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.

Students: We have found several ways of making the game fair for three people. For example, Player 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 an 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 fair if all the possible sums are to be someone's winning numbers.

Please direct questions and comments about this project to Addison-Wesley math@aw.com
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