For any even n, say n = 2 m, a row complete Latin square of order n can be formed by writing down 0 , 1 , 2 m − 1 , 2 , 2 m − 2 , 3 , … , m + 1 , mas the first row and then developing subsequent rows by adding 1 modulo n.I'm not quite clear on how that goes, such as what the difference is between 2 m and m. 2 m stands for n, does m stand for modulo? I'd like to see and example with an even number or two, such as 4, and 6.I tried, 2 m = 4 0 , 1 , 3 , 2 , 2 …wait, that can't be right.

Question

For any even   n, say   n  =  2  m, a row complete Latin square of order   n can be formed by writing down  0  ,  1  ,  2  m  −  1  ,  2  ,  2  m  −  2  ,  3  ,  …  ,  m  +  1  ,  mas the first row and then developing subsequent rows by adding   1 modulo   n.I&#039;m not quite clear on how that goes, such as what the difference is between   2  m and   m.   2  m stands for   n, does   m stand for modulo? I&#039;d like to see and example with an even number or two, such as   4, and   6.I tried,   2  m  =  4  0  ,  1  ,  3  ,  2  ,  2  …wait, that can&#039;t be right.

Bruce Bridges · Accepted Answer

If I understand your description correctly, for   2  m  =  4:      [                            0                          1                          3                          2                                      1                          2                          0                          3                                      2                          3                          1                          0                                      3                          0                          2                          1                      ]  For   2  m  =  6:      [                            0                          1                          5                          2                          4                          3                                      1                          2                          0                          3                          5                          4                                      2                          3                          1                          4                          0                          5                                      3                          4                          2                          5                          1                          0                                      4                          5                          3                          0                          2                          1                                      5                          0                          4                          1                          3                          2                      ]  In each case, the first row follows the pattern you gave; in subsequent rows, each number is one more than the number above it, but wrapping around at the size of the square (e.g. in the case where   n  =  2  m  =  6, after   5 comes   0).Let&#039;s look at   n  =  2  m  =  12   (  m  =  6  ) a bit more closely. The pattern that you gave said:  0  ,  1  ,  2  m  −  1  ,  2  ,  2  m  −  2  ,  3  ,  …  ,  m  +  1  ,  mSo, the first row is:  0  ,  1  ,  12  −  1  ,  2  ,  12  −  2  ,  3  ,  …  ,  6  +  1  ,  6which is  0  ,  2  +  1  =  3  ,  9  +  1  =  10  ,  4  +  1  =  5  ,  8  +  1  =  9  ,  5  +  1  =  6  ,  7  +  1  =  8  ,  6  +  1  =  7Continuing gives:      [                            0                          1                          11                          2                          10                          3                          9                          4                          8                          5                          7                          6                                      1                          2                          0                          3                          11                          4                          10                          5                          9                          6                          8                          7                                      2                          3                          1                          4                          0                          5                          11                          6                          10                          7                          9                          8                                      3                          4                          2                          5                          1                          6                          0                          7                          11                          8                          10                          9                                                                              ⋮                      ]

For any even n , say n = 2 m , a row complete Latin square of order n can be formed

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