Prasad brought work to a halt earlier this week by issuing the following little conundrum. One indepdenent fair 6-sided die is rolled for each of n people. Each person’s number is written on a card and stuck on their head so that they can see everyone else’s number but their own. The people are not allowed to communicate after the numbers are assigned, and from just viewing the others’ numbers they must guess their own. They can come up with any rule that they like for doing this.
What rule should you choose to that is the probability that all of them guess correctly is maximized, and what is that probability?
This is a variation on the “colored hats game,” but the criterion you want to maximize is slightly different than in some instances of the game.