Sealed Solution

Solutions prior to 2013
Sam sent us this solution:

The possible ways of making the numbers on the envelopes are:

3: 1+2 or 0+3
14: 5+9 or 6+8
7: 0+7 or 1+6 or 2+5 or 3+4
8: 0+8 or 1+7 or 2+6 or 3+5
13: 4+9 or 5+8 or 6+7.

Let's try 3=1+2 and 14=5+9. Then we have to have 13=6+7, and 8=0+8 (and 7=3+4).

Let's try 3=1+2 and 14=6+8. Then we must have 13=4+9, and 7=0+7, so 8=3+5.

Let's try 3=0+3 and 14=5+9. Then we must have 13=6+7, and 7=1+6, and this isn't allowed, so this isn't possible.

Let's try 3=0+3 and 14=6+8. Then we must have 8=1+7 (and 7=2+5).

So the possible pairs that are in envelope 8 are 0 and 8, or 3 and 5, or 1 and 7.