CAT 2024 Slot 2 · DILR · Question 3

Solving the set

Step 1 – Coaches who can train each player. Yuki trains only even players and Zara only odd players. Xena can train anyone.

• Player-1 (odd) and Player-4 (even) have the same coach (fact 2). Yuki cannot take Player-1 and Zara cannot take Player-4, so both are with Xena.

• Players 2, 3 and 5 have three different coaches (fact 2). Player-2 can be with Xena or Yuki; Players 3 and 5 can be with Xena or Zara. Since only one of them can be with Xena, Players 3 and 5 take Xena and Zara between them, and Player-2 goes to Yuki.

Step 2 – Players 5 and 7 have the same coach (fact 3). If Player-5 were with Xena, Player-7 would also be with Xena and Player-3 with Zara. Zara could then take no other odd player (1, 5 and 7 are with Xena), so Zara would train only one player, which is not allowed. Hence Players 5 and 7 are with Zara and Player-3 is with Xena. Zara trains exactly Players 5 and 7.

Step 3 – Ratings. There are 8 players but only 7 possible ratings, and only Players 5 and 7 share a rating. So the ratings are 1, 2, …, 7 once each, plus one repeated value v for Players 5 and 7. The total is 8 × 4 = 32 (fact 4), so 28 + v = 32 and v = 4. Player-5 = Player-7 = 4.

Step 4 – Player-2 has the highest rating, 7 (fact 5). Player-4 = 2 × Player-8 and Player-4 < Player-5 = 4, so Player-4 = 2 and Player-8 = 1.

Step 5 – Averages (fact 6). Zara's average = (4 + 4)/2 = 4, so Yuki's average = 4 + 2 = 6 and Xena's average = 6/2 = 3.

Step 6 – Yuki has Player-2 (7) and needs at least one more even player: 6 or 8. Including Player-8 (rating 1) cannot give an average of 6 (with Player-6 as well, 7 + 1 + P6 = 18 would need P6 = 10). So Yuki has Players 2 and 6, with (7 + P6)/2 = 6, i.e., Player-6 = 5. Player-8 goes to Xena.

Step 7 – Xena has Players 1, 3, 4 and 8 (4 players, more than Yuki's 2 ✓). The ratings left are 3 and 6 for Players 1 and 3. Xena's total = 3 + 6 + 2 + 1 = 12, average 3 ✓, whichever way 3 and 6 are assigned.

Player	Coach	Rating	Reason
1	Xena	3 or 6	Same coach as Player-4 (fact 2)
2	Yuki	7	Highest rating (fact 5)
3	Xena	6 or 3	Different coach from 2 and 5
4	Xena	2	2 × Player-8 and less than 4 (fact 7)
5	Zara	4	Repeated rating (facts 3, 4)
6	Yuki	5	Makes Yuki's average 6
7	Zara	4	Same coach and rating as Player-5
8	Xena	1	Half of Player-4's rating

Coach	Players	Ratings	Total	Average
Xena	1, 3, 4, 8	3/6, 6/3, 2, 1	12	3
Yuki	2, 6	7, 5	12	6
Zara	5, 7	4, 4	8	4

This question

From Step 6, Yuki trains Players 2 and 6 and averages 6: (7 + Player-6)/2 = 6, so Player-6 = 5.

Hence, the answer is 5.