How many squares are there on a chessboard?

Simple answer. 64 right? This is the thought problem I gave my year 9 class for their end of Friday problem solving half hour.

On the face of it 64 is the right answer. there are 64 squares on the board

But what about the 2x2 squares and the 3 x 3 squares.... and so on? Apart from one girl who finished the task in around five minutes who's explanation was to point to the board and say " Can't you see them?" ( I think we have a genius there!) the class attacked it with gusto. One or two piled in to systematically counting them all ( We'd just finished a lesson on listing combinations systematically then finding probabilities so I blame myself) 

A more interesting line of thought was looking for patterns on the board and linking them with square numbers. There were some excellent visual examples of this from the class and this is my less than stellar example

And this is the breakthough for solving the problem. There is a link between the square numbers and the number of squares on a chess board. There are

1, 8x8 square
4, 7x7 squares
9, 6x6 squares
16, 5x5 squares
25, 4x4 squares
36, 3x3 squares
49, 2x2 squares
64, 1x1 squares

Therefor there are 204 squares on a chessboard.  

There is a formula for the sum of squares of the integers 1^2 + 2^2 + 3^2 + ...  + n^2

                   n(n+1)(2n+1)
         Sum  = ------------
                         6

In our case, with n = 8, this formula gives 8 x 9 x 17/6 = 204. Following the pattern we could find how many squares are on any subdivided square. For example, a 9x9 square would produce 285 squares

The interesting thing is that all the answers from the pupils came with the same error in arithmetic(211) I'm still trying to work out why this is the case. Any thoughts on this point will be gratefully received.