Question 1
Eight persons are sitting in two concentric circles.
Inner circle: C, D, A and B (4 persons) facing the center.
Outer circle: P, R, 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).
P faces C. D sits immediately to the right of C in the inner circle.
B sits opposite to D in the inner circle. R 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) β all occupied.
Outer circle: 6 positions O1βO6 clockwise β 4 occupied, 2 empty.
Facing alignment: I1βO1, I2βO2, I3βO4, I4βO5 (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)
β C and D are adjacent. Possible: C=I1, D=I2 or C=I2, D=I3, etc.
- B sits opposite to D:
If D=I2, then B=I4. Then C=I1 (left of I2), A=I3 (remaining).
All constraints consistent β.
Inner: I1=C, I2=D, I3=A, I4=B
Step 4: Place outer circle persons
- P faces C(I1) β P at O1 β
- 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,S at O2, O4, O5.
R=O2 β Q=O4, S=O5 β
R=O4 β Q=O6(empty) β
R=O5 β Q=O1=P β
β΄ Q=O2, R=O4, S=O5 β
- S(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(R) β NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) β.
Step 5: Answer the question
D is at I2. Opposite of I2 = I4 = B.
Final Arrangements:
Inner (CW): I1=C, I2=D, I3=A, I4=B
Outer (CW): O1=P, O2=Q, O3=empty, O4=R, O5=S, O6=empty
Verification:
β P faces C (I1βO1)
β D immediately right of C (I2 is CW from I1)
β B opposite D (I4 opposite I2)
β R second right of Q (O4 = O2 + 2 CW)
β S(O5) immediately left of empty O6
Answer: B sits opposite to D.
Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) β all occupied.
Outer circle: 6 positions O1βO6 clockwise β 4 occupied, 2 empty.
Facing alignment: I1βO1, I2βO2, I3βO4, I4βO5 (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)
β C and D are adjacent. Possible: C=I1, D=I2 or C=I2, D=I3, etc.
- B sits opposite to D:
If D=I2, then B=I4. Then C=I1 (left of I2), A=I3 (remaining).
All constraints consistent β.
Inner: I1=C, I2=D, I3=A, I4=B
Step 4: Place outer circle persons
- P faces C(I1) β P at O1 β
- 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,S at O2, O4, O5.
R=O2 β Q=O4, S=O5 β
R=O4 β Q=O6(empty) β
R=O5 β Q=O1=P β
β΄ Q=O2, R=O4, S=O5 β
- S(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(R) β NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) β.
Step 5: Answer the question
D is at I2. Opposite of I2 = I4 = B.
Final Arrangements:
Inner (CW): I1=C, I2=D, I3=A, I4=B
Outer (CW): O1=P, O2=Q, O3=empty, O4=R, O5=S, O6=empty
Verification:
β P faces C (I1βO1)
β D immediately right of C (I2 is CW from I1)
β B opposite D (I4 opposite I2)
β R second right of Q (O4 = O2 + 2 CW)
β S(O5) immediately left of empty O6
Answer: B sits opposite to D.