When two standard 6-sided dice are rolled, there are 36 possible outcomes for the sum of the two rolls: one sum of 2, two sums of 3, and so on, up to one sum of 12.
Find all possible ways of numbering two 6-sided dice with positive integers (not necessarily distinct), so that when they are rolled, the 36 possible outcomes for the sum of the two rolls are the same as the 36 possible outcomes for the sum of two standard 6-sided dice.
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1 - (1+1)=2,
2 - (1+2)=3,
3 - (2+1)=3,
4 - (1+3)=4,
5 - (3+1)=4,
6 - (2+2)=4,
7 - (1+4)=5,
8 - (4+1)=5,
9 - (2+3)=5,
10 - (3+2)=5,
11 - (1+5)=6,
12 - (5+1)=6,
13 - (2+4)=6,
14 - (4+2)=6,
15 - (3+3)=6,
16 - (1+6)=7,
17 - (6+1)=7,
18 - (2+5)=7,
19 - (5+2)=7,
20 - (3+4)=7,
21 - (4+3)=7,
22 - (2+6)=8,
23 - (6+2)=8,
24 - (3+5)=8,
25 - (5+3)=8,
26 - (4+4)=8,
27 - (3+6)=9,
28 - (6+3)=9,
29 - (4+5)=9,
30 - (5+4)=9,
31 - (4+6)=10,
32 - (6+4)=10,
33 - (5+5)=10,
34 - (5+6)=11,
35 - (6+5)=11,
36 - (6+6)=12
