Squares on a Chessboard Challenge

On Monday, we were asked to find the number of squares in a standard 8x8 chessboard.

How many squares are there?

Consider a standard 8x8 chessboard. The maximum dimension of a square found on the chessboard is 8x8 squares, and the minimum dimension is 1x1 squares. Beginning with maximum dimensions and working toward the smaller dimensions, we find the following:

For an 8x8 square, there is 1 possible square (the board itself)
For a 7x7 square, there are 4 possible squares (the 7x7 squares originating on each side of the board).

All remaining squares are identified by finding the number of squares of a given dimension which can fit in each row and in each column. For a 6x6 square, for instance, it can fit into the upper-left corner, in the same corner but moved one square to the right, then another moved one square right from the previous, hence filling up all "horizontal" possibilities. The same can be done vertically. The number of possible placements are 3 horizontally and 3 vertically, so 3x3 gives all 9 placements for a 6x6 square. Continuing this pattern, we find that the total number of squares in this board is:

 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 8*(8+1)*(2*8+1)/6 = 8*9*17/6 = 4*3*17*6/6 = 204

1) From a student's point of view, what did you do to solve this?

I decided that I didn't want to count the squares, and I hoped that there would be a faster way. I began with a list of all the dimensions of squares so that I didn't miss any. I also didn't want to have to count them all, so I started off with the largest dimension and worked my way down, since I realized that counting a smaller number of squares (that is, finding a square with larger dimensions) would be easier than counting/finding the squares making up a square with smaller dimensions.

2) From a teacher's point of view,

     a) Where do you think students get stuck?

I think that without a systematic approach, it would be very easy to make mistakes in this question. It's not a difficult task, necessarily, to find squares, but to find all the squares would require a precise algorithm. I feel that students may get stuck if they don't realize that they can find squares with dimensions greater than 1x1. I also think that some students may want to find every square just to be sure of their answer, or that they can simply draw the squares that they think are the only squares present. They may be overwhelmed by the potential combinations and their placements on the board.

     b) What would you do to assist them?

In order to help students complete this question, it would make sense to remind them that there are squares with various dimensions, not just 1x1 squares. I would also create a chart for them to fill out of each dimension of the squares, such as:

Dimensions	Number of Squares
8x8
7x7
6x6
5x5
4x4
3x3
2x2
1x1
TOTAL

Dimensions	Number of Squares
8x8
7x7
6x6
5x5
4x4
3x3
2x2
1x1
TOTAL

I feel that it is easier to approach the problem if you start with the squares of larger dimension first, and then work to find the squares with the next dimension. It's also much less frustrating to find larger quares, since identifying the pattern (if it happens) will likely occur sooner.

Dimensions	Number of Squares
8x8
7x7
6x6
5x5
4x4
3x3
2x2
1x1
TOTAL

    c) How would you extend this task for students ready for more of a challenge?

For students who were not feeling challenged, I liked Andrew's suggestion of asking them to find the number of rectangles in a chessboard. One of the questions from graph theory which I encountered involved finding whether or not a path (a manner of visiting every desired location and coming back to the starting point) can be drawn on a chessboard. It turns out that this depends on the board, which would be an investigative activity.