Squares

1. Squares By: Gloria Gonzalez And Ieni Vargas

2. 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 1 8^2 = 64 2 x 2 7^2 = 49 2 x 2 7^2 = 49 4 x 4 5^2 = 25 5 x 5 4^2 = 16 6 x 6 3^2 = 9 7 x 7 2^2 = 4 8 x 8 1^2 = 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. Tanks for you time Gloria Gonzalez and Ieni Vargas