A knock-out tournament is being held with 2n tennis players. This means that for each

Chapter 1, Problem 5

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A knock-out tournament is being held with 2n tennis players. This means that for each round, the winners move on to the next round and the losers are eliminated, until only one person remains. For example, if initially there are 24 = 16 players, then there are 8 games in the first round, then the 8 winners move on to round 2, then the 4 winners move on to round 3, then the 2 winners move on to round 4, the winner of which is declared the winner of the tournament. (There are various systems for determining who plays whom within a round, but these do not matter for this problem.) (a) How many rounds are there? (b) Count how many games in total are played, by adding up the numbers of games played in each round. (c) Count how many games in total are played, this time by directly thinking about it without doing almost any calculation. Hint: How many players need to be eliminated