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Math Help - word problem sequence / puzzle help

  1. #1
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    word problem sequence / puzzle help

    A 2 x n rectangular area, where n is a positive integer, is to be titled using non-overlapping 2x1 rectangles.

    The number of ways in which a 2 x n rectangular area can be tiled is denoted by  t_{n}

    find an expression for  t_{n}
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  2. #2
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    Re: word problem sequence / puzzle help

    Quote Originally Posted by Tweety View Post
    A 2 x n rectangular area, where n is a positive integer, is to be titled using non-overlapping 2x1 rectangles.
    The number of ways in which a 2 x n rectangular area can be tiled is denoted by  t_{n}
    find an expression for  t_{n}
    Are the tiling rectangles each identical or different?
    What difference does the orientation of the tile make?
    We need more information.
    Thanks from Tweety
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  3. #3
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    Re: word problem sequence / puzzle help

    I don't really understand the problem my self, but my sheet shows this example,

    one tiling for a 2 x 5 area.

    and since all rectangles are 2 x 1, i am assuming they are identical ?
    Attached Thumbnails Attached Thumbnails word problem sequence   / puzzle help-image.jpg  
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    Re: word problem sequence / puzzle help

    Quote Originally Posted by Tweety View Post
    I don't really understand the problem my self, but my sheet shows this example,
    one tiling for a 2 x 5 area.
    and since all rectangles are 2 x 1, i am assuming they are identical ?
    That diagram does help.
    $t_1=1$ one vertical tile. $t_2=2$ two vertical tiles or two horizontal tiles.

    What is $t_3=~?$ How are they arranged?
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  5. #5
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    Re: word problem sequence / puzzle help

    Hello, Tweety!

    This problem is bigger than I thought.
    I do have some observations.
    Maybe others can follow up.


    A 2 x n rectangular area, where n is a positive integer,
    is to be tiled using non-overlapping 2x1 rectangles ("dominos").

    The number of ways in which a 2 x n rectangular area can be tiled is denoted by T_n.

    Find an expression for T_n.

    The area can be tiled "horizontally" . . .

    Code:
          *-------*-------* . .
          |   H   |   H   |
          *-------*-------*
          |   H   |   H   |
          *-------*-------* . .
    or "vertically" . . .

    Code:
          *---*---*---*---* . .
          |   |   |   |   |
          | V | V | V | V |
          |   |   |   |   |
          *---*---*---*---* . .
    or any combination thereof.


    I reasoned like this:

    For each 2-by-2 square, there are two choices:
    . . two H's or two V's.


    If n is even, there are \tfrac{n}{2} squares
    . . which can be tiled in 2^{\frac{n}{2}} ways.

    If n is odd, there are \tfrac{n-1}{2} squares
    . . which can be tiled in 2^{\frac{n-1}{2}} ways,
    . . plus one V domino.


    I thought my work was done . . . wrong!
    I did some sketching and found the bigger problem.

    For n=4, there are 5 tilings:

    Code:
          *-------*-------*       *-------*---*---*       *---*-------*---*
          |       |       |       |       |   |   |       |   |       |   |
          *-------*-------*       *-------*   |   |       |   *-------*   |
          |       |       |       |       |   |   |       |   |       |   |
          *-------*-------*       *-------*---*---*       *---*-------*---*
    
    
                          *---*---*-------*       *---*---*---*---*
                          |   |   |       |       |   |   |   |   |
                          |   |   *-------*       |   |   |   |   |
                          |   |   |       |       |   |   |   |   |
                          *---*---*-------*       *---*---*---*---*

    I did some more sketching and found this familiar pattern:

    . . \begin{array}{cc} n & T_n \\ \hline 2 & 2 \\ 3 & 3 \\ 4 & 5 \\ 5 & 8 \\ 6 & 13 \\7 & 21 \\ \vdots & \vdots \end{array}

    But I have no proof that the Fibonnaci sequence is involved here.
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