Suppose Alice and Bob are playing a dice game. They each hold a six sided die and a cup. They shake their die in the cup, flip the cup on the table and reveal the roll at the same time. The result is the modular arithmetic of the sum of each die, so 2 + 3 = 5, 3 + 4 = 1, 5 + 3 = 2. The result is a value between 1 and 6 (the values of a single die), so any sum above 6 is "wrapped around" like a clock.
Now, Bob wants to cheat. He thinks he can manipulate the outcome of the roll by setting his die. If Alice continues to roll her dice fairly, can Bob change the outcome of the game?
So far, I don't think so. If Bob always rolled a 5, for example, Alice can roll 1 to get 6, roll 2 to get 1, 3 to get 2 and so on. Alice can effectively hit all the numbers. If Bob loaded the die to only roll 1, 3, 5, Alice can still hit 1-6 by rolling fairly.
Second question if the above is true: If the outcome is fair when at least one player acts fairly, can this fairness be extended to n-players? Example: if 25 people roll, and 24 of them set their die, can the 25th player ensure fairness by rolling fairly?
There's probably a simple proof or property out there that easily answers this, but I've had a bear of a time trying to find it.