CAT 2024 Slot 1 — DILR Question 12
Answer the following questions based on the information given below.
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 1 in Round 7?
Answer & solution
Answer: 3
Hard
With the groups fixed (X={1,5,6}, Y={2,3,4}) and the grid built, Round 7 equals Round 4 (Fact 2). Find Round 4’s match-ups, then read Team 1’s opponent.
Full schedule (Group X={1,5,6}, Group Y={2,3,4}):
| Round | Games |
|---|---|
| 1 | 1–2, 3–5, 4–6 |
| 2 | 1–5, 2–3, 4–6 |
| 3 | 1–2, 3–4, 5–6 |
| 4 | 1–3, 2–6, 4–5 |
| 5 | 1–4, 2–5, 3–6 |
| 6 | 1–6, 2–4, 3–5 |
| 7 | 1–3, 2–6, 4–5 |
| 8 | 1–4, 2–5, 3–6 |
Locate the remaining inter-group games for R4/R7. After placing every fixed and intra-group game, the only pairs left to schedule (each still owed two meetings) are 1–3, 4–5 and 2–6 — and together they use all six teams.
Fact 2 fills R4 and R7. Those three pairs must occupy both Round 4 and Round 7 (the two rounds with identical match-ups), giving each pair its two meetings.
Read Team 1’s Round 7 opponent. The pair containing 1 is 1–3.