CAT — Games & Tournaments
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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?
Step 1:
There are two groups, each consisting of three teams.
Within each group, every team plays the other two teams once. In addition, each team plays every team from the other group exactly twice.
Hence, the number of matches played by each team is:
= 2 × 1 + 3 × 2 = 8
Since there are six teams in total, the total number of matches played is:
= (8 × 6)/2 = 24
Step 2:
From condition (3), Teams 4 and 6 belong to different groups.
From condition (4), Teams 1 and 5 are placed in the same group.
Conditions (5) and (6) allow several fixtures to be filled immediately.
Condition (1) states that Team 1 faced Team 4 in Round 8.
Therefore, the groups must be:
- Group 1: Teams (4, 2, 3)
- Group 2: Teams (6, 1, 5)
Using condition (2), Rounds 5 and 8 have identical fixtures.
The schedule at this stage is:
- Round 1: 4 vs 6
- Round 2: 4 vs 6, 1 vs 5
- Round 3: 3 vs 4
- Round 5: 3 vs 6, 2 vs 5, 1 vs 4
- Round 6: 1 vs 6
- Round 8: 3 vs 6, 2 vs 5, 1 vs 4
Step 3:
Applying condition (2) once again, Rounds 4 and 7 are also identical.
The complete schedule is therefore:
- Round 1: 4 vs 6, 1 vs 2, 3 vs 5
- Round 2: 4 vs 6, 1 vs 5, 2 vs 3
- Round 3: 3 vs 4, 1 vs 2, 5 vs 6
- Round 4: 4 vs 5, 2 vs 6, 1 vs 3
- Round 5: 3 vs 6, 2 vs 5, 1 vs 4
- Round 6: 1 vs 6, 4 vs 2, 3 vs 5
- Round 7: 4 vs 5, 2 vs 6, 1 vs 3
- Round 8: 3 vs 6, 2 vs 5, 1 vs 4
What is the number of the team that played Team 1 in Round 5?
Step 1:
There are two groups, each consisting of three teams.
Within each group, every team plays the other two teams once. In addition, each team plays every team from the other group exactly twice.
Hence, the number of matches played by each team is:
= 2 × 1 + 3 × 2 = 8
Since there are six teams in total, the total number of matches played is:
= (8 × 6)/2 = 24
Step 2:
From condition (3), Teams 4 and 6 belong to different groups.
From condition (4), Teams 1 and 5 are placed in the same group.
Conditions (5) and (6) allow several fixtures to be filled immediately.
Condition (1) states that Team 1 faced Team 4 in Round 8.
Therefore, the groups must be:
- Group 1: Teams (4, 2, 3)
- Group 2: Teams (6, 1, 5)
Using condition (2), Rounds 5 and 8 have identical fixtures.
The schedule at this stage is:
- Round 1: 4 vs 6
- Round 2: 4 vs 6, 1 vs 5
- Round 3: 3 vs 4
- Round 5: 3 vs 6, 2 vs 5, 1 vs 4
- Round 6: 1 vs 6
- Round 8: 3 vs 6, 2 vs 5, 1 vs 4
Step 3:
Applying condition (2) once again, Rounds 4 and 7 are also identical.
The complete schedule is therefore:
- Round 1: 4 vs 6, 1 vs 2, 3 vs 5
- Round 2: 4 vs 6, 1 vs 5, 2 vs 3
- Round 3: 3 vs 4, 1 vs 2, 5 vs 6
- Round 4: 4 vs 5, 2 vs 6, 1 vs 3
- Round 5: 3 vs 6, 2 vs 5, 1 vs 4
- Round 6: 1 vs 6, 4 vs 2, 3 vs 5
- Round 7: 4 vs 5, 2 vs 6, 1 vs 3
- Round 8: 3 vs 6, 2 vs 5, 1 vs 4
Which team among the teams numbered 2, 3, 4, and 5 was not part of the same group?
5
3
2
4
5
Step 1:
There are two groups, each consisting of three teams.
Within each group, every team plays the other two teams once. In addition, each team plays every team from the other group exactly twice.
Hence, the number of matches played by each team is:
= 2 × 1 + 3 × 2 = 8
Since there are six teams in total, the total number of matches played is:
= (8 × 6)/2 = 24
Step 2:
From condition (3), Teams 4 and 6 belong to different groups.
From condition (4), Teams 1 and 5 are placed in the same group.
Conditions (5) and (6) allow several fixtures to be filled immediately.
Condition (1) states that Team 1 faced Team 4 in Round 8.
Therefore, the groups must be:
- Group 1: Teams (4, 2, 3)
- Group 2: Teams (6, 1, 5)
Using condition (2), Rounds 5 and 8 have identical fixtures.
The schedule at this stage is:
- Round 1: 4 vs 6
- Round 2: 4 vs 6, 1 vs 5
- Round 3: 3 vs 4
- Round 5: 3 vs 6, 2 vs 5, 1 vs 4
- Round 6: 1 vs 6
- Round 8: 3 vs 6, 2 vs 5, 1 vs 4
Step 3:
Applying condition (2) once again, Rounds 4 and 7 are also identical.
The complete schedule is therefore:
- Round 1: 4 vs 6, 1 vs 2, 3 vs 5
- Round 2: 4 vs 6, 1 vs 5, 2 vs 3
- Round 3: 3 vs 4, 1 vs 2, 5 vs 6
- Round 4: 4 vs 5, 2 vs 6, 1 vs 3
- Round 5: 3 vs 6, 2 vs 5, 1 vs 4
- Round 6: 1 vs 6, 4 vs 2, 3 vs 5
- Round 7: 4 vs 5, 2 vs 6, 1 vs 3
- Round 8: 3 vs 6, 2 vs 5, 1 vs 4
What is the number of the team that played Team 1 in Round 7?
Step 1:
There are two groups, each consisting of three teams.
Within each group, every team plays the other two teams once. In addition, each team plays every team from the other group exactly twice.
Hence, the number of matches played by each team is:
= 2 × 1 + 3 × 2 = 8
Since there are six teams in total, the total number of matches played is:
= (8 × 6)/2 = 24
Step 2:
From condition (3), Teams 4 and 6 belong to different groups.
From condition (4), Teams 1 and 5 are placed in the same group.
Conditions (5) and (6) allow several fixtures to be filled immediately.
Condition (1) states that Team 1 faced Team 4 in Round 8.
Therefore, the groups must be:
- Group 1: Teams (4, 2, 3)
- Group 2: Teams (6, 1, 5)
Using condition (2), Rounds 5 and 8 have identical fixtures.
The schedule at this stage is:
- Round 1: 4 vs 6
- Round 2: 4 vs 6, 1 vs 5
- Round 3: 3 vs 4
- Round 5: 3 vs 6, 2 vs 5, 1 vs 4
- Round 6: 1 vs 6
- Round 8: 3 vs 6, 2 vs 5, 1 vs 4
Step 3:
Applying condition (2) once again, Rounds 4 and 7 are also identical.
The complete schedule is therefore:
- Round 1: 4 vs 6, 1 vs 2, 3 vs 5
- Round 2: 4 vs 6, 1 vs 5, 2 vs 3
- Round 3: 3 vs 4, 1 vs 2, 5 vs 6
- Round 4: 4 vs 5, 2 vs 6, 1 vs 3
- Round 5: 3 vs 6, 2 vs 5, 1 vs 4
- Round 6: 1 vs 6, 4 vs 2, 3 vs 5
- Round 7: 4 vs 5, 2 vs 6, 1 vs 3
- Round 8: 3 vs 6, 2 vs 5, 1 vs 4
What is the number of the team that played Team 6 in Round 3?
Step 1:
There are two groups, each consisting of three teams.
Within each group, every team plays the other two teams once. In addition, each team plays every team from the other group exactly twice.
Hence, the number of matches played by each team is:
= 2 × 1 + 3 × 2 = 8
Since there are six teams in total, the total number of matches played is:
= (8 × 6)/2 = 24
Step 2:
From condition (3), Teams 4 and 6 belong to different groups.
From condition (4), Teams 1 and 5 are placed in the same group.
Conditions (5) and (6) allow several fixtures to be filled immediately.
Condition (1) states that Team 1 faced Team 4 in Round 8.
Therefore, the groups must be:
- Group 1: Teams (4, 2, 3)
- Group 2: Teams (6, 1, 5)
Using condition (2), Rounds 5 and 8 have identical fixtures.
The schedule at this stage is:
- Round 1: 4 vs 6
- Round 2: 4 vs 6, 1 vs 5
- Round 3: 3 vs 4
- Round 5: 3 vs 6, 2 vs 5, 1 vs 4
- Round 6: 1 vs 6
- Round 8: 3 vs 6, 2 vs 5, 1 vs 4
Step 3:
Applying condition (2) once again, Rounds 4 and 7 are also identical.
The complete schedule is therefore:
- Round 1: 4 vs 6, 1 vs 2, 3 vs 5
- Round 2: 4 vs 6, 1 vs 5, 2 vs 3
- Round 3: 3 vs 4, 1 vs 2, 5 vs 6
- Round 4: 4 vs 5, 2 vs 6, 1 vs 3
- Round 5: 3 vs 6, 2 vs 5, 1 vs 4
- Round 6: 1 vs 6, 4 vs 2, 3 vs 5
- Round 7: 4 vs 5, 2 vs 6, 1 vs 3
- Round 8: 3 vs 6, 2 vs 5, 1 vs 4
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