How many squares are there 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