Understanding the Setup

We are given a scenario involving eight individuals named A, B, P, C, D, O, Q, and E. They are seated around a circular table, and importantly, they are all facing the center of the table. Our goal is to determine their exact positions based on the provided clues and then identify who is sitting immediately to the left of person B.

Analyzing the Constraints

Let's break down the information provided:

• People: A, B, P, C, D, O, Q, E (8 total).
• Table: Circular, 8 positions, all facing inward.
• Vowel Rule: Individuals whose names start with a vowel (A, E, O) cannot sit in adjacent seats.
• Clue 1: A is positioned 3rd to the right of D. Mathematically, if D's position is $P_D$, then A's position is $P_A = (P_D + 3) \pmod 8$.
• Clue 2: There are exactly two people seated between P and A. This implies the distance between P and A is 3 positions.
• Clue 3: B is seated next to O (adjacent).
• Clue 4: O is positioned 4th to the right of E. If E's position is $P_E$, then O's position is $P_O = (P_E + 4) \pmod 8$.
• Clue 5: Q is positioned 3rd to the left of O. If O's position is $P_O$, then Q's position is $P_Q = (P_O - 3) \pmod 8$.

Step-by-Step Deduction of the Seating Arrangement

1. Placing E and O: We start with Clue 4: O sits 4th to the right of E. Let's assign E the first position, so $P_E = 1$. Calculating O's position: $P_O = (P_E + 4) \pmod 8 = (1 + 4) \pmod 8 = 5$. O is at position 5.
   Current layout: E (Pos 1), _ (Pos 2), _ (Pos 3), _ (Pos 4), O (Pos 5), _ (Pos 6), _ (Pos 7), _ (Pos 8).
2. Placing Q: Next, use Clue 5: Q sits 3rd to the left of O. Since O is at position 5 ($P_O = 5$), we calculate Q's position: $P_Q = (P_O - 3) \pmod 8 = (5 - 3) \pmod 8 = 2$. Q is at position 2.
   Current layout: E (Pos 1), Q (Pos 2), _ (Pos 3), _ (Pos 4), O (Pos 5), _ (Pos 6), _ (Pos 7), _ (Pos 8).
3. Identifying Possible Seats for A: A is a vowel. The Vowel Rule states A cannot be adjacent to E (Pos 1) or O (Pos 5). This eliminates positions 2, 8 (next to E) and 4, 6 (next to O). The only remaining seats for A are position 3 or position 7.
4. Using the D-A Relationship (Clue 1): A sits 3rd to the right of D ($P_A = (P_D + 3) \pmod 8$).
   • If A is at position 3 ($P_A = 3$), then $3 = (P_D + 3) \pmod 8$. Solving for $P_D$, we get $P_D = 0 \pmod 8$, which means D is at position 8.
   • If A is at position 7 ($P_A = 7$), then $7 = (P_D + 3) \pmod 8$. Solving for $P_D$, we get $P_D = 4 \pmod 8$, which means D is at position 4.
5. Testing the Scenarios for A:
   • Scenario 1: Assume $P_A = 3$. Based on the deduction above, $P_D = 8$. The arrangement is: D (Pos 8), E (Pos 1), Q (Pos 2), A (Pos 3), _ (Pos 4), O (Pos 5), _ (Pos 6), _ (Pos 7).
     Now, let's use Clue 2: Two people between P and A. Since $P_A = 3$, P must be 3 positions away. This means $P_P = (3 + 3) \pmod 8 = 6$. P is at position 6.
     Layout: D (8), E (1), Q (2), A (3), _ (4), O (5), P (6), _ (7).
     Next, use Clue 3: B sits adjacent to O. O is at position 5. Adjacent seats are 4 and 6. P is already at 6, so B must be at position 4 ($P_B = 4$).
     Layout: D (8), E (1), Q (2), A (3), B (4), O (5), P (6), _ (7).
     The only remaining person is C, who must be in the last seat, position 7 ($P_C = 7$).
     This gives the full arrangement: D, E, Q, A, B, O, P, C (clockwise). Let's verify all clues: Vowel rule (A, E, O not adjacent) holds. D-A relation holds ($P_A = P_D + 3$). P-A distance holds (2 people B, O between them). B-O adjacency holds. E-O relation holds ($P_O = P_E + 4$). O-Q relation holds ($P_Q = P_O - 3$). This scenario is valid.
   • Scenario 2: Assume $P_A = 7$. Based on the deduction above, $P_D = 4$. The arrangement is: E (Pos 1), Q (Pos 2), _ (Pos 3), D (Pos 4), O (Pos 5), A (Pos 7), _ (Pos 6), _ (Pos 8).
     Now, let's use Clue 2: Two people between P and A. With $P_A = 7$, P must be at position 4 ($P_P = (7 - 3) \pmod 8 = 4$) or position 2 ($P_P = (7 + 3) \pmod 8 = 10 \pmod 8 = 2$).
     However, position 4 is occupied by D, and position 2 is occupied by Q. This directly contradicts Clue 2. Therefore, this scenario is impossible.
6. Final Seating Arrangement: Based on the analysis, the only consistent arrangement (reading clockwise) is: E, Q, A, B, O, P, C, D.
   Here is the arrangement shown in a table format:

   Position	Person
   1	E
   2	Q
   3	A
   4	B
   5	O
   6	P
   7	C
   8	D

Identifying the Person to B's Immediate Left

The question asks who is sitting at the immediate left of B.

In our confirmed arrangement, B is located at position 4 ($P_B = 4$).

The position immediately to the left of B is found by moving counter-clockwise. This corresponds to position 3 ($P_{Left\_of\_B} = (P_B - 1) \pmod 8 = (4 - 1) \pmod 8 = 3$).

Looking at our table, the person at position 3 is A.

Thus, A is the person sitting immediately to the left of B.