Two six sided dice are marked with 12 different integers, so that any number from 1 to 36 can be derived when both dice are thrown and the showing face-up values are added.

What is the lowest possible value of the highest number on one of the dice faces?

Well, if duplicates were allowed, then the answer would be 6.

The dice would be

1,2,3,4,5,6

and

0,6,12,18,24,30

But duplicates are not allowed, and this problem is difficulty 4, so this thought is probably not even helpful.

First real thought:

One of the dice must have a 0, and the other must have both a 1 and a 2. There is no other way to sum to 1 and 2.

But then how is 3 formed? It can only be from a 3 on the same die as the 1 and 2, because otherwise a number would be duplicated. But the same argument says next that 4 and then 5 and then 6 and then 7 must be on that die, and there isn't enough room. So my thinking is off, someplace.

Things that don't seem to help:

1) Negative numbers

2) Two numbers on one side and no numbers on another.

So, in short, I'm going to look forward to how this puzzle proceeds.