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CAT 2024 Slot 1 — DILR Question 13

Round Robin TournamentEasy
Passage / Data

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 6 in Round 3?

Answer & solution

Answer: 5

Solution

Medium

Round 3 is partly fixed: Team 3 plays Team 4 (Fact 5). The other four teams (1, 2, 5, 6) must pair up under the constraints, which forces Team 6’s opponent.

Full schedule (Group X={1,5,6}, Group Y={2,3,4}):

RoundGames
11–2, 3–5, 4–6
21–5, 2–3, 4–6
31–2, 3–4, 5–6
...(remaining rounds as derived)
1

Start from Fact 5. In Round 3, Team 3 plays Team 4. That leaves teams 1, 2, 5 and 6 to form two pairs.

R3 contains 3-4(Fact 5)⇒ pair up {1,2,5,6}\begin{aligned} &\text{R3 contains } 3\text{-}4 \quad\text{(Fact 5)}\\ &\Rightarrow\ \text{pair up } \{1,2,5,6\} \end{aligned}
2

Eliminate the impossible pairings. 1–5 happens only in Round 2 (Fact 4), so not in R3. 1–6 happens only in Round 6 (Fact 5), so not in R3. Hence Team 1 must play Team 2 in R3.

1-5 barred (Fact 4),1-6 barred (Fact 5)⇒ Team 1 plays Team 2\begin{aligned} &1\text{-}5\ \text{barred (Fact 4)},\quad 1\text{-}6\ \text{barred (Fact 5)}\\ &\Rightarrow\ \text{Team 1 plays Team 2} \end{aligned}
3

What is left meets Team 6. With 1–2 and 3–4 placed, the remaining teams 5 and 6 must play each other.

R3={3-4, 1-2, 5-6} ⇒ Team 6 plays Team 5\begin{aligned} &\text{R3}=\{3\text{-}4,\ 1\text{-}2,\ 5\text{-}6\}\ \Rightarrow\ \text{Team 6 plays Team 5} \end{aligned}
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CAT 2024 Slot 1 DILR Q13: What is the number of the team that played Team 6 in Round 3? — Solution | TheCATExam