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Math Help - Checkerboard

  1. #1
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    Exclamation Checkerboard

    How many squares are there on a checkerboard?
    Last edited by phillyfan09; April 6th 2009 at 07:58 AM.
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  2. #2
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    Quote Originally Posted by phillyfan09 View Post
    How many squares are there on a checkerboard?
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  3. #3
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    Squares on a Checkerboard

    Hello phillyfan09
    Quote Originally Posted by phillyfan09 View Post
    How many squares are there on a checkerboard?
    Of course, there are 8^2=64 little squares, but we suspect that this isn't the answer that we're supposed to come up with!

    There's one very big square (8\times 8), of course, and there will be some number between 1 and 64 squares with a size between these two. So how do we find out how many there are in an organised way? Well, look first at the next size square down: (7 \times 7). You'll see that you can find 4 of these.

    Then look at a size (6 \times 6) square. There's one up in the top left-hand corner, and two more along the top row of the board. If we move down a square there are another 3; and a final row of 3 along the bottom of the board. That's 3^2 = 9 altogether.

    Can you see a pattern emerging? We have

    1^2 = 1\, (8\times 8) square.

    2^2 = 4\, (7\times 7) squares

    3^2 = 9\, (6\times 6) squares

    ...

    8^2 = 64\, (1\times 1) squares

    So all you have to do is work out the numbers in between and add them together to get your answer. If you want to use a formula, you might like to know that the sum of the first n square numbers is

    \tfrac{1}{6}n(n+1)(2n+1)

    Grandad
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