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Squares on a Chessboard Challenge

    • Thread Starter

    Hey guys! :wavey:

    I have an interesting puzzle/challenge here for you guys! There is no required knowledge for this, so anyone can have a go.

    This is a standard 8x8 chessboard. If I asked you how many squares were on there, you may count up all of the black and white small squares and end up with 64.

    However you need to consider that other squares can be formed using the smaller squares, such as a 2x2 square as demonstrated below:

    Name:  square.png
Views: 34
Size:  1.6 KB

    The questions are:
    How many squares on a 8x8 chessboard?
    How many squares on are on a nxn chessboard?
    Is there a way to work out how many rectangles there are?

    Please post your answers in the following format:

    [spoiler]Post your answer here[/spoiler]



    8x8 204 squares

    nxn add up n squared from 1 to n

    yes there definitely is but that's as far as I've got


    maybe an answer to the 3rd part of the question, although i dont have much confidence in it :dontknow:
    Start in the top left square with a square of 1x1. The box can only be increased in size by dragging the right edge to the right or the bottom edge down

    we can make 64 squares/rectangles by starting in the top left
    starting in the one below the top left gives us 7*8 different shapes
    starting two below the top left gives us 6*8 different shapes
    starting in the one to the right of the top left gives us 8*7 shapes
    starting in the diagonal from the top left gives us 7*7 shapes
    carry this on until you reach the far right column and work down to the bottom right where you have 1*1 shape which you can make

    On a 8x8 we get (1+2+3+4+...+8)*8 + (1+2+3+4+...+8)*7 + ... (1+2+3+4+...8)*1 shapes (including squares)

    so (1+2+3+4+...+8) ^2 total shapes on the 8x8 board

    So number of rectangles on 8x8 is number of shapes - number of squares

     \displaystyle (1+2+3+4+...+8) ^2 - [8^2 + 7^2 + 6^2 +...+ 1^2] = 1092

    So number of rectangles on an nxn board is  \displaystyle \left( \frac{n(n+1)}{2} \right) ^2 - \left(\frac{n(n+1)(2n+1)}{6} \right ) or  \displaystyle \sum_1^n r^3 - \sum_1^n r^2


    (Original post by DylanJ42)
    So number of rectangles on 8x8 is number of shapes - number of squares
    Um, a square is a rectangle, so there's no reason to exclude the squares...

    (Original post by DFranklin)
    Um, a square is a rectangle, so there's no reason to exclude the squares...
    oh i didnt know, whoops

    I guess ignore the "minus squares" part then
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