Transcript Squares

Squares
By: Gloria Gonzalez
And Ieni Vargas

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.
We can summarize the results as follows:
Sizes of squares
1x1
2x2
2x2
4x4
5x5
6x6
7x7
8x8
Numbers of squares
8^2 = 64
7^2 = 49
7^2 = 49
5^2 = 25
4^2 = 16
3^2 = 9
2^2 = 4
1^2 = 1
Total = 204
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.
Tanks for you time
Gloria Gonzalez
and Ieni Vargas