Name: Tom

Who is asking: Teacher
Level: Secondary

Question:
It was once claimed that there are 204 squares on an ordinary chessboard (8sq. x 8sq.) Can you justify this claim? "PLEASE" include pictures.

How many rectangles are there on an ordinary chessboard? (8sq. x 8sq.) "PLEASE" include pictures.

The 204 is correct. What is best to do is to think of the chessboard as being drawn in the plane by 9 vertical and 9 horizontal equally spaced line segments so that the lines meet at (0,0), (0,1), ... (8,8) (that's 9x9 points altogether).

To count the number of squares think of placing a 1x1 square on the grid to cover one of the unit squares. Where can the upper right hand corner of this square be? At (1,1), (1,2), ... , (8,8) giving 8x8 choices. How about 2x2 squares? The upper right hand corner must appear at (2,2), (2,3), ... , (8,8) giving 7x7 choices. If you continue this until you place an 8x8 square you will end up with just 1x1 choice for the upper right hand corner. Thus the total number of (integer sided) squares that exist on the board is 8x8 + 7x7 + ... + 2x2 + 1x1 = 204.

For an nxn board you get nxn + (n-1)x(n-1) + ... + 1x1 = n(n+1)(2n+1)/2 in all.

To find how many rectangles you need the same idea but this time you need to look at

1x1 rectangles, 1x2's, ..., 1x8's and then
2x1 rectangles, 2x2's , ... 2x8's and then
.
.
.
8x1 rectangles, 8x2's , ..., 8x8's

and how they can be placed. I think you might get 1296 (which is  ₉C₂ x  ₉C₂).

What would you get for general n?

Penny