# Checkerboard

• Apr 6th 2009, 06:28 AM
phillyfan09
Checkerboard
How many squares are there on a checkerboard?
• Apr 6th 2009, 08:32 AM
stapel
Quote:

Originally Posted by phillyfan09
How many squares are there on a checkerboard?

• Apr 6th 2009, 10:46 AM
Squares on a Checkerboard
Hello phillyfan09
Quote:

Originally Posted by phillyfan09
How many squares are there on a checkerboard?

Of course, there are $8^2=64$ 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 $(8\times 8)$, of course, and there will be some number between $1$ and $64$ 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: $(7 \times 7)$. You'll see that you can find $4$ of these.

Then look at a size $(6 \times 6)$ 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 $3$; and a final row of $3$ along the bottom of the board. That's $3^2 = 9$ altogether.

Can you see a pattern emerging? We have

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

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

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

...

$8^2 = 64\, (1\times 1)$ 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 $n$ square numbers is

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