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In a single elimination tournament, any a player is eliminated with a single loss. The tournament is played in multiple rounds subject to the following rules :
(a) If the number of players, say n, in any round is even, then the players are grouped into n/2 pairs. The players in each pair play a match against each other and the winner moves on to the next round.
(b) If the number of players, say n, in any round is odd, then one of them is given a bye, that is he automatically moves on to the next round. The remaining (n–1) players are grouped into (n–1)/2 pairs. The players in each pair play a match against each other and the winner moves on to the next round. No player gets more than one bye in the entire tournament.
Thus, if n is even, then n/2 players move on to the next round while if n is odd, then (n+1)/2 players move on to the next round. The process is continued till the final round, which obviously is played between two players. The winner in the final round is the champion of the tournament.
If the number of players, say n, in the first round was between 65 and 128, then what is the exact value of n?
A. Exactly one player received a bye in the entire tournament.
B. One player received a bye while moving on to the fourth round from the third round.
Using only statement A, we cannot say anything because the player might have received the bye in any round and the total number of players differs if the round in which the player got a bye changes.
Using only statement B, we cannot saty anything because there might be players who received a bye in other rounds.
Using both the statements, we know that only 1 player got a bye in the whole tournament and that happened in round 3.
If the number of players were between 65 and 128, then the number of rounds is 7.
=> In the fourth round there would be 16 people.
One person got a bye here => 31 people in round 3 => 62 people in round 2 => 124 people in round 1.
Hence, we can ansswer the question using both the statements together.
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