Winning Cake

A large group of people play an unusual lottery in the hopes of winning a cake. All write on a piece of paper their name and a positive integer greater than or equal to 1. All the entries are then collected and sorted through. If two or more people enter the same number, they are disqualified from the lottery. Only the unique numbers submitted are considered. The highest unique number wins, and the prize is that fraction of the cake! Thus if someone wins with the number 20 they win one twentieth of the cake and no one else will receive any cake. If you were to play this game, what strategy would you employ?

Comment. Try playing this game multiple times to experiment with alternative strategies. Cupcakes make good prizes.

Unexpected Winner

Some students write their names on individual ballots and place them into a hat. The professor selects one ballot at random to determine the winner of a fabulous chocolate cake. However, the professor suddenly makes this surprising announcement: “I am going to wait two minutes before announcing the name of the winner. No-one, except me of course, knows who has won the fabulous cake. You have no way of guessing who the winner could be, and it will remain that way for the next two minutes. The name of the winner will be a complete surprise to you all. John has won the cake.”