Question 1
Six boxes A, B, C, D, E, and F are stacked vertically (positions 1 to 6 from bottom to top).
The box at stack position n is then placed at seat position n in a circular arrangement of 6 chairs (seats 1 to 6 in clockwise order).
Conditions:
- Box A is at the bottom of the stack (position 1)
- Box B is at the top of the stack (position 6)
- Box C is immediately above box D in the stack
- In the circular arrangement, box E is opposite box F
- Box C is at seat 4 in the circular arrangement
Which box sits opposite box A in the circular arrangement?
Step-by-step Solution:
1. Stack to Circle mapping:
- Seat n = Box at stack position n
2. Fixed assignments:
- Stack pos1 = A → Seat1 = A
- Stack pos6 = B → Seat6 = B
- Box C at seat4 (given) → Stack pos4 = C
3. Box C immediately above Box D in stack:
- C at pos4 → D must be at pos3
- Therefore: Stack pos3 = D
4. Remaining positions in stack: pos2 and pos5 for boxes E and F
- Stack pos2 → Seat2
- Stack pos5 → Seat5
5. In circular arrangement, E opposite F:
- In a 6-seat circle, opposite pairs are: (1,4), (2,5), (3,6)
- Seat1 = A, Seat4 = C, Seat3 = D, Seat6 = B
- For E and F to be opposite, they must occupy seats (2,5)
- Therefore: Seat2 and Seat5 = E and F (in some order)
6. Determine which is which:
- Seat2 = Stack pos2 = either E or F
- Seat5 = Stack pos5 = the other one
- Both arrangements are valid
7. Complete the arrangement:
- Stack: pos1=A, pos2=E/F, pos3=D, pos4=C, pos5=F/E, pos6=B
- Circle: Seat1=A, Seat2=E/F, Seat3=D, Seat4=C, Seat5=F/E, Seat6=B
8. Find box opposite A:
- A is at Seat1
- Opposite seat to Seat1 is Seat4 (since 1+3=4)
- Seat4 = C
9. Answer: Box C sits opposite Box A
Final verification:
- A at bottom ✓
- B at top ✓
- C(pos4) immediately above D(pos3) ✓
- E and F at seats 2 and 5 (opposite) ✓
- C at seat4 ✓
Answer: Box C
1. Stack to Circle mapping:
- Seat n = Box at stack position n
2. Fixed assignments:
- Stack pos1 = A → Seat1 = A
- Stack pos6 = B → Seat6 = B
- Box C at seat4 (given) → Stack pos4 = C
3. Box C immediately above Box D in stack:
- C at pos4 → D must be at pos3
- Therefore: Stack pos3 = D
4. Remaining positions in stack: pos2 and pos5 for boxes E and F
- Stack pos2 → Seat2
- Stack pos5 → Seat5
5. In circular arrangement, E opposite F:
- In a 6-seat circle, opposite pairs are: (1,4), (2,5), (3,6)
- Seat1 = A, Seat4 = C, Seat3 = D, Seat6 = B
- For E and F to be opposite, they must occupy seats (2,5)
- Therefore: Seat2 and Seat5 = E and F (in some order)
6. Determine which is which:
- Seat2 = Stack pos2 = either E or F
- Seat5 = Stack pos5 = the other one
- Both arrangements are valid
7. Complete the arrangement:
- Stack: pos1=A, pos2=E/F, pos3=D, pos4=C, pos5=F/E, pos6=B
- Circle: Seat1=A, Seat2=E/F, Seat3=D, Seat4=C, Seat5=F/E, Seat6=B
8. Find box opposite A:
- A is at Seat1
- Opposite seat to Seat1 is Seat4 (since 1+3=4)
- Seat4 = C
9. Answer: Box C sits opposite Box A
Final verification:
- A at bottom ✓
- B at top ✓
- C(pos4) immediately above D(pos3) ✓
- E and F at seats 2 and 5 (opposite) ✓
- C at seat4 ✓
Answer: Box C