# Is it possible to change numbers on two die $\in \left\{ 1,\ldots ,6\right\}$ so that the probability distribution of their sum remains unchanged?

I have been thinking about the problem of ' Is it possible to change the numbers on two six sided dice to other positive numbers so that the probability distribution of their sum remains unchanged?'

I can see that taking away one on all the values of one die, and adding one to all the values on the other gives the same probability distribution.

Also 1,2,2,3,3,4 and the other 1,3,4,5,6,8 is another example.

I wondered how one can generate other solutions to adding a value to one die, and taking away that value on the other, and if there is a solution set, where all the values are still 1,2,3,4,5,6?

• "if there is a solution set, where all the values are still 1,2,3,4,5,6?" I dont think so... You clearly have to have one 1 and one 6 on each die (because of the probability of getting 2 or 12). I would assume the rest follows similarily (although I haven't looked into it). I wonder about other solutions, though. – Arthur Jun 13 '18 at 8:45
• @Arthur, Yes agreed, that you have to have 1 or a 6 on both, then you would need at least one repeat of one number. And I am interested about the general solutions – JimSi Jun 13 '18 at 10:24
• 1,2,2,3,3,4 and 1,3,4,5,6,8 are called "Sicherman dice" and are the only non-standard dice that work if we require the faces of the dice to be labeled with positive integers. Two references: math.stackexchange.com/questions/934624/… and cut-the-knot.org/arithmetic/combinatorics/Sicherman.shtml – awkward Jun 13 '18 at 12:45