Squares

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Squares

• By Gloria Gonzalez
• And Ieni Vargas

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• 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.

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We can summarize the results as follows

• Sizes of squares Numbers of squares
• 1 x 1 82
  64
• 2 x 2 72 49
• 2 x 2 72 49
• 4 x 4 52 25
• 5 x 5 42 16
• 6 x 6 32 9
• 7 x 7 22 4
• 8 x 8 12 1
• Total
  204

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• There is a formula for the sum of squares of the
  integers 12 22 32 ... n2
• N (n1) (2n1)
  
• 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
  362 1296.

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Tanks for you time

• Gloria Gonzalez
• and Ieni Vargas