CAT 2024 Slot 1 — DILR Question 10

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 1 in Round 5?

Answer & solution

Answer: 4

Solution

Hard

First settle the two groups, then build the 8-round grid from the fixed fixtures and the “identical match-up” facts (R4=R7, R5=R8). Round 5 is then read straight off the grid.

Groups. Teams 1–5 play only once (Fact 4) ⇒ same group; 4–6 play twice (Fact 3) ⇒ different groups. Round 8 is all inter-group (Fact 1) with 3–6 and 2–5 (Fact 6), leaving 1–4, so 1 and 4 are in different groups. With 1–6 playing once (Fact 5, R6) they share a group. This forces:

Group X = {1, 5, 6} Group Y = {2, 3, 4}.

Full schedule (3 games per round):

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

Round 5 = Round 8 (Fact 2). Round 8 is given as 3–6, 2–5 (Fact 6), and the leftover pair 1–4. Identical match-ups make Round 5 the same.

\begin{aligned} &\text{R8}=\{3\text{-}6,\ 2\text{-}5,\ 1\text{-}4\} \quad\text{(Fact 6 + leftover)}\\ &\Rightarrow\ \text{R5}=\{3\text{-}6,\ 2\text{-}5,\ 1\text{-}4\} \quad\text{(R5}=\text{R8, Fact 2)} \end{aligned}

Read Team 1’s Round 5 opponent. In Round 5 the pair containing 1 is 1–4.

\begin{aligned} &\text{Team 1 plays Team 4 in Round 5} \end{aligned}

4