Question 1
Seven boxes A, B, C, D, E, F, and G are stacked vertically (positions 1 to 7 from bottom to top).
Given conditions:
- Box C is at position 3
- The product of the positions of boxes C and D is 12
- Box E is at a prime-numbered position
- Box F is at a position that is 5 more than twice the position of box G
- The sum of the positions of boxes A and B is 8, with box A below box B
- Box D is at an even-numbered position
- Box G is at an odd-numbered position
What is the position of box A?
Step-by-step Solution:
1. Start with fixed information:
- Position 3 = C (given)
2. Product condition: pos(C) × pos(D) = 12
- 3 × pos(D) = 12
- pos(D) = 4
3. Box D is at even position: 4 is even ✓
4. Prime position for Box E: Prime positions are 2, 3, 5, 7
- Position 3 is C, so E can be at 2, 5, or 7
5. F and G relationship: pos(F) = 5 + 2 × pos(G)
- Possible (G, F) pairs: (1,7), (2,9-invalid), (3,11-invalid)
- Therefore: G at 1, F at 7
6. Sum condition: pos(A) + pos(B) = 8, with pos(A) < pos(B) (A below B)
- Possible pairs: (1,7), (2,6), (3,5)
- Position 1 is G, position 3 is C, position 7 is F
- Therefore: (1,7) and (3,5) invalid
- Only possibility: A at 2, B at 6
7. Box E must be at prime position:
- Remaining position after placing A(2), B(6), C(3), D(4), F(7), G(1)
- Only position left is 5 for E
- 5 is prime ✓
8. Final arrangement (bottom to top):
- Position 1: G
- Position 2: A
- Position 3: C
- Position 4: D
- Position 5: E
- Position 6: B
- Position 7: F
9. Answer: Box A is at position 2
Verification of all conditions:
- C at position 3 ✓
- C(3) × D(4) = 12 ✓
- E at position 5 (prime) ✓
- F(7) = 5 + 2×G(1) ✓
- A(2) + B(6) = 8, A below B (2 < 6) ✓
- D at even position (4) ✓
- G at odd position (1) ✓
Answer: Position 2
1. Start with fixed information:
- Position 3 = C (given)
2. Product condition: pos(C) × pos(D) = 12
- 3 × pos(D) = 12
- pos(D) = 4
3. Box D is at even position: 4 is even ✓
4. Prime position for Box E: Prime positions are 2, 3, 5, 7
- Position 3 is C, so E can be at 2, 5, or 7
5. F and G relationship: pos(F) = 5 + 2 × pos(G)
- Possible (G, F) pairs: (1,7), (2,9-invalid), (3,11-invalid)
- Therefore: G at 1, F at 7
6. Sum condition: pos(A) + pos(B) = 8, with pos(A) < pos(B) (A below B)
- Possible pairs: (1,7), (2,6), (3,5)
- Position 1 is G, position 3 is C, position 7 is F
- Therefore: (1,7) and (3,5) invalid
- Only possibility: A at 2, B at 6
7. Box E must be at prime position:
- Remaining position after placing A(2), B(6), C(3), D(4), F(7), G(1)
- Only position left is 5 for E
- 5 is prime ✓
8. Final arrangement (bottom to top):
- Position 1: G
- Position 2: A
- Position 3: C
- Position 4: D
- Position 5: E
- Position 6: B
- Position 7: F
9. Answer: Box A is at position 2
Verification of all conditions:
- C at position 3 ✓
- C(3) × D(4) = 12 ✓
- E at position 5 (prime) ✓
- F(7) = 5 + 2×G(1) ✓
- A(2) + B(6) = 8, A below B (2 < 6) ✓
- D at even position (4) ✓
- G at odd position (1) ✓
Answer: Position 2