CAT 2024 Slot 1 · DILR · Question 10

CAT 2024 Slot 1 previous year question on Games and Tournaments, with the answer and a step-by-step solution.

Games and TournamentsLOGICAL REASONING - High LevelType in the answer (TITA)Medium

Set 3: QUIET Tournament Schedule — this question belongs to a set. Read the full set with all 5 questions →

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?

Answer & solution

Answer: 8

Solving the set

Step 1 – Count the rounds. With groups of 3, each group has 3 same-group pairs (played once): 6 games. There are 3 × 3 = 9 cross-group pairs (played twice): 18 games. Total = 24 games. Each round has 3 games (six teams, each playing once), so there are 24/3 = 8 rounds.

Step 2 – Find the groups.

• Teams 1 and 5 played only once (fact 4), so they are in the same group.

• Round 8 has 3-6 and 2-5, so its third game is 1-4. Every Round 8 game is cross-group (fact 1), so 1 and 4 are in different groups.

• Teams 4 and 6 played twice (fact 3), so they are in different groups. Hence 6 is in the same group as 1.

Group	Teams
Group I	1, 5, 6
Group II	2, 3, 4

Check: 3-6 and 2-5 in Round 8 are both cross-group, as required.

Step 3 – Place the six same-group games (1-5, 1-6, 5-6, 2-3, 2-4, 3-4).

• Round 2 has 4-6 and 1-5, so its third game is 2-3.

• Round 3 has 3-4. Teams 1, 2, 5 and 6 remain. 1-5 (only in Round 2) and 1-6 (only in Round 6) are not possible, so Round 3 is 1-2 and 5-6.

• Round 6 has 1-6. Teams 2, 3, 4 and 5 remain. 2-3 and 3-4 are already used, so Round 6 is 2-4 and 3-5.

All six same-group games are now in Rounds 2, 3 and 6, so Rounds 1, 4, 5, 7 and 8 contain only cross-group games.

Step 4 – Round 1 has 4-6. Teams 1, 2, 3 and 5 remain. 1-5 is not possible. The other choices are (1-2, 3-5) or (1-3, 2-5). But 2-5 is already played twice (Rounds 5 and 8, since Round 5 = Round 8). So Round 1 is 1-2 and 3-5.

Step 5 – Round 5 = Round 8 = 1-4, 2-5, 3-6. Counting the cross-group games placed so far, the pairs still needing two games each are 1-3, 4-5 and 2-6. These six games fill Rounds 4 and 7, which must be identical: 1-3, 4-5, 2-6.

Round	Game 1	Game 2	Game 3	Type of games
1	1 v 2	3 v 5	4 v 6	All cross-group
2	1 v 5	2 v 3	4 v 6	Two same-group + one cross
3	1 v 2	3 v 4	5 v 6	Two same-group + one cross
4	1 v 3	4 v 5	2 v 6	All cross-group
5	1 v 4	2 v 5	3 v 6	All cross-group
6	1 v 6	2 v 4	3 v 5	Two same-group + one cross
7	1 v 3	4 v 5	2 v 6	All cross-group
8	1 v 4	2 v 5	3 v 6	All cross-group

Step 6 – Check every pair: each same-group pair appears once and each cross-group pair appears twice.

Cross-group pair	Rounds played	Same-group pair	Round played
1-2	1, 3	1-5	2
1-3	4, 7	1-6	6
1-4	5, 8	5-6	3
5-2	5, 8	2-3	2
5-3	1, 6	2-4	6
5-4	4, 7	3-4	3
6-2	4, 7
6-3	5, 8
6-4	1, 2

This question

From Step 1: 6 same-group games + 18 cross-group games = 24 games, at 3 games per round, so 8 rounds. This also matches the facts, which mention Round 8 as the last round used.

Hence, the answer is 8.

CAT 2024 Slot 1 · DILR · Question 10

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