Sports tournaments are often conducted to determine the best player or team at a particular level. Consider the following format for running a tournament. The tournament is made up of rounds. Suppose that at the beginning of round i, there are ki teams. When ki > 1, the first and second team, the third and fourth team etc., play against each other. The winner moves on to the next round; the loser drops off from the tournament. When ki is odd, the last team also moves on to the next round without playing a game. If ki = 1, the tournament ends; the remaining team is declared the champion. Please answer the questions below.

a. When there are 5 teams, it takes 3 rounds of play before a champion is declared. For n teams, how many rounds of play are needed? Why?

b. When there are 5 teams, a total of 4 games are played before a champion is declared. For n teams, how many games have to be played before a champion is declared? Why?

c. As an tournament official, you feel uneasy that during a tournament round, it is possible for a team to move on to the next round without actually playing a game while every other team plays a game. You decide at the beginning that youll only run the tournament if this unfair situation will never arise. Clearly, the answer depends on n. What values of n will you accept? Why? Make sure that you explain how the unfair situation will arise for the values you are not accepting.