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Math Help - latin square conjecture

  1. #1
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    Cool latin square conjecture

    I have been struggling with a conjecture about latin square, hoping maybe someone on this forum might enlighten me.

    Assume positive integers m,n,k and m=kn (k=2,3,...). Let's define an m-by-n matrix A with two constraints:
    (1) each row of A is a permutation on {1,2,...,n} (these permutations are not necessarily distinct)
    (2) each column of A contains exactly k instances of number j (j=1,2,...,n)

    For example, m=6,n=3, k=2
    1 2 3 *
    3 1 2 *
    1 3 2
    3 2 1
    2 1 3
    2 3 1 *
    each row is a permutation on {1,2,3}, each column contains 1/2/3 k=2 times

    The thing is, I found that for such an A matrix, I can always pick out n rows out of its m rows, so that these n rows constitute a latin square of order n. (e.g., the rows with * in the above example)
    I tried many scenarios and found no exceptions.

    So is this conjecture true? How to prove it please?
    I think it might relate with birkhoff von neumann theorem or theorems in graph theory, but cannot relate those with my problem.

    Any idea??? Many thanks in advance!
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  2. #2
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    HongKong
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    Re: latin square conjecture

    solved... find a counter-example:

    1 2 4 3
    1 4 3 2
    4 2 3 1
    2 4 1 3
    2 3 4 1
    3 1 2 4
    3 1 2 4
    4 3 1 2
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