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Math Help - Pairs and pieces

  1. #1
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    Pairs and pieces

    Find all such pairs of positive integers (m,n), for which a rectangle with dimensions of m x n can be built from such pieces created from 6 squares marked as "O" (note that you can both rotate and flip the pieces):
    Code:
    O O
    OOOO
    (if it's not clear, it's a F-type shape)

    OK. So we now that we can create a 3x4 rectangle by simply putting together this shape and the one rotated by 180^:
    Code:
    OOOO
     O O
    As we can create 3x4, we can also create 4a x 3a for a positive integer a. But what else? I'm kind of clueless...
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  2. #2
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    Re: Pairs and pieces

    Hello, GGPaltrow!

    Find all such pairs of positive integers (m,n),
    for which a rectangle with dimensions m\!\times\!n can be built
    from pieces created from 6 squares as shown.
    (Note that you can both rotate and flip the pieces):

    . . \begin{array}{cc}\square \;\,\square \;\; \\ [-2mm]\square\! \square\! \square\! \square \end{array}


    OK, so we know that we can create a 3x4 rectangle
    . . by simply putting together this shape and the one rotated by 180^o.

    . . \begin{array}{c}\blacksquare\! \blacksquare\!\blacksquare\! \blacksquare \\ [-2mm] \square \! \blacksquare\! \square\! \blacksquare \\ [-2mm] \square \! \square \! \square \! \square  \end{array}


    We can also create 4a\times 3b rectangles for positive integers a and b.
    But what else? .I'm kind of clueless ...

    Me too . . .

    I've tried to find other possible rectangles, but have failed so far.
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  3. #3
    Junior Member
    Joined
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    Re: Pairs and pieces

    Hmm.. maybe sth like this:

    Suppose we have:
    Code:
    O O
    OOOO
    in which the four squares form an edge. Then, let's try to put sth beside it (as putting above leads to a 3x4 rectangle):

    Code:
       OO
        O
    O OOO
    OOOOO
    This can't be transformed into a rectangle as our "F" can't be put inside it in any way. The same goes with this flipped:
    Code:
        OO
        O
    O O OO
    OOOOO
    Which is even more wrong as we leave our black F laying and blocked from the "open" side so we can only complete it to a 4x3.

    Making any F lie on its "back" beside the black one doesn't make sense cause it leads to completing each of them to a 4x3 which means we eveantully end with 4ax3b.

    What about this?
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  4. #4
    Junior Member
    Joined
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    Posts
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    Re: Pairs and pieces

    Is this the correct solution? Or it needs other formal handling? Or maybe it's wrong?
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