Divisibility

Mentor: Let us make the game with two dice fair. Can you assign every possible sum to one of our three players in such a way that the game is fair? Each will get several winning numbers.

Student: We have found several ways of making the game fair for three people. For example, Player 1 can get the sums of 2, 3, 4, and 7, Player 2 the sums of 8, 9, 11 and 12, and Player 3 all the rest: 5, 6, and 10. The important thing is, each player has to get exactly 12 outcomes.

Student: I have made it fair for four people. Each gets 9 outcomes.

Mentor: Can you make it fair for two people assigning all the numbers?

Student: Sure, each one will get 18 outcomes.

Mentor: What other numbers of people can play the game fairly if someone wins in every outcome?

Student: The numbers that divide 36. Three can do it. Five can't, because you can't divide 36 outcomes between five people fairly. Six people can have a fair game, but not seven or eight. It looks like nine people can play it if each of them gets four outcomes.

Mentor: OK, we got into some pure abstractions here. Can you please set up the winning numbers for nine people to make the game fair?

Student (trying to do it): Not really, because whoever gets 7 as a winning number has six outcomes already! Actually, no more than six people can play the game fairly if all of the possible sums are to be someone's winning numbers.