Six people A, B, C, X, Y,Z sat in two rows of three seats each. Three people in one row sat opposite to the three people in another row. X sat to the right of Y. B was sitting opposite to X. A sat diagonally opposite to B. C is sitting diagonally opposite to X but not opposite to Y. Which of the following options was true in this case?

To solve this seating arrangement problem, let's break it down step-by-step using the given clues.

We need to arrange six people \( A, B, C, X, Y, Z \) in two rows with three seats each. We will label the seats in the following way:
- Row 1: \( A1, A2, A3 \)
- Row 2: \( B1, B2, B3 \)

1. **Clue 1: X sat to the right of Y.**
   This means if Y is at \( A2 \), X must be at \( A3 \) (assuming \( A1 \) is the leftmost seat).

2. **Clue 2: B was sitting opposite to X.**
   Since X is in \( A3 \), B must be in \( B3 \).

3. **Clue 3: A sat diagonally opposite to B.**
   This means A can be in either \( A1 \) or \( A2 \) (since B is in \( B3 \)).

4. **Clue 4: C is sitting diagonally opposite to X but not opposite to Y.**
   Since X is in \( A3 \), diagonally opposite to X can be either \( B1 \) or \( B2 \). However, it cannot be \( B2 \) because that would place Y directly opposite to X. Therefore, C must be in \( B1 \).

Now we have the following seating:
- \( A3 = X \)
- \( B3 = B \)
- \( B1 = C \)

We still need to place \( A, Y, Z \):
- \( Y \) is in \( A2 \) (since X is to the right of Y).
- \( A \) should be diagonally opposite to \( B \). Since \( B \) is in \( B3 \), \( A \) should be in \( A1 \).

Finally, \( Z \) will take the remaining seat, which is \( B2 \).

The final seating arrangement is:
- Row 1: \( A1 = A, A2 = Y, A3 = X \)
- Row 2: \( B1 = C, B2 = Z, B3 = B \)

### Verification:
- \( X \) is to the right of \( Y \): \( Y = A2, X = A3 \).
- \( B \) is opposite \( X \): \( X = A3, B = B3 \).
- \( A \) is diagonally opposite \( B \): \( A = A1, B = B3 \).
- \( C \) is diagonally opposite \( X \) but not opposite \( Y \): \( C = B1, X = A3, Y = A2 \).

Therefore, the final seating arrangement satisfies all the given conditions.

### Conclusion:
The following statements are true:
- \( A \) is in \( A1 \).
- \( Y \) is in \( A2 \).
- \( X \) is in \( A3 \).
- \( C \) is in \( B1 \).
- \( Z \) is in \( B2 \).
- \( B \) is in \( B3 \).

Would you like more details or have any questions? Here are some related questions:

1. What other possible seating arrangements could exist with different assumptions?
2. How can we represent the seating arrangement graphically?
3. What strategies can be used to solve similar logical arrangement problems?
4. How does the position of \( A \) affect the positions of others?
5. Can the seating arrangement be changed if the row configurations are swapped?
6. What is the importance of the word "diagonally" in such problems?
7. How does understanding the relative positions help in solving logical puzzles?
8. What are other types of logical arrangement puzzles?

**Tip:** When dealing with seating arrangements, always visualize the setup and use elimination methods to narrow down possibilities.

Learn how to solve a complex logical seating arrangement problem involving six people: A, B, C, X, Y, Z. This problem requires logical reasoning skills to determine the positions based on given clues. Suitable for high school students and logical puzzle enthusiasts.

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