9$ altogether.

Can you see a pattern emerging? We have

$\displaystyle 1^2 = 1\, (8\times 8)$ square.

$\displaystyle 2^2 = 4\, (7\times 7)$ squares

$\displaystyle 3^2 = 9\, (6\times 6)$ squares

...

$\displaystyle 8^2 = 64\, (1\times 1)$ squares

So all you have to do is work out the numbers in between and add them together to get your answer. If you want to use a formula, you might like to know that the sum of the first $\displaystyle n$ square numbers is

$\displaystyle \tfrac{1}{6}n(n+1)(2n+1)$

Grandad