# Combinations on superimposed latin squares

Let's have two latin squares. When we superimpose them we obtain a 6x6 superimposed square as shown in the diagram.

By visual inspection we see this contains 15 combinations and their repetitions which are the following.

combinations             number of repetitions
1,2                      2
1,3                      1
1,4                      2
1,5                      3
1,6                      4
2,3                      3
2,4                      3
2,5                      2
2,6                      2
3,4                      4
3,5                      2
3,6                      2
4,5                      2
4,6                      1
5,6                      3


Can someone find two latin squares which, when superimposed give 13 combinations with 2 repetitions and 2 combinations with 5 repetitions?

Nope. This can be seen by counting individual numbers. Two latin squares have 6 copies each of each the numbers 1-6, together 12 copies of each. The actual number 12 won't matter, what's important is that it is the same number of copies for each number 1-6. Now let's look at the pairs we are asked to create. Out of the numbers 1-6 15 distinct pairs can be formed. The prescription thus asks for 2 full sets of pairs plus an excess of 3 copies of 2 pairs. Because of symmetry the full sets of pairs when broken up into the numbers they are made from will yield the same count of each. The excess copies cannot preserve this balance contradicting our initial observation on latin squares.

• I'm talking about latin squares not magic squares. In addition to that, are you saying that it is impossible to construct a 6x6 superimposed latin square from two 6x6 latin squares, which obeys the conditions I stipulate in the question? Nov 7, 2020 at 21:00
• Yes, the three or four digits used by the pairs that are repeated 5 times occur much more often than the others, whereas all 6 digits have to occur the same number of times. Nov 7, 2020 at 21:10
• @Jaap Scherphuis. Your conclusion is that it is impossible to construct such superimposed latin squares obeying the rules I set, which agrees with my own conclusion. Nov 7, 2020 at 21:34

Can someone find two latin squares which, when superimposed give 13 combinations with 2 repetitions and 2 combinations with 5 repetitions?

It is not possible to have exactly 5 repetitions: suppose (a,b) appears 5 times, then the Latin squares (up to permutation of the rows and columns) contain cells equivalent to

a.....  b.....
.a....  .b....
..a...  ..b...
...a..  ...b..
....a.  ....b.
.....?  .....?


In order for them both to be Latin squares, we must have an additional (a,b):

a.....  b.....
.a....  .b....
..a...  ..b...
...a..  ...b..
....a.  ....b.
.....a  .....b