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Let's have a double 6x6 Latin Square (see Figure 1). You can see that this Latin Square has thirteen combinations (see Figure 2). Can you make a double Latin Square that contains the maximum number of combinations and another with a minimum number of combinations? Only numbers from 1 to 6 are allowed. double Latin Square with combinations

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  • $\begingroup$ To be clear, your example in Figure 1 contains 13 pairs (all but 2,6 and 3,5)? $\endgroup$
    – RobPratt
    Jan 2 at 22:41
  • $\begingroup$ Is 1,2 and 2,1 two different combinations or is that considered the same combination? $\endgroup$
    – Amorydai
    Jan 2 at 23:24
  • $\begingroup$ 1,2 and 2,1 is one combination. $\endgroup$ Jan 2 at 23:27
  • $\begingroup$ Where does 2,6 appear in your example? $\endgroup$
    – RobPratt
    Jan 2 at 23:29
  • $\begingroup$ My example is for demonstration purposes. It is up to you to find the maximum number of combinations. $\endgroup$ Jan 2 at 23:34

1 Answer 1

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Minimal:

3 distinct unordered pairs:

     16 25 34 43 52 61
     25 34 43 52 61 16
     34 43 52 61 16 25
     43 52 61 16 25 34
     52 61 16 25 34 43
     61 16 25 34 43 52
 

Maximal:

21 distinct unordered pairs (which is all of them):

     11 22 33 44 55 66
     23 34 45 56 61 12
     35 46 51 62 13 24
     52 63 14 25 36 41
     44 55 66 11 22 33
     66 11 22 33 44 55
 

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  • $\begingroup$ Confirmed via integer linear programming. $\endgroup$
    – RobPratt
    Jan 3 at 1:08

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