# Maximum and minimum numbers of combinations in a double Latin Square

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.

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

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


• Confirmed via integer linear programming. Jan 3 at 1:08