1. By: Gloria Gonzalez And Ieni Vargas
Squares
By: Gloria Gonzalez
And Ieni Vargas

2. Consider the left hand vertical edge of a square of size 1 x 1
Consider the left hand vertical edge of a square of size 1 x 1. This edge can be in any one of 8 positions. Similarly, the top edge Can occupy any one of 8 positions for a 1 x 1 square. So the total Number of 1 x 1 squares = 8 x 8 = 64.
For a 2 x 2 square the left hand edge can occupy 7 positions and the Top edge 7 positions, giving 7 x 7 = 49 squares of size 2 x 2.
Continuing in this way we get squares of size 3 x 3, 4 x 4 and so on.

3. We can summarize the results as follows:
Sizes of squares Numbers of squares
1 x ^2 = 64
2 x ^2 = 49
4 x ^2 = 25
5 x ^2 = 16
6 x ^2 = 9
7 x ^2 = 4
8 x ^2 = 1
Total = 204

4. 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.
As an extension to this problem, you might want to calculate the Number of rectangles that can be drawn on a chessboard.
There are 9 vertical lines and 9 horizontal lines. To form a rectangle You must choose 2 of the 9 vertical lines, and 2 of the 9 horizontal Lines. Each of these can be done in 9C2 ways = 36 ways. So the number Of rectangles is given by 36^2 = 1296.

5. Gloria Gonzalez and Ieni Vargas
Tanks for you time
Gloria Gonzalez
and Ieni Vargas