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.

What is the number of the team that played Team 6 in Round 3

The correct answer is 5.

Let us break down the schedule and match-ups step-by-step to understand why Team 5 played Team 6 in Round 3.

Step 1: Determine the number of rounds and group structure
The tournament consists of six teams (1, 2, 3, 4, 5, and 6) divided equally into two groups of three teams each. Let these groups be Group A and Group B.
Each team plays every other team in the same group exactly once (2 games) and each team in the other group exactly twice (3 × 2 = 6 games). Therefore, each team plays a total of 8 games (2 + 6 = 8). Since every team plays exactly one game in each round, the tournament must have exactly 8 rounds.

Step 2: Group the teams
From Fact 3, Team 4 played Team 6 twice (in Round 1 and Round 2). Since teams in the same group only play each other once, Team 4 and Team 6 must belong to different groups.
From Fact 4, Team 1 played Team 5 only once (in Round 2). Thus, Team 1 and Team 5 must be in the same group.
From Fact 6, in Round 8, the matches played were:
• Team 3 vs Team 6
• Team 2 vs Team 5
Since Fact 1 states that each team played against a team from the other group in Round 8, these pairs must consist of teams from different groups. Thus:
• Team 3 and Team 6 are in different groups.
• Team 2 and Team 5 are in different groups.
• Consequently, the remaining match-up in Round 8 must be Team 1 vs Team 4, meaning Team 1 and Team 4 are also in different groups.
Combining these relationships:
• Team 1 and Team 5 are in the same group.
• Team 4 is in a different group than Team 1, and Team 2 is in a different group than Team 5. This places Team 2 and Team 4 in the second group.
• Team 6 is in a different group than Team 4, placing Team 6 in the first group.
• Team 3 is in a different group than Team 6, placing Team 3 in the second group.
Thus, the groups are uniquely determined as:
• Group A = {1, 5, 6}
• Group B = {2, 3, 4}

Step 3: Analyze Round 3 matchups
From Fact 5, Team 3 played Team 4 in Round 3. Both of these teams belong to Group B. This accounts for the single same-group match within Group B.
The remaining four teams that must play in Round 3 are {1, 2, 5, 6}. Within this remaining set, Team 2 belongs to Group B, while Teams 1, 5, and 6 belong to Group A.
Since Team 2 is the only Group B team available, only one of the Group A teams can play against Team 2. The other two Group A teams must play against each other in a same-group match.
Let's look at the possible same-group matches for Group A (Teams 1, 5, and 6):
1. Team 1 vs Team 5: Fact 4 states that they played only once, and that was in Round 2. Therefore, they cannot play in Round 3.
2. Team 1 vs Team 6: Fact 5 states that Team 1 played Team 6 in Round 6. Since they only play each other once, they cannot play in Round 3.
3. Team 5 vs Team 6: This is the only remaining same-group match for Group A that has not been scheduled in other rounds.

Therefore, the same-group match for Group A in Round 3 must be between Team 5 and Team 6. Consequently, the team that played Team 6 in Round 3 is Team 5.