Chapter 8.3, Problem 55E

Elimination Tournaments In an elimination tournament the teams are arranged in opponent pairs for the first round, and the winner of each round goes on to the next round until the champion emerges. The following diagram illustrates a 16-team tournament bracket, in which the 16 participating teams are arranged on the left under Round 1 and the winners of each round are added as the tournament progresses. The top team in each game is considered the “home” team, so the top-to-bottom order matters.

To seed a tournment means to select which teams to play each other in the first round according to their preliminary ranking. For instance, in professional tennis and NCAA basketball the seeding is set up in the following order based on the preliminary rankings: 1 versus 16, 8 versus 9, 5 versus 12, 4 versus 13, 6 versus 11, 3 versus 14, 7 versus 10, and 2 versus 15.³⁵ Exercises 53–56 are based on various types of elimination tournaments.

In 2014, after the NCAA basketball 64-team tournament had already been seeded, Quicken Loans, backed by investor Warren Buffett, offered a billion-dollar prize for picking all the winners.³⁶ If 50,000,000 people entered the contest by picking winners at random, how much money did Quicken Loans expect to have to pay out? [hinT: See Exercise 39 in Section 8.4.]