How many squares are there on a checkerboard?
There's one very big square , of course, and there will be some number between and 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: . You'll see that you can find of these.
Then look at a size 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 ; and a final row of along the bottom of the board. That's altogether.
Can you see a pattern emerging? We have
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 square numbers is