How many matches will be played in Round 1, if 33 teams are participating in a knock out tournament ?

Question

Accepted Answer

The correct option is (3):
In a knock-out tournament format, when the number of participating teams (N) is not a power of two (like 2, 4, 8, 16, 32, 64, etc.), a system of 'byes' must be implemented to ensure a balanced bracket for the subsequent rounds. A 'bye' is a privilege that allows a team to advance to the next round without having to play a first-round match. To calculate the arrangement for N=33 teams, we must first find the next highest power of two, which is 64. The total number of byes is calculated by subtracting the number of participating teams from this next power of two ($64 - 33 = 31$ byes). The number of teams that will actually participate in a match in the first round is found by subtracting the byes from the total teams ($33 - 31 = 2$ teams). Since two teams are playing in the first round, there will be only one match ($\frac{2}{2}$ teams), with the 31 teams receiving a bye directly advancing to the second round. Therefore, the total number of matches in Round 1 is 1.

The incorrect option is (1):
The value 16 is incorrect because it is half of 32, which is the total number of teams in the next lower power of two. This number would represent the number of matches if 32 teams were playing in Round 1, or if the total number of teams was 32 and there were no byes. For a tournament with 33 teams, the arrangement necessitates a high number of byes (31) to reach the bracket size of 64, significantly limiting the number of teams that participate in the opening round.

The incorrect option is (2):
The number 32 represents the total number of matches that will be played throughout the *entire* knock-out tournament. The formula for total matches in a knock-out tournament is $N-1$, where $N$ is the number of teams. In this case, $33 - 1 = 32$ total matches. This figure includes matches from Round 1 up to the final, but it is not the specific number of matches for the first round alone.

The incorrect option is (4):
The number 3 has no logical connection to the standard method of calculating the number of matches in the first round of a knock-out tournament with 33 teams. The calculation is driven by the need to adjust the teams to a power-of-two number through the allocation of byes. The calculated number of teams playing in Round 1 is only 2, which results in a single match.