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.
Let's consider the possible outcomes for the sum of two standard 6-sided dice:
- There is one way to get a sum of 2: rolling two 1's.
- There are two ways to get a sum of 3: rolling a 1 and a 2, or rolling a 2 and a 1.
- There are three ways to get a sum of 4: rolling a 1 and a 3, rolling a 2 and a 2, or rolling a 3 and a 1.
- There are four ways to get a sum of 5: rolling a 1 and a 4, rolling a 2 and a 3, rolling a 3 and a 2, or rolling a 4 and a 1.
- There are five ways to get a sum of 6: rolling a 1 and a 5, rolling a 2 and a 4, rolling a 3 and a 3, rolling a 4 and a 2, or rolling a 5 and a 1.
- There are six ways to get a sum of 7: rolling a 1 and a 6, rolling a 2 and a 5, rolling a 3 and a 4, rolling a 4 and a 3, rolling a 5 and a 2, or rolling a 6 and a 1.
- There are five ways to get a sum of 8: rolling a 2 and a 6, rolling a 3 and a 5, rolling a 4 and a 4, rolling a 5 and a 3, or rolling a 6 and a 2.
- There are four ways to get a sum of 9: rolling a 3 and a 6, rolling a 4 and a 5, rolling a 5 and a 4, or rolling a 6 and a 3.
- There are three ways to get a sum of 10: rolling a 4 and a 6, rolling a 5 and a 5, or rolling a 6 and a 4.
- There are two ways to get a sum of 11: rolling a 5 and a 6, or rolling a 6 and a 5.
- There is one way to get a sum of 12: rolling two 6's.
We want to 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.
One way to approach this problem is to consider the number of ways to get each possible sum using the two numbered dice. For example, if we number the dice with the positive integers a and b, then the number of ways to get a sum of 2 is the number of ways to roll two 1's, which is equal to the number of ways to choose two 1's from the numbers a and b. In other words, the number of ways to get a sum of 2 is equal to the number of ways to choose two 1's from the set {a, b}.
Using this approach, we can generate a table of the number of ways to get each possible sum using the two numbered dice:
| Sum | Number of ways |
|-----|---------------|
| 2 | 1 |
| 3 | 2 |
| 4 | 3 |
| 5 | 4 |
| 6 | 5 |
| 7 | 6 |
| 8 | 5 |
| 9 | 4 |
| 10 | 3 |
| 11 | 2 |
| 12 | 1 |
We can see that the number of ways to get each possible sum using the two numbered dice must match the number of ways to get each possible sum using two standard 6-sided dice. Therefore, we can use this table to generate a system of equations:
a + b = 2 (one way to get a sum of 2)
a + 2b = 3 (two ways to get a sum of 3)
2a + 2b = 4 (three ways to get a sum of 4)
2a + 3b = 5 (four ways to get a sum of 5)
3a + 3b = 6 (five ways to get a sum of 6)
3a + 4b = 7 (six ways to get a sum of 7)
4a + 4b = 8 (five ways to get a sum of 8)
4a + 5
