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Math Help - nxn grid problem.

  1. #1
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    nxn grid problem.

    Hello.

    I asked about this a while ago, but I'm still unsure. Imagine you have a nxn grid (3x3 for argument). I want to know how I can work out the total number of patterns I can make with 2 colours, say black and white. An example would be:

    x = black o = white

    ooo xoo xxo
    ooo ooo oxx
    ooo ooo oxo

    So, pattern one is all white, patter two has one bit black etc. I was told for this it would be 2 to the 9th, giving me 512, which can't be correct! Is it 9 squared, giving me 81? What if I had a 6x8 grid, or 17x31 grid?

    I did try to host a nicer picture than my x and o, but sadly I can't get imageshack to work.

    Many thanks.

    N.
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  2. #2
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    Quote Originally Posted by theNoodler View Post
    Hello.

    I asked about this a while ago, but I'm still unsure. Imagine you have a nxn grid (3x3 for argument). I want to know how I can work out the total number of patterns I can make with 2 colours, say black and white. An example would be:

    x = black o = white

    ooo xoo xxo
    ooo ooo oxx
    ooo ooo oxo

    So, pattern one is all white, patter two has one bit black etc. I was told for this it would be 2 to the 9th, giving me 512, which can't be correct! Is it 9 squared, giving me 81? What if I had a 6x8 grid, or 17x31 grid?

    I did try to host a nicer picture than my x and o, but sadly I can't get imageshack to work.

    Many thanks.

    N.
    An n x n grid with m possible colours has  m^{n \times n} possible combinations.
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  3. #3
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    Quote Originally Posted by theNoodler View Post
    Imagine you have a nxn grid (3x3 for argument). I want to know how I can work out the total number of patterns I can make with 2 colours, say black and white. An example would be:
    So, pattern one is all white, patter two has one bit black etc. I was told for this it would be 2 to the 9th, giving me 512, which can't be correct!
    But that is correct.
    Look at that attached graph. There are two 3\times 3 grids.
    Are they different colorings?

    Maybe not. Rotate I 90^o counter-clockwise. We get II.
    Now are they different?

    If I & II are different then there are 2^9=512 possible colorings.

    If I & II are not different then you must tell us why?
    Attached Thumbnails Attached Thumbnails nxn grid problem.-untitled.gif  
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  4. #4
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    Hello, theNoodler!

    Imagine you have a n\times n grid (3x3 for argument).
    How can I work out the total number of patterns made with 2 colours, say black and white.

    An example would be: \begin{array}{c}X \,=\, \text{black} \\ O \,=\,\text{white}\end{array}

    . . \begin{array}{c}OOO\\OOO\\OOO\end{array} \qquad \begin{array}{c} XOO\\ OOO\\OOO \end{array} \qquad \begin{array}{c}XXO\\OXX \\ OXO \end{array}\qquad \hdots\text{ etc.}

    I was told for this it would be 2^9 \,=\,512, which can't be correct!
    . . Why not?

    For each of the 9 cells, you have two choices: place a Black or place a White.

    So you have 9 decisions with 2 options each.
    . . There will be: . 2^9 \,=\,512 possible choices you can make.



    What if I had a 6x8 grid grid?

    You have 48 cells to fill with 2 options each.

    There will be: . 2^{48} possible choices.

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  5. #5
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    Hello.

    Thanks for the help. I guess it just doesn't look like there could be that many combinations. So for a 6x8 grid, there would be 281,474,976,710,656 combinations. That's just nuts.
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