Dice With All Pairwise Sums Represented
You want to design a 6-sided die whose faces take values from $\{1, 2, 3, 4, 5, 6\}$ (with repeats allowed). The requirement is that when you roll two identical copies of this die, every integer sum from
$ through 2$ must be achievable with positive probability.
Two dice are considered the same if they have the same multiset of face values -- i.e., the same number of faces showing each value, regardless of physical orientation. Each of the 6 faces is equally likely on each roll.
How many distinct dice satisfy this property?
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