Eight gymnastics players numbered 1 through 8 underwent a training camp where they were coached by three coaches- Xena, Yuki, and Zara. Each coach trained at least two players. Yuki trained only even-numbered players, while Zara trained only odd-numbered players. After the camp, the coaches evaluated the players and gave integer ratings to the respective players trained by them on a scale of 1 to 7, with 1 being the lowest rating and 7 the highest. The following additional information is known:

1. Xena trained more players than Yuki.

2. Player-1 and Player-4 were trained by the same coach, while the coaches who trained Player-2, Player-3, and Player-5 were all different.

3. Player-5 and Player-7 were trained by the same coach and got the same rating. All other players got a unique rating.

4. The average of the ratings of all the players was 4.

5. Player-2 got the highest rating.

6. The average of the ratings of the players trained by Yuki was twice that of the players trained by Xena and two more than that of the players trained by Zara.

7. Player-4’s rating was double of Player-8’s and less than Player-5’s.

Who all were the players trained by Xena?

The correct option is Player-1, Player-3, Player-4, Player-8.

Let us break down the logical reasoning step-by-step to determine the players trained by Xena:

Step 1: Understand the coach constraints for even and odd players
- Yuki (Y) trained only even-numbered players (from {2, 4, 6, 8}).
- Zara (Z) trained only odd-numbered players (from {1, 3, 5, 7}).
- Xena (X) has no such restrictions and can train both odd and even-numbered players.
- Each coach trained at least two players.

Step 2: Identify the coach of Player-1 and Player-4
- Clue 2 states that Player-1 (odd) and Player-4 (even) were trained by the same coach.
- Since Yuki cannot train Player-1 (odd) and Zara cannot train Player-4 (even), the only coach who could have trained both of them is Xena.
- Therefore, Player-1 and Player-4 were trained by Xena.

Step 3: Analyze the coaches of Player-2, Player-3, and Player-5
- Clue 2 states that the coaches who trained Player-2, Player-3, and Player-5 were all different.
- Player-2 is even, so Player-2's coach is either Yuki or Xena.
- Player-3 and Player-5 are odd, so their coaches are either Zara or Xena.
- Since the three coaches must be different:
- One of them is trained by Yuki. Since only Player-2 is even, Player-2 must be trained by Yuki. - One of them is trained by Xena. - One of them is trained by Zara.

Step 4: Determine the assignments of Player-5 and Player-7
- Clue 3 states that Player-5 and Player-7 were trained by the same coach.
- Since Player-5 and Player-7 are odd, their coach is either Zara or Xena.
- If they were trained by Xena, then the coach of Player-5 would be Xena. This would force Player-3 to be trained by Zara (to ensure coaches of 2, 3, and 5 are different).
- If they were trained by Zara, then the coach of Player-5 is Zara. This forces Player-3 to be trained by Xena.
- Comparing the options, if Player-3 is trained by Xena, we have:
- Xena's players: Player-1, Player-3, Player-4, and Player-8. - Zara's players: Player-5 and Player-7 (satisfying Clue 3). - Yuki's players: Player-2 and Player-6.
- Let us check the number of players for each coach under this assignment:
- Xena trained 4 players (Player-1, Player-3, Player-4, Player-8). - Yuki trained 2 players (Player-2, Player-6). - Zara trained 2 players (Player-5, Player-7).
- This perfectly satisfies Clue 1 (Xena trained more players than Yuki, as 4 > 2) and the condition that each coach trained at least two players.

Thus, the players trained by Xena are Player-1, Player-3, Player-4, Player-8.