Circular Seating Arrangement Guide & Practice

Question

Six students—Aman, Bina, Charan, Divya, Esha, and Fiza—are sitting around a circular table facing towards the center.

1. Bina is sitting between Fiza and Charan.
2. Aman is sitting between Esha and Divya.
3. Fiza is sitting to the immediate right of Divya.

Determine the seating arrangement and find who sits opposite to Divya.

Solution

1. Set up the Positions: Draw a circle with 6 seats. Number them clockwise from Seat 1 (bottom/6 o'clock) to Seat 6 (4 o'clock):

   • Seat 1 (6 o'clock)
   • Seat 2 (8 o'clock)
   • Seat 3 (10 o'clock)
   • Seat 4 (12 o'clock)
   • Seat 5 (2 o'clock)
   • Seat 6 (4 o'clock) Since all face the center, clockwise is Left and anti-clockwise is Right.

2. Apply Clue 3: "Fiza is sitting to the immediate right of Divya."

   • Place Divya at Seat 1.
   • Since Divya faces inward, her immediate right (anti-clockwise) is Seat 6.
   • Place Fiza at Seat 6.

3. Apply Clue 1: "Bina is sitting between Fiza and Charan."

   • Fiza is at Seat 6. Bina must sit next to Fiza.
   • The seat adjacent to Seat 6 is either Seat 1 (occupied by Divya) or Seat 5.
   • Thus, place Bina at Seat 5.
   • To place Bina between Fiza and Charan, Charan must sit at Seat 4.

4. Apply Clue 2: "Aman is sitting between Esha and Divya."

   • Divya is at Seat 1. The seat adjacent to Seat 1 on the other side is Seat 2.
   • Place Aman at Seat 2.
   • To place Aman between Esha and Divya, Esha must sit at Seat 3.

5. Final Seating Verification:

   • Seat 1: Divya
   • Seat 2: Aman
   • Seat 3: Esha
   • Seat 4: Charan
   • Seat 5: Bina
   • Seat 6: Fiza All clues are satisfied.

6. Identify the Opposite Person:

   • In a 6-person circle, the opposite of Seat $i$ is Seat $i + 3$.
   • The seat opposite to Divya (Seat 1) is Seat 4, which is occupied by Charan.

Question

Eight friends—A, B, C, D, E, F, G, and H—are sitting around a circular table. Some face inside (towards the center) and others face outside (away from the center). They each play a different sport: Polo, Cricket, Football, Golf, Badminton, Chess, Tennis, and Hockey.

1. A sits third to the right of H, who likes Polo. A likes Chess.
2. B sits second to the left of A.
3. F sits third to the left of A and is not an immediate neighbour of A.
4. E sits opposite B. E faces outside and does not like Golf.
5. C sits second to the left of B. The one who likes Football sits to the immediate left of C.
6. D likes Badminton and sits second to the left of F.
7. The one who likes Tennis is not an immediate neighbour of H or B.
8. G sits adjacent to A.
9. The one who likes Golf and the one who likes Hockey sit opposite to each other.
10. The immediate neighbours of A face the direction opposite to that of B.
11. The immediate neighbours of H face the same direction as F.

Determine the seating arrangement, facing directions, and sports of all eight friends.

Solution

1. Set up the Positions: Draw a circle with 8 seats, labeled 0 to 7 in a clockwise direction. Let's place A at Seat 0. A likes Chess.
2. Determine B's position: "B sits second to the left of A."
   • Case 1 (A faces inside): Left is clockwise. B is placed at Seat 2.
   • Case 2 (A faces outside): Left is anti-clockwise. B is placed at Seat 6.
3. Determine F and E's positions:
   • "F sits third to the left of A."
     • In Case 1: F is placed at Seat 3 (since A faces inside).
     • In Case 2: F is placed at Seat 5 (since A faces outside).
   • "E sits opposite B."
     • In Case 1: Since B is at Seat 2, E sits opposite at Seat 6.
     • In Case 2: Since B is at Seat 6, E sits opposite at Seat 2.
4. Determine C's position and B's direction: "C sits second to the left of B."
   • In Case 1 (B at Seat 2):
     • If B faces inside, second to the left is Seat 4.
     • If B faces outside, second to the left is Seat 0 (already occupied by A).
     • Thus, B must face inside and C is placed at Seat 4.
   • In Case 2 (B at Seat 6):
     • If B faces inside, second to the left is Seat 0 (occupied by A).
     • If B faces outside, second to the left is Seat 4.
     • Thus, B must face outside and C is placed at Seat 4.
5. Determine H's position and direction: "A sits third to the right of H, who likes Polo."
   • In Case 1: The remaining empty seats are 1, 5, and 7. Since H sits 3 seats away from A (Seat 0), H must be at Seat 5.
     • For A (Seat 0) to be third to the right of H (Seat 5): H must face outside (right is clockwise; 5 -> 6 -> 7 -> 0).
   • In Case 2: The remaining empty seats are 1, 3, and 7. Thus, H must be at Seat 3.
     • For A (Seat 0) to be third to the right of H (Seat 3): H must face inside (right is anti-clockwise; 3 -> 2 -> 1 -> 0).
6. Determine D's position and F's direction: "D likes Badminton and sits second to the left of F."
   • In Case 1: F is at Seat 3. The empty seats are 1 and 7. D must sit at Seat 1 (second to the left of F). For Seat 1 to be to the left of Seat 3, F must face outside (left is anti-clockwise; 3 -> 2 -> 1).
   • In Case 2: F is at Seat 5. The empty seats are 1 and 7. D must sit at Seat 7. For Seat 7 to be to the left of Seat 5, F must face inside (left is clockwise; 5 -> 6 -> 7).
7. Position G:
   • In Case 1: G sits at the remaining Seat 7.
   • In Case 2: G sits at the remaining Seat 1.
8. Determine Sports and Eliminate Case 2:
   • "The one who likes Tennis is not an immediate neighbour of H or B."
     • In Case 1: H is at Seat 5, B is at Seat 2. Neighbors of H are 4 (C) and 6 (E). Neighbors of B are 1 (D) and 3 (F).
     • The Tennis player cannot be C, E, D, or F. Thus, G (Seat 7) must play Tennis.
     • In Case 2: H is at Seat 3, B is at Seat 6. Neighbors of H are 2 (E) and 4 (C). Neighbors of B are 5 (F) and 7 (D).
     • The Tennis player cannot be E, C, F, or D. Thus, G (Seat 1) must play Tennis.
   • "The one who likes Golf and the one who likes Hockey sit opposite to each other."
     • In both cases, the only opposite pair available with unassigned sports is B and E.
     • E does not like Golf, so E likes Hockey and B likes Golf.
   • "The immediate neighbours of A face the direction opposite to that of B."
     • In Case 1: B faces inside. Neighbors of A (G at Seat 7 and D at Seat 1) must face outside.
     • In Case 2: B faces outside. Neighbors of A (G at Seat 1 and D at Seat 7) must face inside.
   • "The immediate neighbours of H face the same direction as F."
     • In Case 1: H is at Seat 5. Neighbors of H are C(4) and E(6). Since F faces outside, C and E must face outside. This matches the clue "E faces outside."
     • In Case 2: H is at Seat 3. Neighbors of H are E(2) and C(4). Since F faces inside, E and C must face inside. This contradicts the clue "E faces outside."
     • Thus, Case 2 is eliminated.
9. Final Sport Assignments for Case 1:
   • "The one who likes Football sits to the immediate left of C."
     • C is at Seat 4 and faces outside. Its immediate left (anti-clockwise) is Seat 3.
     • Thus, F at Seat 3 likes Football.
     • The remaining sport, Cricket, is assigned to C at Seat 4.

Final Seating Matrix (Moderate Example)

Question

Ten friends—five boys (P, Q, R, S, T) and five girls (A, E, I, O, U)—are sitting around a circular table with equal spacing. Four of them (two boys and two girls) face outside the center, while the other six face towards the center.

1. A is fourth to the left of O, who is third to the right of P.
2. P and E sit opposite to each other and face each other.
3. R sits second to the right of T, who is not sitting next to E.
4. There are exactly three friends sitting between I and R.
5. S sits second to the left of U and faces towards the center.
6. I sits second to the left of Q.
7. A faces outside the center.
8. U is not facing outside the center (faces inside).
9. Gender Constraint: No more than three boys or three girls can sit in consecutive seats.
10. Facing Constraint: No more than two persons facing outside the center can sit in consecutive seats.

Determine the exact seating sequence, gender of each seat, and their facing directions.

Solution

1. Define Positions & Opposite Pair:

   • Let the seats be numbered 0 to 9 in a clockwise direction.
   • P and E sit opposite each other and face each other (meaning both face inside).
   • Place P (Boy) at Seat 0 (faces inside).
   • Place E (Girl) at Seat 5 (faces inside).

2. Position O and A:

   • "O is third to the right of P."
     • P is at Seat 0 and faces inside. Right is anti-clockwise. Thus, O is placed at Seat 7.
   • "A is fourth to the left of O."
     • A is a girl and faces outside.
     • Case A (O faces inside): Left is clockwise. Fourth to the left of O (Seat 7) is Seat 1. A is placed at Seat 1.
     • Case B (O faces outside): Left is anti-clockwise. Fourth to the left of O (Seat 7) is Seat 3. A is placed at Seat 3.

3. Analyze Case A (A at Seat 1, O faces inside):

   • S sits second to the left of U. Both face inside (since U does not face outside).
   • In clockwise numbering, $S = U + 2$ because U faces inside (left is clockwise).
   • Vacant seats for U and S are among {2, 3, 4, 6, 8, 9}.
     • If U = 2: S = 4.
     • If U = 4: S = 6.
     • If U = 6: S = 8.
   • Let's test these sub-cases:
     • Sub-case A1: U = 2, S = 4.
       • Occupied seats: P(0), A(1), U(2), S(4), E(5), O(7).
       • Vacant seats: {3, 6, 8, 9}.
       • T is not next to E(5), so T cannot be at Seat 6. T must be at Seat 3, 8, or 9.
       • "R sits second to the right of T."
         • If T = 8 (faces inside): R must be Seat 6.
         • All other placements for T yield occupied positions for R.
         • So, T = 8, R = 6.
       • Now check: "There are exactly three friends sitting between I and R."
         • Since R is at Seat 6, I must be at Seat 2 or Seat 0 (both already occupied).
         • Sub-case A1 is invalid.
     • Sub-case A2: U = 4, S = 6.
       • Occupied seats: P(0), A(1), U(4), E(5), S(6), O(7).
       • Vacant seats: {2, 3, 8, 9}.
       • T cannot be at Seat 4 or 6. T must be at Seat 2, 3, 8, or 9.
       • Test "R sits second to the right of T":
         • No placement for T yields a valid empty seat for R.
         • Sub-case A2 is invalid.
     • Sub-case A3: U = 6, S = 8.
       • Occupied seats: P(0), A(1), E(5), U(6), O(7), S(8).
       • Vacant seats: {2, 3, 4, 9}.
       • T cannot be at Seat 4. T must be at Seat 2, 3, or 9.
       • Test "R sits second to the right of T":
         • If T = 2 (faces outside): R must sit at Seat 4.
         • So, T = 2, R = 4.
       • Check: "There are exactly three friends sitting between I and R."
         • Since R is at Seat 4, I must sit at Seat 8 or Seat 0 (both occupied).
         • Sub-case A3 is invalid.
   • Since all sub-cases fail, Case A is eliminated.

4. Analyze Case B (A at Seat 3, O faces outside):

   • A is at Seat 3 (faces outside). O is at Seat 7 (faces outside).
   • Occupied seats: P(0, inside), E(5, inside), A(3, outside), O(7, outside).
   • S sits second to the left of U ($S = U + 2$ clockwise). U and S face inside.
   • Vacant seats are among {1, 2, 4, 6, 8, 9}.
     • Let's test U and S:
       • Sub-case B1: U = 9, S = 1.
         • Occupied seats: P(0), S(1), A(3), E(5), O(7), U(9).
         • Vacant seats: {2, 4, 6, 8}.
         • T cannot be at Seat 4 or 6. Thus, T must be at Seat 2 or 8.
         • Test "R sits second to the right of T":
           • Option 1: T = 2 (faces outside), R = 4.
           • Option 2: T = 8 (faces inside), R = 6.
         • Test Option 1 (T = 2, R = 4):
           • "There are exactly three friends sitting between I and R."
             • R is at Seat 4. Thus, I must sit at Seat 8 (since Seat 0 is occupied by P).
             • The remaining vacant seat for Q is Seat 6.
           • Check: "I sits second to the left of Q."
             • I is at Seat 8, Q is at Seat 6.
             • For I to be second to the left of Q, Q must face outside (left is anti-clockwise; 6 -> 7 -> 8).
           • Let's count facing directions:
             • Facing Outside (4 total): A(3), O(7), T(2), Q(6). (Matches the constraint).
             • Facing Inside (6 total): P(0), S(1), R(4), E(5), I(8), U(9).
             • Check consecutive outside facing seats: Seats 2 & 3 (two), Seats 6 & 7 (two). No more than two adjacent seats face outside. (Constraint satisfied).
           • Let's assign genders and check consecutive gender constraints:
             • Boys (5): P(0), S(1), T(2), R(4), Q(6).
             • Girls (5): A(3), E(5), O(7), I(8), U(9).
             • Let's read genders clockwise starting from Seat 0:
               • Seat 0: Boy
               • Seat 1: Boy
               • Seat 2: Boy
               • Seat 3: Girl
               • Seat 4: Boy
               • Seat 5: Girl
               • Seat 6: Boy
               • Seat 7: Girl
               • Seat 8: Girl
               • Seat 9: Girl
             • Maximum consecutive boys is 3 (Seats 0, 1, 2).
             • Maximum consecutive girls is 3 (Seats 7, 8, 9).
             • Gender constraint is satisfied.
         • Test Option 2 (T = 8, R = 6):
           • R is at Seat 6. For three friends between I and R, I must sit at Seat 2.
           • Q must sit at the remaining Seat 4.
           • Check: "I sits second to the left of Q."
             • I is at Seat 2, Q is at Seat 4.
             • For I to be second to the left of Q, Q must face inside (left is clockwise; 4 -> 3 -> 2).
           • Let's count genders:
             • Boys (5): P(0), S(1), Q(4), R(6), T(8).
             • Girls (5): I(2), A(3), E(5), O(7), U(9).
             • Genders clockwise: Boy(0), Boy(1), Girl(2), Girl(3), Boy(4), Girl(5), Boy(6), Girl(7), Boy(8), Girl(9).
             • Max consecutive is 2. (Gender constraint satisfied).
           • Let's check facing directions:
             • Outside (4 total): A(3, outside), O(7, outside).
             • Two boys and two girls must face outside. Currently, we have 2 girls (A and O) facing outside. The other two outside-facing persons must be boys.
             • The boys are P(inside), S(inside), Q(inside), R, T. Thus, two of {R, T} must face outside.
             • But R sits second to the right of T.
               • If T(8) faces inside: Right is anti-clockwise. Second right of T is Seat 6 (R).
               • If T(8) faces outside: Right is clockwise. Second right of T is Seat 0 (occupied by P).
               • Thus, T must face inside.
               • If T faces inside, then only one boy (R) can face outside. This violates the rule that exactly two boys and two girls face outside.
               • Thus, Option 2 is eliminated.

5. Conclusion: Option 1 of Sub-case B1 is the unique valid seating configuration.

Final Seating Matrix (Hard Example)