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Math Help - Tiled square

  1. #1
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    Tiled square

    A square is assembled out of 2-by-1 tiles. Show that there exists a 2-by-2 square assembled out of some two tiles.
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    tiled square

    Quote Originally Posted by atreyyu View Post
    A square is assembled out of 2-by-1 tiles. Show that there exists a 2-by-2 square assembled out of some two tiles.
    Bonjour mon ami,

    Are you leaving something out or do you need to restate the question?

    bjh
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  3. #3
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    Erm, no, I need to prove that in such a square I can pick two tiles that themselves form a 2-by-2 square.
    Please, anyone?
    Last edited by atreyyu; April 2nd 2010 at 12:28 PM.
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  4. #4
    Senior Member MacstersUndead's Avatar
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    A square is assembled out of 2-by-1 tiles. Show that there exists a 2-by-2 square assembled out of some two tiles. "[]" will be 2x1 orientation while "=" will be 1x2 orientiation
    --

    The base case is using two 2x1 tiles to make a 2x2 square. (side by side. row = 2, column = 1 + 1 = 2)
    [][] (2x2 square)

    Now all you need to show is that for any square, you have a 2x2 square. (induction?)

    For example, take the next case, using 8 = 2^3 tiles. ((EDIT//: is it possible with 4? you can get something close like 3 2x1 side by side and then 1 1x2 on the bottom))

    [][][][]
    [][][][] (4x4 square)

    The pattern seems to be, then, that you can only construct a square with 2x1 tiles if and only if you have exactly 2^odd tiles.

    hope that helps.
    - MacstersUndead
    Last edited by MacstersUndead; April 2nd 2010 at 09:08 PM.
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  5. #5
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    What about 6x6 square? It can be laid out properly with 18 tiles, and 18 \neq 2^odd.
    I still don't know how to tackle this.
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  6. #6
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    Quote Originally Posted by atreyyu View Post
    What about 6x6 square? It can be laid out properly with 18 tiles, and 18 \neq 2^odd.
    I still don't know how to tackle this.
    WHY are you bringing this in, since your original post was:
    "A square is assembled out of 2-by-1 tiles.
    Show that there exists a 2-by-2 square assembled out of some two tiles."
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  7. #7
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    I'm bringing this in because it is wrong, as you can build a square using 18 tiles, and 18 is not an odd power of 2.
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  8. #8
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    Ok...not sure what you're asking or what your point is, BUT:

    with these tiles (1 by 2), only even-sided squares can be built

    if n = side length of squares, then number of tiles to build =
    n^2 / 2 ; like your 6by6 = 6^2 / 2 = 18

    odd-sided squares can be built, BUT one 1by1 square always required.

    C'est pas ma faute, tabarnak (en bon canadien francais) !
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  9. #9
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    Hello, atreyyu!

    I haven't found a proof yet,
    but let me explain the problem further.


    A square is assembled out of 2-by-1 tiles.
    Show that there exists a 2-by-2 square assembled out of some two tiles.
    Let n = side of the square.

    Since the square is composed of "dominos", n must be even.
    . .
    And that is the only constraint.


    Prove: in every possible tiling of an n\times n square,

    . . . . . there will be a pair of tiles oriented like:
    Code:
          *---*---*        *-------*
          |   |   |        |       |
          |   |   |   or   *-------*
          |   |   |        |       |
          *---*---*        *-------*

    I've tried various approaches (including proof by contradiction)
    . . but have had no luck so far.

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  10. #10
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    OK I thought I was making myself clear all along:

    Using the 2x1 tiles, we make a square. Of course it's length can only be an even number, and only an even number of tiles could have been used; that's a fact.
    The point is to show that you can pick such a 2x2 square inside the big square (that is of any size nxn, n=even) that is formed using only two tiles.

    EDIT: Soroban, thanks, that's exactly what I meant.
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