Count On
Explorer
Squares on a Chessboard
You don't need to know anything about the game of chess for today's Mathematical Morsel.

At first, it looks like the answer is 64. However, in this case, I want ALL the squares with edges along the lines of the chessboard. So there are not only single squares (of which there are 64) but also 2by2 sized squares, and 3by3 sized ones, all the way up to the 8by8 square board itself. So:
In mathematics, a useful way to tackle such a problem is to take a smaller case as an example. So, suppose on a distant planet, chess was played on a board which was 2by2.
Also, to help our investigations let's organize our search.
What sizes of square are possible on this small 2by2 board?
Single squares and 2by2 squares only!
Let's put this in a table and you can write in your answers:



Have you spotted the pattern in the results? [Hint: we are counting SQUARES.]
Now can you tell me how many squares there are in total in the 8by8 chessboard if all the squares have sides along the rows or columns on the board?

Reference
John Mason's Thinking Mathematically, published by AddisonWesley, 1985, ISBN 0201102382 is a good book for further investigations of this type, but needs some familiarity with schoollevel algebra.
Patterns