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Concentric Circles Arrangement - Absolute-Beginner Level: core concept mastery Concentric Circles Arrangement ABSOLUTE BEGINNER

This skill primer 🌟 worksheet focuses on Concentric Circles Arrangement - a key topic in Seating Arrangement. You'll solve 20 absolute-beginner-level problems (Worksheet 1 of 10). The primary focus is on core concept mastery. Master concentric circles arrangement problems, concentric circles arrangement reasoning questions, and concentric circles arrangement practice through systematic practice.

📝 Worksheet 1 of 10 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Absolute Beginner level

What you'll learn in this worksheet:
Your progress through Concentric Circles Arrangement
Worksheet 1 of 10 (0% complete)

Question 1

Eight persons are sitting in two concentric circles. Inner circle: A, D, C and B (4 persons) facing the center. Outer circle: P, Q, S and R (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). P faces A. D sits immediately to the right of A in the inner circle. B sits opposite to D in the inner circle. Q sits second to the right of S in the outer circle. R sits immediately to the left of an empty seat in the outer circle. Who sits opposite to D in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- D sits immediately to the right of A (inner circle, facing center: right=CW)
→ A and D are adjacent. Possible: A=I1, D=I2 or A=I2, D=I3, etc.
- B sits opposite to D:
If D=I2, then B=I4. Then A=I1 (left of I2), C=I3 (remaining).
All constraints consistent ✓.

Inner: I1=A, I2=D, I3=C, I4=B

Step 4: Place outer circle persons
- P faces A(I1) → P at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- Q second to right of S (outer, facing center: right=CW, step 2):
Remaining outer persons: Q,S,R at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ S=O2, Q=O4, R=O5 ✓
- R(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(Q) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
D is at I2. Opposite of I2 = I4 = B.

Final Arrangements:
Inner (CW): I1=A, I2=D, I3=C, I4=B
Outer (CW): O1=P, O2=S, O3=empty, O4=Q, O5=R, O6=empty

Verification:
✓ P faces A (I1↔O1)
✓ D immediately right of A (I2 is CW from I1)
✓ B opposite D (I4 opposite I2)
✓ Q second right of S (O4 = O2 + 2 CW)
✓ R(O5) immediately left of empty O6

Answer: B sits opposite to D.

Question 2

Eight persons are sitting in two concentric circles. Inner circle: D, B, A and C (4 persons) facing the center. Outer circle: Q, R, P and S (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). Q faces D. B sits immediately to the right of D in the inner circle. C sits opposite to B in the inner circle. R sits second to the right of P in the outer circle. S sits immediately to the left of an empty seat in the outer circle. Who sits opposite to B in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- B sits immediately to the right of D (inner circle, facing center: right=CW)
→ D and B are adjacent. Possible: D=I1, B=I2 or D=I2, B=I3, etc.
- C sits opposite to B:
If B=I2, then C=I4. Then D=I1 (left of I2), A=I3 (remaining).
All constraints consistent ✓.

Inner: I1=D, I2=B, I3=A, I4=C

Step 4: Place outer circle persons
- Q faces D(I1) → Q at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- R second to right of P (outer, facing center: right=CW, step 2):
Remaining outer persons: R,P,S at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ P=O2, R=O4, S=O5 ✓
- S(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(R) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
B is at I2. Opposite of I2 = I4 = C.

Final Arrangements:
Inner (CW): I1=D, I2=B, I3=A, I4=C
Outer (CW): O1=Q, O2=P, O3=empty, O4=R, O5=S, O6=empty

Verification:
✓ Q faces D (I1↔O1)
✓ B immediately right of D (I2 is CW from I1)
✓ C opposite B (I4 opposite I2)
✓ R second right of P (O4 = O2 + 2 CW)
✓ S(O5) immediately left of empty O6

Answer: C sits opposite to B.

Question 3

Eight persons are sitting in two concentric circles. Inner circle: C, B, D and A (4 persons) facing the center. Outer circle: R, S, P and Q (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). R faces C. B sits immediately to the right of C in the inner circle. A sits opposite to B in the inner circle. S sits second to the right of P in the outer circle. Q sits immediately to the left of an empty seat in the outer circle. Who sits opposite to B in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- B sits immediately to the right of C (inner circle, facing center: right=CW)
→ C and B are adjacent. Possible: C=I1, B=I2 or C=I2, B=I3, etc.
- A sits opposite to B:
If B=I2, then A=I4. Then C=I1 (left of I2), D=I3 (remaining).
All constraints consistent ✓.

Inner: I1=C, I2=B, I3=D, I4=A

Step 4: Place outer circle persons
- R faces C(I1) → R at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- S second to right of P (outer, facing center: right=CW, step 2):
Remaining outer persons: S,P,Q at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ P=O2, S=O4, Q=O5 ✓
- Q(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(S) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
B is at I2. Opposite of I2 = I4 = A.

Final Arrangements:
Inner (CW): I1=C, I2=B, I3=D, I4=A
Outer (CW): O1=R, O2=P, O3=empty, O4=S, O5=Q, O6=empty

Verification:
✓ R faces C (I1↔O1)
✓ B immediately right of C (I2 is CW from I1)
✓ A opposite B (I4 opposite I2)
✓ S second right of P (O4 = O2 + 2 CW)
✓ Q(O5) immediately left of empty O6

Answer: A sits opposite to B.

Question 4

Eight persons are sitting in two concentric circles. Inner circle: B, D, C and A (4 persons) facing the center. Outer circle: R, S, Q and P (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). R faces B. D sits immediately to the right of B in the inner circle. A sits opposite to D in the inner circle. S sits second to the right of Q in the outer circle. P sits immediately to the left of an empty seat in the outer circle. Who sits opposite to D in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- D sits immediately to the right of B (inner circle, facing center: right=CW)
→ B and D are adjacent. Possible: B=I1, D=I2 or B=I2, D=I3, etc.
- A sits opposite to D:
If D=I2, then A=I4. Then B=I1 (left of I2), C=I3 (remaining).
All constraints consistent ✓.

Inner: I1=B, I2=D, I3=C, I4=A

Step 4: Place outer circle persons
- R faces B(I1) → R at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- S second to right of Q (outer, facing center: right=CW, step 2):
Remaining outer persons: S,Q,P at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ Q=O2, S=O4, P=O5 ✓
- P(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(S) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
D is at I2. Opposite of I2 = I4 = A.

Final Arrangements:
Inner (CW): I1=B, I2=D, I3=C, I4=A
Outer (CW): O1=R, O2=Q, O3=empty, O4=S, O5=P, O6=empty

Verification:
✓ R faces B (I1↔O1)
✓ D immediately right of B (I2 is CW from I1)
✓ A opposite D (I4 opposite I2)
✓ S second right of Q (O4 = O2 + 2 CW)
✓ P(O5) immediately left of empty O6

Answer: A sits opposite to D.

Question 5

Eight persons are sitting in two concentric circles. Inner circle: C, B, D and A (4 persons) facing the center. Outer circle: R, P, S and Q (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). R faces C. B sits immediately to the right of C in the inner circle. A sits opposite to B in the inner circle. P sits second to the right of S in the outer circle. Q sits immediately to the left of an empty seat in the outer circle. Who sits opposite to B in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- B sits immediately to the right of C (inner circle, facing center: right=CW)
→ C and B are adjacent. Possible: C=I1, B=I2 or C=I2, B=I3, etc.
- A sits opposite to B:
If B=I2, then A=I4. Then C=I1 (left of I2), D=I3 (remaining).
All constraints consistent ✓.

Inner: I1=C, I2=B, I3=D, I4=A

Step 4: Place outer circle persons
- R faces C(I1) → R at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- P second to right of S (outer, facing center: right=CW, step 2):
Remaining outer persons: P,S,Q at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ S=O2, P=O4, Q=O5 ✓
- Q(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(P) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
B is at I2. Opposite of I2 = I4 = A.

Final Arrangements:
Inner (CW): I1=C, I2=B, I3=D, I4=A
Outer (CW): O1=R, O2=S, O3=empty, O4=P, O5=Q, O6=empty

Verification:
✓ R faces C (I1↔O1)
✓ B immediately right of C (I2 is CW from I1)
✓ A opposite B (I4 opposite I2)
✓ P second right of S (O4 = O2 + 2 CW)
✓ Q(O5) immediately left of empty O6

Answer: A sits opposite to B.

Question 6

Eight persons are sitting in two concentric circles. Inner circle: B, C, A and D (4 persons) facing the center. Outer circle: P, R, S and Q (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). P faces B. C sits immediately to the right of B in the inner circle. D sits opposite to C in the inner circle. R sits second to the right of S in the outer circle. Q sits immediately to the left of an empty seat in the outer circle. Who sits opposite to C in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- C sits immediately to the right of B (inner circle, facing center: right=CW)
→ B and C are adjacent. Possible: B=I1, C=I2 or B=I2, C=I3, etc.
- D sits opposite to C:
If C=I2, then D=I4. Then B=I1 (left of I2), A=I3 (remaining).
All constraints consistent ✓.

Inner: I1=B, I2=C, I3=A, I4=D

Step 4: Place outer circle persons
- P faces B(I1) → P at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- R second to right of S (outer, facing center: right=CW, step 2):
Remaining outer persons: R,S,Q at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ S=O2, R=O4, Q=O5 ✓
- Q(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(R) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
C is at I2. Opposite of I2 = I4 = D.

Final Arrangements:
Inner (CW): I1=B, I2=C, I3=A, I4=D
Outer (CW): O1=P, O2=S, O3=empty, O4=R, O5=Q, O6=empty

Verification:
✓ P faces B (I1↔O1)
✓ C immediately right of B (I2 is CW from I1)
✓ D opposite C (I4 opposite I2)
✓ R second right of S (O4 = O2 + 2 CW)
✓ Q(O5) immediately left of empty O6

Answer: D sits opposite to C.

Question 7

Eight persons are sitting in two concentric circles. Inner circle: D, B, A and C (4 persons) facing the center. Outer circle: Q, S, R and P (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). Q faces D. B sits immediately to the right of D in the inner circle. C sits opposite to B in the inner circle. S sits second to the right of R in the outer circle. P sits immediately to the left of an empty seat in the outer circle. Who sits opposite to B in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- B sits immediately to the right of D (inner circle, facing center: right=CW)
→ D and B are adjacent. Possible: D=I1, B=I2 or D=I2, B=I3, etc.
- C sits opposite to B:
If B=I2, then C=I4. Then D=I1 (left of I2), A=I3 (remaining).
All constraints consistent ✓.

Inner: I1=D, I2=B, I3=A, I4=C

Step 4: Place outer circle persons
- Q faces D(I1) → Q at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- S second to right of R (outer, facing center: right=CW, step 2):
Remaining outer persons: S,R,P at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ R=O2, S=O4, P=O5 ✓
- P(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(S) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
B is at I2. Opposite of I2 = I4 = C.

Final Arrangements:
Inner (CW): I1=D, I2=B, I3=A, I4=C
Outer (CW): O1=Q, O2=R, O3=empty, O4=S, O5=P, O6=empty

Verification:
✓ Q faces D (I1↔O1)
✓ B immediately right of D (I2 is CW from I1)
✓ C opposite B (I4 opposite I2)
✓ S second right of R (O4 = O2 + 2 CW)
✓ P(O5) immediately left of empty O6

Answer: C sits opposite to B.

Question 8

Eight persons are sitting in two concentric circles. Inner circle: D, B, C and A (4 persons) facing the center. Outer circle: S, P, R and Q (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). S faces D. B sits immediately to the right of D in the inner circle. A sits opposite to B in the inner circle. P sits second to the right of R in the outer circle. Q sits immediately to the left of an empty seat in the outer circle. Who sits opposite to B in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- B sits immediately to the right of D (inner circle, facing center: right=CW)
→ D and B are adjacent. Possible: D=I1, B=I2 or D=I2, B=I3, etc.
- A sits opposite to B:
If B=I2, then A=I4. Then D=I1 (left of I2), C=I3 (remaining).
All constraints consistent ✓.

Inner: I1=D, I2=B, I3=C, I4=A

Step 4: Place outer circle persons
- S faces D(I1) → S at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- P second to right of R (outer, facing center: right=CW, step 2):
Remaining outer persons: P,R,Q at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ R=O2, P=O4, Q=O5 ✓
- Q(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(P) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
B is at I2. Opposite of I2 = I4 = A.

Final Arrangements:
Inner (CW): I1=D, I2=B, I3=C, I4=A
Outer (CW): O1=S, O2=R, O3=empty, O4=P, O5=Q, O6=empty

Verification:
✓ S faces D (I1↔O1)
✓ B immediately right of D (I2 is CW from I1)
✓ A opposite B (I4 opposite I2)
✓ P second right of R (O4 = O2 + 2 CW)
✓ Q(O5) immediately left of empty O6

Answer: A sits opposite to B.

Question 9

Eight persons are sitting in two concentric circles. Inner circle: B, A, D and C (4 persons) facing the center. Outer circle: P, S, R and Q (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). P faces B. A sits immediately to the right of B in the inner circle. C sits opposite to A in the inner circle. S sits second to the right of R in the outer circle. Q sits immediately to the left of an empty seat in the outer circle. Who sits opposite to A in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- A sits immediately to the right of B (inner circle, facing center: right=CW)
→ B and A are adjacent. Possible: B=I1, A=I2 or B=I2, A=I3, etc.
- C sits opposite to A:
If A=I2, then C=I4. Then B=I1 (left of I2), D=I3 (remaining).
All constraints consistent ✓.

Inner: I1=B, I2=A, I3=D, I4=C

Step 4: Place outer circle persons
- P faces B(I1) → P at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- S second to right of R (outer, facing center: right=CW, step 2):
Remaining outer persons: S,R,Q at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ R=O2, S=O4, Q=O5 ✓
- Q(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(S) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
A is at I2. Opposite of I2 = I4 = C.

Final Arrangements:
Inner (CW): I1=B, I2=A, I3=D, I4=C
Outer (CW): O1=P, O2=R, O3=empty, O4=S, O5=Q, O6=empty

Verification:
✓ P faces B (I1↔O1)
✓ A immediately right of B (I2 is CW from I1)
✓ C opposite A (I4 opposite I2)
✓ S second right of R (O4 = O2 + 2 CW)
✓ Q(O5) immediately left of empty O6

Answer: C sits opposite to A.

Question 10

Eight persons are sitting in two concentric circles. Inner circle: B, C, D and A (4 persons) facing the center. Outer circle: S, Q, P and R (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). S faces B. C sits immediately to the right of B in the inner circle. A sits opposite to C in the inner circle. Q sits second to the right of P in the outer circle. R sits immediately to the left of an empty seat in the outer circle. Who sits opposite to C in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- C sits immediately to the right of B (inner circle, facing center: right=CW)
→ B and C are adjacent. Possible: B=I1, C=I2 or B=I2, C=I3, etc.
- A sits opposite to C:
If C=I2, then A=I4. Then B=I1 (left of I2), D=I3 (remaining).
All constraints consistent ✓.

Inner: I1=B, I2=C, I3=D, I4=A

Step 4: Place outer circle persons
- S faces B(I1) → S at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- Q second to right of P (outer, facing center: right=CW, step 2):
Remaining outer persons: Q,P,R at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ P=O2, Q=O4, R=O5 ✓
- R(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(Q) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
C is at I2. Opposite of I2 = I4 = A.

Final Arrangements:
Inner (CW): I1=B, I2=C, I3=D, I4=A
Outer (CW): O1=S, O2=P, O3=empty, O4=Q, O5=R, O6=empty

Verification:
✓ S faces B (I1↔O1)
✓ C immediately right of B (I2 is CW from I1)
✓ A opposite C (I4 opposite I2)
✓ Q second right of P (O4 = O2 + 2 CW)
✓ R(O5) immediately left of empty O6

Answer: A sits opposite to C.

Question 11

Eight persons are sitting in two concentric circles. Inner circle: B, C, D and A (4 persons) facing the center. Outer circle: R, S, Q and P (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). R faces B. C sits immediately to the right of B in the inner circle. A sits opposite to C in the inner circle. S sits second to the right of Q in the outer circle. P sits immediately to the left of an empty seat in the outer circle. Who sits opposite to C in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- C sits immediately to the right of B (inner circle, facing center: right=CW)
→ B and C are adjacent. Possible: B=I1, C=I2 or B=I2, C=I3, etc.
- A sits opposite to C:
If C=I2, then A=I4. Then B=I1 (left of I2), D=I3 (remaining).
All constraints consistent ✓.

Inner: I1=B, I2=C, I3=D, I4=A

Step 4: Place outer circle persons
- R faces B(I1) → R at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- S second to right of Q (outer, facing center: right=CW, step 2):
Remaining outer persons: S,Q,P at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ Q=O2, S=O4, P=O5 ✓
- P(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(S) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
C is at I2. Opposite of I2 = I4 = A.

Final Arrangements:
Inner (CW): I1=B, I2=C, I3=D, I4=A
Outer (CW): O1=R, O2=Q, O3=empty, O4=S, O5=P, O6=empty

Verification:
✓ R faces B (I1↔O1)
✓ C immediately right of B (I2 is CW from I1)
✓ A opposite C (I4 opposite I2)
✓ S second right of Q (O4 = O2 + 2 CW)
✓ P(O5) immediately left of empty O6

Answer: A sits opposite to C.

Question 12

Eight persons are sitting in two concentric circles. Inner circle: A, D, B and C (4 persons) facing the center. Outer circle: R, P, Q and S (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). R faces A. D sits immediately to the right of A in the inner circle. C sits opposite to D in the inner circle. P sits second to the right of Q in the outer circle. S sits immediately to the left of an empty seat in the outer circle. Who sits opposite to D in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- D sits immediately to the right of A (inner circle, facing center: right=CW)
→ A and D are adjacent. Possible: A=I1, D=I2 or A=I2, D=I3, etc.
- C sits opposite to D:
If D=I2, then C=I4. Then A=I1 (left of I2), B=I3 (remaining).
All constraints consistent ✓.

Inner: I1=A, I2=D, I3=B, I4=C

Step 4: Place outer circle persons
- R faces A(I1) → R at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- P second to right of Q (outer, facing center: right=CW, step 2):
Remaining outer persons: P,Q,S at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ Q=O2, P=O4, S=O5 ✓
- S(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(P) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
D is at I2. Opposite of I2 = I4 = C.

Final Arrangements:
Inner (CW): I1=A, I2=D, I3=B, I4=C
Outer (CW): O1=R, O2=Q, O3=empty, O4=P, O5=S, O6=empty

Verification:
✓ R faces A (I1↔O1)
✓ D immediately right of A (I2 is CW from I1)
✓ C opposite D (I4 opposite I2)
✓ P second right of Q (O4 = O2 + 2 CW)
✓ S(O5) immediately left of empty O6

Answer: C sits opposite to D.

Question 13

Eight persons are sitting in two concentric circles. Inner circle: C, D, B and A (4 persons) facing the center. Outer circle: P, R, S and Q (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). P faces C. D sits immediately to the right of C in the inner circle. A sits opposite to D in the inner circle. R sits second to the right of S in the outer circle. Q sits immediately to the left of an empty seat in the outer circle. Who sits opposite to D in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- D sits immediately to the right of C (inner circle, facing center: right=CW)
→ C and D are adjacent. Possible: C=I1, D=I2 or C=I2, D=I3, etc.
- A sits opposite to D:
If D=I2, then A=I4. Then C=I1 (left of I2), B=I3 (remaining).
All constraints consistent ✓.

Inner: I1=C, I2=D, I3=B, I4=A

Step 4: Place outer circle persons
- P faces C(I1) → P at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- R second to right of S (outer, facing center: right=CW, step 2):
Remaining outer persons: R,S,Q at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ S=O2, R=O4, Q=O5 ✓
- Q(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(R) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
D is at I2. Opposite of I2 = I4 = A.

Final Arrangements:
Inner (CW): I1=C, I2=D, I3=B, I4=A
Outer (CW): O1=P, O2=S, O3=empty, O4=R, O5=Q, O6=empty

Verification:
✓ P faces C (I1↔O1)
✓ D immediately right of C (I2 is CW from I1)
✓ A opposite D (I4 opposite I2)
✓ R second right of S (O4 = O2 + 2 CW)
✓ Q(O5) immediately left of empty O6

Answer: A sits opposite to D.

Question 14

Eight persons are sitting in two concentric circles. Inner circle: C, B, A and D (4 persons) facing the center. Outer circle: S, R, Q and P (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). S faces C. B sits immediately to the right of C in the inner circle. D sits opposite to B in the inner circle. R sits second to the right of Q in the outer circle. P sits immediately to the left of an empty seat in the outer circle. Who sits opposite to B in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- B sits immediately to the right of C (inner circle, facing center: right=CW)
→ C and B are adjacent. Possible: C=I1, B=I2 or C=I2, B=I3, etc.
- D sits opposite to B:
If B=I2, then D=I4. Then C=I1 (left of I2), A=I3 (remaining).
All constraints consistent ✓.

Inner: I1=C, I2=B, I3=A, I4=D

Step 4: Place outer circle persons
- S faces C(I1) → S at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- R second to right of Q (outer, facing center: right=CW, step 2):
Remaining outer persons: R,Q,P at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ Q=O2, R=O4, P=O5 ✓
- P(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(R) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
B is at I2. Opposite of I2 = I4 = D.

Final Arrangements:
Inner (CW): I1=C, I2=B, I3=A, I4=D
Outer (CW): O1=S, O2=Q, O3=empty, O4=R, O5=P, O6=empty

Verification:
✓ S faces C (I1↔O1)
✓ B immediately right of C (I2 is CW from I1)
✓ D opposite B (I4 opposite I2)
✓ R second right of Q (O4 = O2 + 2 CW)
✓ P(O5) immediately left of empty O6

Answer: D sits opposite to B.

Question 15

Eight persons are sitting in two concentric circles. Inner circle: A, C, B and D (4 persons) facing the center. Outer circle: Q, S, R and P (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). Q faces A. C sits immediately to the right of A in the inner circle. D sits opposite to C in the inner circle. S sits second to the right of R in the outer circle. P sits immediately to the left of an empty seat in the outer circle. Who sits opposite to C in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- C sits immediately to the right of A (inner circle, facing center: right=CW)
→ A and C are adjacent. Possible: A=I1, C=I2 or A=I2, C=I3, etc.
- D sits opposite to C:
If C=I2, then D=I4. Then A=I1 (left of I2), B=I3 (remaining).
All constraints consistent ✓.

Inner: I1=A, I2=C, I3=B, I4=D

Step 4: Place outer circle persons
- Q faces A(I1) → Q at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- S second to right of R (outer, facing center: right=CW, step 2):
Remaining outer persons: S,R,P at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ R=O2, S=O4, P=O5 ✓
- P(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(S) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
C is at I2. Opposite of I2 = I4 = D.

Final Arrangements:
Inner (CW): I1=A, I2=C, I3=B, I4=D
Outer (CW): O1=Q, O2=R, O3=empty, O4=S, O5=P, O6=empty

Verification:
✓ Q faces A (I1↔O1)
✓ C immediately right of A (I2 is CW from I1)
✓ D opposite C (I4 opposite I2)
✓ S second right of R (O4 = O2 + 2 CW)
✓ P(O5) immediately left of empty O6

Answer: D sits opposite to C.

Question 16

Eight persons are sitting in two concentric circles. Inner circle: C, A, D and B (4 persons) facing the center. Outer circle: Q, P, R and S (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). Q faces C. A sits immediately to the right of C in the inner circle. B sits opposite to A in the inner circle. P sits second to the right of R in the outer circle. S sits immediately to the left of an empty seat in the outer circle. Who sits opposite to A in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- A sits immediately to the right of C (inner circle, facing center: right=CW)
→ C and A are adjacent. Possible: C=I1, A=I2 or C=I2, A=I3, etc.
- B sits opposite to A:
If A=I2, then B=I4. Then C=I1 (left of I2), D=I3 (remaining).
All constraints consistent ✓.

Inner: I1=C, I2=A, I3=D, I4=B

Step 4: Place outer circle persons
- Q faces C(I1) → Q at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- P second to right of R (outer, facing center: right=CW, step 2):
Remaining outer persons: P,R,S at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ R=O2, P=O4, S=O5 ✓
- S(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(P) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
A is at I2. Opposite of I2 = I4 = B.

Final Arrangements:
Inner (CW): I1=C, I2=A, I3=D, I4=B
Outer (CW): O1=Q, O2=R, O3=empty, O4=P, O5=S, O6=empty

Verification:
✓ Q faces C (I1↔O1)
✓ A immediately right of C (I2 is CW from I1)
✓ B opposite A (I4 opposite I2)
✓ P second right of R (O4 = O2 + 2 CW)
✓ S(O5) immediately left of empty O6

Answer: B sits opposite to A.

Question 17

Eight persons are sitting in two concentric circles. Inner circle: A, C, B and D (4 persons) facing the center. Outer circle: S, R, Q and P (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). S faces A. C sits immediately to the right of A in the inner circle. D sits opposite to C in the inner circle. R sits second to the right of Q in the outer circle. P sits immediately to the left of an empty seat in the outer circle. Who sits opposite to C in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- C sits immediately to the right of A (inner circle, facing center: right=CW)
→ A and C are adjacent. Possible: A=I1, C=I2 or A=I2, C=I3, etc.
- D sits opposite to C:
If C=I2, then D=I4. Then A=I1 (left of I2), B=I3 (remaining).
All constraints consistent ✓.

Inner: I1=A, I2=C, I3=B, I4=D

Step 4: Place outer circle persons
- S faces A(I1) → S at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- R second to right of Q (outer, facing center: right=CW, step 2):
Remaining outer persons: R,Q,P at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ Q=O2, R=O4, P=O5 ✓
- P(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(R) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
C is at I2. Opposite of I2 = I4 = D.

Final Arrangements:
Inner (CW): I1=A, I2=C, I3=B, I4=D
Outer (CW): O1=S, O2=Q, O3=empty, O4=R, O5=P, O6=empty

Verification:
✓ S faces A (I1↔O1)
✓ C immediately right of A (I2 is CW from I1)
✓ D opposite C (I4 opposite I2)
✓ R second right of Q (O4 = O2 + 2 CW)
✓ P(O5) immediately left of empty O6

Answer: D sits opposite to C.

Question 18

Eight persons are sitting in two concentric circles. Inner circle: C, B, D and A (4 persons) facing the center. Outer circle: S, Q, P and R (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). S faces C. B sits immediately to the right of C in the inner circle. A sits opposite to B in the inner circle. Q sits second to the right of P in the outer circle. R sits immediately to the left of an empty seat in the outer circle. Who sits opposite to B in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- B sits immediately to the right of C (inner circle, facing center: right=CW)
→ C and B are adjacent. Possible: C=I1, B=I2 or C=I2, B=I3, etc.
- A sits opposite to B:
If B=I2, then A=I4. Then C=I1 (left of I2), D=I3 (remaining).
All constraints consistent ✓.

Inner: I1=C, I2=B, I3=D, I4=A

Step 4: Place outer circle persons
- S faces C(I1) → S at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- Q second to right of P (outer, facing center: right=CW, step 2):
Remaining outer persons: Q,P,R at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ P=O2, Q=O4, R=O5 ✓
- R(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(Q) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
B is at I2. Opposite of I2 = I4 = A.

Final Arrangements:
Inner (CW): I1=C, I2=B, I3=D, I4=A
Outer (CW): O1=S, O2=P, O3=empty, O4=Q, O5=R, O6=empty

Verification:
✓ S faces C (I1↔O1)
✓ B immediately right of C (I2 is CW from I1)
✓ A opposite B (I4 opposite I2)
✓ Q second right of P (O4 = O2 + 2 CW)
✓ R(O5) immediately left of empty O6

Answer: A sits opposite to B.

Question 19

Eight persons are sitting in two concentric circles. Inner circle: A, B, C and D (4 persons) facing the center. Outer circle: Q, R, P and S (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). Q faces A. B sits immediately to the right of A in the inner circle. D sits opposite to B in the inner circle. R sits second to the right of P in the outer circle. S sits immediately to the left of an empty seat in the outer circle. Who sits opposite to B in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- B sits immediately to the right of A (inner circle, facing center: right=CW)
→ A and B are adjacent. Possible: A=I1, B=I2 or A=I2, B=I3, etc.
- D sits opposite to B:
If B=I2, then D=I4. Then A=I1 (left of I2), C=I3 (remaining).
All constraints consistent ✓.

Inner: I1=A, I2=B, I3=C, I4=D

Step 4: Place outer circle persons
- Q faces A(I1) → Q at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- R second to right of P (outer, facing center: right=CW, step 2):
Remaining outer persons: R,P,S at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ P=O2, R=O4, S=O5 ✓
- S(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(R) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
B is at I2. Opposite of I2 = I4 = D.

Final Arrangements:
Inner (CW): I1=A, I2=B, I3=C, I4=D
Outer (CW): O1=Q, O2=P, O3=empty, O4=R, O5=S, O6=empty

Verification:
✓ Q faces A (I1↔O1)
✓ B immediately right of A (I2 is CW from I1)
✓ D opposite B (I4 opposite I2)
✓ R second right of P (O4 = O2 + 2 CW)
✓ S(O5) immediately left of empty O6

Answer: D sits opposite to B.

Question 20

Eight persons are sitting in two concentric circles. Inner circle: B, C, D and A (4 persons) facing the center. Outer circle: R, P, S and Q (4 persons) facing the center (2 positions in outer circle are empty). Each inner person directly faces one outer seat (4 outer seats occupied, 2 empty). R faces B. C sits immediately to the right of B in the inner circle. A sits opposite to C in the inner circle. P sits second to the right of S in the outer circle. Q sits immediately to the left of an empty seat in the outer circle. Who sits opposite to C in the inner circle?
Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- C sits immediately to the right of B (inner circle, facing center: right=CW)
→ B and C are adjacent. Possible: B=I1, C=I2 or B=I2, C=I3, etc.
- A sits opposite to C:
If C=I2, then A=I4. Then B=I1 (left of I2), D=I3 (remaining).
All constraints consistent ✓.

Inner: I1=B, I2=C, I3=D, I4=A

Step 4: Place outer circle persons
- R faces B(I1) → R at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- P second to right of S (outer, facing center: right=CW, step 2):
Remaining outer persons: P,S,Q at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ S=O2, P=O4, Q=O5 ✓
- Q(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(P) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
C is at I2. Opposite of I2 = I4 = A.

Final Arrangements:
Inner (CW): I1=B, I2=C, I3=D, I4=A
Outer (CW): O1=R, O2=S, O3=empty, O4=P, O5=Q, O6=empty

Verification:
✓ R faces B (I1↔O1)
✓ C immediately right of B (I2 is CW from I1)
✓ A opposite C (I4 opposite I2)
✓ P second right of S (O4 = O2 + 2 CW)
✓ Q(O5) immediately left of empty O6

Answer: A sits opposite to C.
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