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Math Help - Euler's Officers

  1. #1
    Senior Member Sampras's Avatar
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    Euler's Officers

     36 officers are given, belonging to six regiments and holding six ranks (so that each combination of rank and regiment corresponds to just one officer). Can the officers be paraded in a  6 \times 6 array so that in any line (row or column) of the array, each regiment and each rank occurs precisely once?

    If you put the  6 regiments at the first row and the  6 ranks in the first column, isn't this a solution to the problem?
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  2. #2
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    123456
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    or am I interpreting the question wrongly? it's like a mini sudoku :P
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    Quote Originally Posted by TiRune View Post
    123456
    234561
    345612
    456123
    561234
    612345

    or am I interpreting the question wrongly? it's like a mini sudoku :P
    No, you are misinterpreting the problem.

    You have 36 ordered pairs (regiment, rank), where you might label the regiments from 1 to 6 and the ranks from 1 to 6, so the pairs are of the form (i,j), i = 1 to 6, j = 1 to 6. You have to arrange the pairs in a 6 by 6 array so that if you consider the regiments alone, each regiment appears exactly once in each row and column, and if you consider the ranks alone, each rank appears exactly once in each row and column.

    Here is a solution for the 3 by 3 case:

    Code:
    (1,1)  (2,2)  (3,3)
    (3,2)  (1,3)  (2,1)
    (2,3)  (3,1)  (1,2)
    As you may know, the 6 by 6 case has been shown to be impossible.
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  4. #4
    Senior Member Sampras's Avatar
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    Quote Originally Posted by awkward View Post
    No, you are misinterpreting the problem.

    You have 36 ordered pairs (regiment, rank), where you might label the regiments from 1 to 6 and the ranks from 1 to 6, so the pairs are of the form (i,j), i = 1 to 6, j = 1 to 6. You have to arrange the pairs in a 6 by 6 array so that if you consider the regiments alone, each regiment appears exactly once in each row and column, and if you consider the ranks alone, each rank appears exactly once in each row and column.

    Here is a solution for the 3 by 3 case:

    Code:
    (1,1)  (2,2)  (3,3)
    (3,2)  (1,3)  (2,1)
    (2,3)  (3,1)  (1,2)
    As you may know, the 6 by 6 case has been shown to be impossible.
    Yeah so its basically each "coordinate" in the ordered pair has to appear only once in each row and column. Not the whole ordered pair itself.
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