Understanding the Circular Table Arrangement Puzzle

This problem involves arranging eight people (A, B, P, C, D, O, Q, E) around a circular table facing inward. We need to determine the seating arrangement based on several given conditions and then identify who sits opposite person O. The key constraints are:

• Eight persons: A, B, P, C, D, O, Q, E.
• Circular table, all facing inward.
• Vowels (A, E, O) do not sit adjacent to each other.
• A is seated $3^{rd}$ to the right of D.
• Exactly two people sit between P and A.
• B is seated adjacent to O.
• O is seated $4^{th}$ to the right of E.
• Q is seated $3^{rd}$ to the left of O.

Step-by-Step Deduction

Let's determine the seating arrangement step by step:

1. Positioning E and O: We are given that "O sits $4^{th}$ to the right of E". Let's assume E is at position 1 in a clockwise arrangement (positions 1 to 8). "Right" means clockwise. Therefore, O is 4 positions clockwise from E. If E is at position 1, O is at position $1 + 4 = 5$. The arrangement looks like: E _ _ _ O _ _ _
2. Positioning Q: We know "Q sits $3^{rd}$ to the left of O". "Left" means counter-clockwise. From O's position (5), the $3^{rd}$ position counter-clockwise is $5 - 3 = 2$. So, Q is at position 2. The arrangement now looks like: E Q _ _ O _ _ _
3. Positioning B: The condition states "B sits adjacent to O". O is at position 5. The adjacent positions are 4 and 6. This means B could be at position 4 or position 6.
4. Applying the Vowel Constraint: Persons with vowel names (A, E, O) cannot sit next to each other. We have E at position 1 and O at position 5. A cannot be adjacent to E, meaning A cannot be at position 2 (occupied by Q) or position 8. A cannot be adjacent to O, meaning A cannot be at position 4 or position 6. Considering the available spots (3, 4, 6, 7, 8), A cannot be at 2, 4, 6, or 8. Thus, A must be at either position 3 or position 7.
5. Testing Possibility 1 (B at position 4): Let's assume B is at position 4. The arrangement is: E Q _ B O _ _ _ * If A is at position 3: The arrangement becomes E Q A B O _ _ _ Now, apply "A sits $3^{rd}$ to the right of D". If A is at position 3, D must be at position $3 - 3 = 0$, which wraps around to position 8. The arrangement becomes: E Q A B O _ _ D Next, apply "Two persons sit between P and A". Since A is at position 3, P must be at position $3 + 3 = 6$ (as position $3-3=0$ i.e. 8 is occupied by D). The arrangement becomes: E Q A B O P _ D The only person left is C, who must occupy the remaining position 7. The complete clockwise arrangement is: E, Q, A, B, O, P, C, D. Let's verify vowel adjacency: E (pos 1) is adjacent to Q (pos 2) and D (pos 8). A (pos 3) is adjacent to Q (pos 2) and B (pos 4). O (pos 5) is adjacent to B (pos 4) and P (pos 6). None of the vowels A, E, O are adjacent to each other. This arrangement satisfies all conditions. * If A is at position 7 (with B at 4): The arrangement is E Q _ B O _ A _ Apply "A sits $3^{rd}$ to the right of D". If A is at position 7, D must be at position $7 - 3 = 4$. However, position 4 is occupied by B. This scenario is invalid.
6. Testing Possibility 2 (B at position 6): Let's assume B is at position 6. The arrangement is: E Q _ _ O B _ _ * If A is at position 3: The arrangement is E Q A _ O B _ _ Apply "A sits $3^{rd}$ to the right of D". If A is at position 3, D must be at position 8. The arrangement is: E Q A _ O B _ D Apply "Two persons sit between P and A". Since A is at position 3, P must be at position $3 + 3 = 6$. But position 6 is occupied by B. This scenario is invalid. * If A is at position 7: The arrangement is E Q _ _ O B A _ Apply "A sits $3^{rd}$ to the right of D". If A is at position 7, D must be at position $7 - 3 = 4$. The arrangement is: E Q _ D O B A _ Apply "Two persons sit between P and A". Since A is at position 7, P must be at position $7 - 3 = 4$. But position 4 is occupied by D. This scenario is invalid.
7. Final Arrangement Confirmation: The only arrangement that satisfies all conditions is E, Q, A, B, O, P, C, D in clockwise order.

Final Seating Arrangement

The confirmed seating arrangement around the circular table, starting from an arbitrary position and moving clockwise, is as follows:

Identifying the Person Opposite O

To find the person sitting opposite O, we look at the person seated exactly 4 positions away in the circular arrangement (since there are 8 people in total, the opposite position is $8 / 2 = 4$ seats away).

O is at position 5.

The person opposite O would be at position $5 + 4 = 9$. Since there are only 8 positions, we wrap around. Position 9 is equivalent to position 1 ($9 \mod 8 = 1$).

The person at position 1 is E.

Therefore, E is sitting opposite O.