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Squares on a Chessboard

You don't need to know anything about the game of chess for today's Mathematical Morsel.

How many squares are there on a chessboard?

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 2-by-2 sized squares, and 3-by-3 sized ones, all the way up to the 8-by-8 square board itself. So:

How many squares are there on a chessboard?

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 2-by-2.

Also, to help our investigations let's organize our search.

What sizes of square are possible on this small 2-by-2 board?

Single squares and 2-by-2 squares only!

Let's put this in a table and you can write in your answers:

Square's sides:12TOTAL
How many?
Now let's try on a 3-by-3 chessboard. This time there are three sizes of squares. Single squares (1-by-1), 2-by-2 and 3-by-3 (the whole board).
Square's sides:123TOTAL
How many?
How about trying another small example:
Square's sides:1234TOTAL
How many?

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 8-by-8 chessboard if all the squares have sides along the rows or columns on the board?

Square's sides: 1234 5678TOTAL
How many?


John Mason's Thinking Mathematically, published by Addison-Wesley, 1985, ISBN 0-201-10238-2 is a good book for further investigations of this type, but needs some familiarity with school-level algebra.

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