The official solution is at:

https://www.research.ibm.com/haifa/p...ember2016.html
That is Motty Porat's math solution (this earned a star for him, the only star solution!).

My sent solution was:

"The minimal is N=42, and one possible solution:

0 13 29

11 12 19

10 11 21

9 10 23

8 9 25

7 8 27

6 7 29

5 18 19

4 17 21

3 16 23

2 15 25

1 14 27

found this in 18 minutes with a backtracking code: for each month we store the possible triplets: if we fix the i-th month's triplet, then we store those triplets in the further months (j=i+1,..,12) for that we don't get a violation for the (i,j) month dual. In this way we ensure that (k,j) month dual will be valid for all k<i (where j>i). If we reach i=12, then obviously we found a solution." [...]

ps. After I have sent this observed that we can use symmetry: we can assume that N1<N2<N3, with this the running time is only 2 seconds... Btw it is the lex. smallest solution.