1. Joined
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    12 Feb '07 00:27
    True or false:

    If you cut a chessboard into 16 pieces, each one a tetris shape made of 4 squares, and 15 of the pieces are L-shaped, then the 16th piece is also L-shaped.
  2. Joined
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    12 Feb '07 16:211 edit
    I'll give this one a go.

    The five different shapes for four square tetris (tetrominoes) are

    a) Stick. e.g. would cover a1, b1, c1 and d1 on a chessboard
    b) L. e.g. would cover a2, a1, b1 and c1.
    c) Square. e.g. would cover a2, b2, a1 and b1.
    d) T. e.g. would cover b2, a1, b1 and c1.
    3) Z. e.g. would cover a2, b2, b1 and c1.

    Note that on an ordinary chessboard, all of these except the T would cover two black squares and two white squares. Thus it follows that a solution of 15 Ls and a T covering a chessboard is not possible.

    Now consider a chessboard marked with black and white bands, e.g. the a-file is all black, the b-file all white, the c-file black etc.

    On this chessboard, the L always covers 3 black squares and a white square, or 3 white squares and a black square. The only other shape for which this is true is the T (this can either cover 2 black and 2 white or a 3/1 split).

    So if we try to cover this board with 15 Ls we will end up with a gap consisting of 3 blacks and a white, or 3 whites and a black (I'm assuming one gap, but if the squares are separate there will be still be a 3 and 1 split of colours).

    The only shapes that can cover this gap are another L or a T. We know that a T will not fit the gap from our attempt to cover an ordinary chessboard with 15 Ls and a T, thus if there is a solution to the problem of 15 Ls + one other piece to cover an 8x8 chessboard then the 16th piece must be another L.

    It is trivial to show that a solution of 16 Ls does indeed exist because 2 Ls can be put together to make a 4x2 rectangle, and thus four Ls can be put together to exactly cover 2 rows of a chessboard.
  3. Account suspended
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    12 Feb '07 17:00
    Originally posted by David113
    True or false:

    If you cut a chessboard into 16 pieces, each one a tetris shape made of 4 squares, and 15 of the pieces are L-shaped, then the 16th piece is also L-shaped.
    false
    i believe...
    just solving it in my head, there seems to be a way you could force a square...........
  4. Standard memberXanthosNZ
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    12 Feb '07 17:19
    Originally posted by rubberjaw30
    false
    i believe...
    just solving it in my head, there seems to be a way you could force a square...........
    Prove it.
  5. Subscribersonhouse
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    12 Feb '07 17:25
    Originally posted by Fat Lady
    I'll give this one a go.

    The five different shapes for four square tetris (tetrominoes) are

    a) Stick. e.g. would cover a1, b1, c1 and d1 on a chessboard
    b) L. e.g. would cover a2, a1, b1 and c1.
    c) Square. e.g. would cover a2, b2, a1 and b1.
    d) T. e.g. would cover b2, a1, b1 and c1.
    3) Z. e.g. would cover a2, b2, b1 and c1.

    Note that on an ordi ...[text shortened]... a 4x2 rectangle, and thus four Ls can be put together to exactly cover 2 rows of a chessboard.
    Nice.
  6. Joined
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    12 Feb '07 20:221 edit
    Originally posted by Fat Lady
    I'll give this one a go.

    The five different shapes for four square tetris (tetrominoes) are

    a) Stick. e.g. would cover a1, b1, c1 and d1 on a chessboard
    b) L. e.g. would cover a2, a1, b1 and c1.
    c) Square. e.g. would cover a2, b2, a1 and b1.
    d) T. e.g. would cover b2, a1, b1 and c1.
    3) Z. e.g. would cover a2, b2, b1 and c1.

    Note that on an ordi a 4x2 rectangle, and thus four Ls can be put together to exactly cover 2 rows of a chessboard.
    Nice solution.

    A variation of your solution:
    The black and white bands also prove that 15 Ls + T is impossible, since the bands can be drawn in two ways (vertical or horizontal), and in one of the ways the T covers 2 black squares and 2 white ones.
  7. SubscriberAThousandYoung
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    12 Feb '07 23:161 edit
    Originally posted by Fat Lady
    I'll give this one a go.

    The five different shapes for four square tetris (tetrominoes) are

    a) Stick. e.g. would cover a1, b1, c1 and d1 on a chessboard
    b) L. e.g. would cover a2, a1, b1 and c1.
    c) Square. e.g. would cover a2, b2, a1 and b1.
    d) T. e.g. would cover b2, a1, b1 and c1.
    3) Z. e.g. would cover a2, b2, b1 and c1.

    Note that on an ordi a 4x2 rectangle, and thus four Ls can be put together to exactly cover 2 rows of a chessboard.
    There are two L's; they are mirror images. I don't know if this matters to your solution though.
  8. Account suspended
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    13 Feb '07 00:47
    Originally posted by XanthosNZ
    Prove it.
    i just tried it, it doesn't work...
    ud have to have two squares...
    so my solution wasn't right...
    or well, it could be...
    but not the way i tried solving it...
  9. Standard memberXanthosNZ
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    13 Feb '07 02:51
    Originally posted by rubberjaw30
    i just tried it, it doesn't work...
    ud have to have two squares...
    so my solution wasn't right...
    or well, it could be...
    but not the way i tried solving it...
    Considering there has been a proof that it must be L shaped posted in this thread you might want to just give up.
  10. Standard memberXanthosNZ
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    13 Feb '07 02:52
    Originally posted by AThousandYoung
    There are two L's; they are mirror images. I don't know if this matters to your solution though.
    But they will always cover 2 of each colour square in diagonal colouring and 3 and 1 in horizontal colouring.
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