1
$\begingroup$

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

$\endgroup$
11
  • $\begingroup$ To be clear, your example in Figure 1 contains 13 pairs (all but 2,6 and 3,5)? $\endgroup$
    – RobPratt
    Commented Jan 2, 2023 at 22:41
  • $\begingroup$ Is 1,2 and 2,1 two different combinations or is that considered the same combination? $\endgroup$
    – Amorydai
    Commented Jan 2, 2023 at 23:24
  • $\begingroup$ 1,2 and 2,1 is one combination. $\endgroup$ Commented Jan 2, 2023 at 23:27
  • $\begingroup$ Where does 2,6 appear in your example? $\endgroup$
    – RobPratt
    Commented Jan 2, 2023 at 23:29
  • $\begingroup$ My example is for demonstration purposes. It is up to you to find the maximum number of combinations. $\endgroup$ Commented Jan 2, 2023 at 23:34

1 Answer 1

4
$\begingroup$

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
 

$\endgroup$
1
  • $\begingroup$ Confirmed via integer linear programming. $\endgroup$
    – RobPratt
    Commented Jan 3, 2023 at 1:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.