Directions (Q.33-Q.37): The game of QUIET is played between two teams. Six teams, numbered 1, 2, 3, 4, 5, and 6, play in a QUIET tournament. These teams are divided equally into two groups. In the tournament, each team plays every other team in the same group only once, and each team in the other group exactly twice. The tournament has several rounds, each of which consists of a few games. Every team plays exactly one game in each round.

The following additional facts are known about the schedule of games in the tournament

1. Each team played against a team from the other group in Round 8.

2. In Round 4 and Round 7, the match-ups, that is the pair of teams playing against each other, were identical. In Round 5 and Round 8, the match-ups were identical.

3. Team 4 played Team 6 in both Round 1 and Round 2.

4. Team 1 played Team 5 ONLY once and that was in Round 2.

5. Team 3 played Team 4 in Round 3. Team 1 played Team 6 in Round 6.

6. In Round 8, Team 3 played Team 6, while Team 2 played Team 5.

How many rounds were there in the tournament?

The correct option is 8.

To find the total number of rounds in the tournament, we can calculate the total number of games that any single team plays. The tournament rules specify how the games are scheduled for the teams:

1. Group Division:
There are 6 teams in total, divided equally into two groups. Therefore, each group contains:
Group size = 3 teams.

2. Intra-group Games (within the same group):
Each team plays against every other team in its own group exactly once. Since there are 3 teams in a group, any single team has 2 other opponents in its group.
So, the number of intra-group games played by a single team is:

$3-1=2$ games

3. Inter-group Games (against the other group):
Each team plays against every team in the other group exactly twice. Since the other group has 3 teams, a team will play 2 games against each of those 3 teams.
So, the number of inter-group games played by a single team is:

$3\times 2=6$ games

4. Total Games and Rounds:
Summing these up, the total number of games played by any individual team throughout the tournament is:

$2+6=8$ games

The problem states that every team plays exactly one game in each round. Since each team must complete exactly 8 games, there must be exactly 8 rounds in the tournament.