The key part is that they have to be fair when any subset of the dice is rolled together, not just when all five are rolled. Also if the dice are allowed to be different sizes then it's easier as well.
This is very much missing from the article. It makes the math problem more interesting, but the practical solution is much easier.
There are possibly other applications in navigation, radio communication, or other domains that might make this a more interesting problem than just figuring out game ordering.
Came here also to point this out, because there is an easy to find solution that lacks only this property. I find this restriction somewhat unsatisfying though, because the easy solution does have the property that, for any smaller number of players, there is a subset that is fair for them (so you could manufacture these dice and resolve any number of players up to the number you have).
Requiring same size doesn't make it harder; you can take any solution and inflate the dice (take the lcm of all the sizes, duplicate the face numbers).
edit: oops i just double checked my construction and while they give equal go-first probabilities they don't give equal permutation probabilities...
Sorry, by harder I meant for the same number of faces. E.g. the article is about 5 60 sided dice, which was a breakthrough, but there are other known solution where all the dice are less than 60 sided, but they are unequally sized.
Comments
The key part is that they have to be fair when any subset of the dice is rolled together, not just when all five are rolled. Also if the dice are allowed to be different sizes then it's easier as well.
This is very much missing from the article. It makes the math problem more interesting, but the practical solution is much easier.
There are possibly other applications in navigation, radio communication, or other domains that might make this a more interesting problem than just figuring out game ordering.
Came here also to point this out, because there is an easy to find solution that lacks only this property. I find this restriction somewhat unsatisfying though, because the easy solution does have the property that, for any smaller number of players, there is a subset that is fair for them (so you could manufacture these dice and resolve any number of players up to the number you have).
Requiring same size doesn't make it harder; you can take any solution and inflate the dice (take the lcm of all the sizes, duplicate the face numbers).
edit: oops i just double checked my construction and while they give equal go-first probabilities they don't give equal permutation probabilities...
Sorry, by harder I meant for the same number of faces. E.g. the article is about 5 60 sided dice, which was a breakthrough, but there are other known solution where all the dice are less than 60 sided, but they are unequally sized.