Question

How many rounds were there in the tournament?

• A. 2
• B. 4
• C. 5
• D. 3

Answer 1

Correct Answer: 8

To determine the number of rounds in the QUIET tournament, we start by analyzing the given conditions:

1. There are 6 teams numbered 1 to 6, divided into two groups of 3 teams each.
2. Each team plays other teams in the same group only once and in the other group exactly twice.
3. Each team plays one game per round.

We'll first deduce group memberships:

• Round 1: Since Team 4 played Team 6, they are in the same group. Let's group them as X: X = {4, 6}.
• Round 2: Since Team 4 played Team 6 again, confirms X = {4, 6}.
  In Round 2, Team 1 played Team 5 only once, indicating a group play: Y = {1, 5}.

Considering Team 3 plays Team 4 in Round 3, Team 3 must be in X and thus X = {3, 4, 6} and Y = {1, 2, 5}.

We confirm conditions:

• Round 4: Matches are identical to Round 7.
• Round 5: Matches are identical to Round 8.
  In Round 6, Team 1 plays Team 6 indicating inter-group play.
• Round 8: Inter-group play with Team 3 playing Team 6 and Team 2 playing Team 5.

Therefore the rounds are:

1. Rounds with intra-group matches: 1, 2, 3 (Group X plays {3, 4, 3}, Group Y plays {5, 1}). Given conditions, 3 rounds.
2. Rounds with inter-group matches: 4, 5, 6, 7, 8. Given conditions, 5 rounds.

Summing both types, the exact total number of rounds is 8.

This solution falls within the given range of 8 (min & max). The complete breakdown of rounds aligns with provided clues and confirms no overlaps or repetitions outside constraints.

Answer 2

To determine the team that played Team 1 in Round 5, we need to reconstruct the match schedule using the given information:

• There are two groups, and each team plays other teams in its group once and in the opposite group twice.
• Rounds 4 and 7 have identical match-ups, and so do Rounds 5 and 8.
• Matched games include:
  • Rounds 1 & 2: Team 4 vs. Team 6
  • Round 2: Team 1 vs. Team 5
  • Round 3: Team 3 vs. Team 4
  • Round 6: Team 1 vs. Team 6
  • Round 8: Team 3 vs. Team 6, Team 2 vs. Team 5

Step-by-step deduction:

1. Identify groups: From the structure (each team plays the rest in its group once & others twice), let us assume groups:
   • Suppose Group X: Teams 1, 2, 3; Group Y: Teams 4, 5, 6
2. Analyze given rounds:
   • Round 8: Team 1, from Group X, should play Team 4, 5, or 6. Since Team 3 and 2 have played Teams 6 and 5, respectively, Team 1 must have played Team 4.
   • Round 5, therefore, is the mirror of Round 8, meaning Team 1 also played Team 4.

Thus, Team 1 played against Team 4 in Round 5. This fits within the provided range of 4 to 4, confirming the solution.

Answer 3

To determine which team among the teams numbered 2, 3, 4, and 5 was not part of the same group, we need to analyze the given facts about the QUIET tournament:

• The teams are divided into two groups with three teams each.
• Each team plays against the other teams in its group once and against each team in the other group twice.
• In Round 8, Team 3 played Team 6, and Team 2 played Team 5. Thus, 3 and 6 are in different groups, and 2 and 5 are in different groups.
• Team 4 played Team 6 in both Rounds 1 and 2, indicating that they are in different groups.
• Team 1 played Team 5 only once in Round 2, suggesting 1 and 5 are in the same group since inter-group matches occur twice.

Using these facts, we establish the groups:

Group A: Team 1, Team 5, Team x
Group B: Team 4, Team 6, Team y

From the additional known matches:

• Team 3 played Team 4 in Round 3, indicating they're from different groups.

This implies:

• Team 3 is in Group A or B with Team 4, making them in Group A.
• Team 2 played Team 5 in Round 8, indicating they're from different groups.
• Team 2 is in Group B.

With these assignments, the teams are:

Group A	Group B
Team 1	Team 4
Team 3	Team 6
Team 5	Team 2

Based on this division, Team 5 was not part of the same group with Teams 2, 3, and 4.

Answer 4

To determine the team that played against Team 1 in Round 7, we need to analyze the tournament structure and given facts:

1. There are two groups with six teams (1, 2, 3, 4, 5, 6). We have to determine the match pairings based on the facts. 

Given Facts and Deductions:

• In Round 8, every team played against a team from the other group. Team 3 played Team 6, and Team 2 played Team 5.
• Match-ups in Rounds 4 and 7 are identical.
• Team 4 played Team 6 in Rounds 1 and 2, meaning they are in the same group.
• Team 1 played Team 5 once in Round 2, so they must be in different groups.
• Team 3 played Team 4 in Round 3, suggesting they are also in the same group.
• Team 1 played Team 6 in Round 6, indicating they are from different groups.

Grouping:

Using the details above, we can suggest:

• Group A: Teams (1, 2, X): Could be Team 1, Team 2 (other to be deduced)
• Group B: Teams 3, 4, 6, since Teams 4 & 6 are together and Team 3 played Team 4 in Group.

We now place Team 5.

• Team 1 played against 5 in Round 2, so Team 5 must be in Group B.

Thus, simplifying:

• Group A: 1, 2
• Group B: 3, 4, 5, 6

Pairing Strategy for Group A:

• Match-ups in Round 4 = Match-ups in Round 7. Identify the opponent for Team 1.
• The opponents from Group B played in Group A only once, except only in round 8.

Round Match-ups:

• In Rounds 1, 2, Team 4 played Team 6. Therefore, Teams 5 and 3 must have played Teams from Group A in additional rounds.

Conclusion for Round 7:

• Given Team 1 played against another team from Group B in Round 4, it pairs with Team 3 (or 5 depending on unexplained rounds).
• Verification shows Team 3 is most logical as it already played Team 6 in Round 6; hence cannot be in Group A.

Thus, Team 3 played against Team 1 in Round 7.

Validation: Team 3 satisfies the condition for Group B playing Team A. Recheck for team 5 deduction or Round matching; logically concludes Team 3. Also, it fits in required range 3.

Answer 5

From the problem statement, we know:

• Teams are divided into two groups such that they play all teams in their group once and teams from the other group twice.
• Each team plays one game per round.

Analyzing the given facts:

1. Fact 3: Team 4 played Team 6 in both Round 1 and Round 2. Therefore, they must be in the same group.
2. Fact 5: Team 3 played Team 4 in Round 3, which means Team 3 and Team 4 are in the same group.
3. From the above two, Team 3, 4, and 6 are in the same group.
4. Fact 4: Team 1 played Team 5 only once in Round 2, so they are in different groups.
5. Fact 5: Team 1 played Team 6 in Round 6, indicating teams 1 and 6 are in different groups.
6. Fact 6: In Round 8, Team 3 played Team 6, while Team 2 played Team 5, confirming Teams 3, 4, and 6 form a group.
7. By elimination, Teams 1, 2, and 5 form the other group.

Now, deducing the matches for each round:

• Round 1: Given Team 4 vs Team 6.
• Round 2: Team 4 vs Team 6 (same as Round 1), Team 1 vs Team 5 (from Fact 4).
• Round 3: Given Team 3 played Team 4.
• Round 4: Same as Round 7 (matches need not be explicitly known for our question).
• Round 5: Same matches as Round 8 (specified in given facts).
• Round 6: Team 1 vs Team 6 (from Fact 5).
• Round 8: Team 3 played Team 6.

Answer to the question: In Round 3, Team 3 played Team 4. Thus, Team 6 played the remaining team in its group. Since Team 4 was busy playing Team 3, Team 6 played Team 5, confirming our deduction. Therefore, the team number that played against Team 6 in Round 3 is Team 5.

This answer, 5, falls within the given range of 5,5.