Sicherman dice are a particular pair of six-sided dice that, when rolled, produce sums with the same probability distribution as a pair of standard six-sided dice. In other words, these dice do not have the arrangement of 1, 2, 3, 4, 5, and 6 on their six sides, but when rolled together, still produce the same distribution of sums:

- One way to roll a 2
- Two ways to roll a 3
- Three ways to roll a 4
- Four ways to roll a 5
- Five ways to roll a 6
- Six ways to roll a 7
- Five ways to roll a 8
- Four ways to roll a 9
- Three ways to roll a 10
- Two ways to roll a 11
- One way to roll a 12

Sicherman dice are the *only *alternative arrangement of six-sided dice with positive integers that produce the same probability distribution as a pair of standard six-sided dice. Can you figure out what numbers belong on Sicherman dice?

Hint: One of the dice has no numbers greater than 4.

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