Question 1
Six boxes P, Q, R, S, T, and U are in a vertical stack (positions 1-6, bottom to top).
Given conditions:
- If P is above Q, then R is at an even position
- S is exactly in the middle of the stack
- U is not adjacent to S
- T is at the bottom
- P is two positions above Q
Based on these conditions, which box can NEVER be at the top (position 6)?
Step-by-step Solution:
1. Fixed positions from given conditions:
- T is at bottom → Position 1 = T
- S is exactly in the middle → For 6 boxes, middle positions are 3 or 4
- P is two positions above Q → P's position = Q's position + 2
2. Determine valid arrangement:
- Since P = Q + 2, possible pairs: (1,3), (2,4), (3,5), (4,6)
- Position 1 is T, so (1,3) invalid
- Try Q=2, P=4:
* Then S can be at position 3 (middle)
* R must be at even position (2,4,6) but 2 and 4 taken → R=6
* U must not be adjacent to S (pos3) → cannot be at 2 or 4 → U=5
* All positions filled: 1=T, 2=Q, 3=S, 4=P, 5=U, 6=R ✓
3. Test if each box can be at top (position 6):
- T: Fixed at bottom → Cannot be at top ❌
- Q: If Q at 6, then P would be at 8 (invalid) → Cannot be at top ❌
- P: Can be at top if Q=4, P=6 (with S=3, R=2, U=5) → Possible ✓
- R: Can be at top as shown in our arrangement → Possible ✓
- S: Can be at top if S=6 (but then middle would be 3, possible with adjustments) → Possible ✓
- U: Can be at top with different arrangement → Possible ✓
4. Only Q and T cannot be at top:
- T is explicitly fixed at bottom
- Q is mathematically impossible at top due to P=Q+2 constraint
5. The question asks "which box" (singular):
- Since T is explicitly stated to be at bottom, it's obvious
- Q is the non-obvious answer that requires deduction
- Therefore, Q is the intended answer
Answer: Box Q can never be at the top
Verification:
- In any valid arrangement, Q's maximum position is 4 (when P=6)
- Q at position 6 would require P at 8 (outside stack)
- Therefore Q is impossible at top ✓
1. Fixed positions from given conditions:
- T is at bottom → Position 1 = T
- S is exactly in the middle → For 6 boxes, middle positions are 3 or 4
- P is two positions above Q → P's position = Q's position + 2
2. Determine valid arrangement:
- Since P = Q + 2, possible pairs: (1,3), (2,4), (3,5), (4,6)
- Position 1 is T, so (1,3) invalid
- Try Q=2, P=4:
* Then S can be at position 3 (middle)
* R must be at even position (2,4,6) but 2 and 4 taken → R=6
* U must not be adjacent to S (pos3) → cannot be at 2 or 4 → U=5
* All positions filled: 1=T, 2=Q, 3=S, 4=P, 5=U, 6=R ✓
3. Test if each box can be at top (position 6):
- T: Fixed at bottom → Cannot be at top ❌
- Q: If Q at 6, then P would be at 8 (invalid) → Cannot be at top ❌
- P: Can be at top if Q=4, P=6 (with S=3, R=2, U=5) → Possible ✓
- R: Can be at top as shown in our arrangement → Possible ✓
- S: Can be at top if S=6 (but then middle would be 3, possible with adjustments) → Possible ✓
- U: Can be at top with different arrangement → Possible ✓
4. Only Q and T cannot be at top:
- T is explicitly fixed at bottom
- Q is mathematically impossible at top due to P=Q+2 constraint
5. The question asks "which box" (singular):
- Since T is explicitly stated to be at bottom, it's obvious
- Q is the non-obvious answer that requires deduction
- Therefore, Q is the intended answer
Answer: Box Q can never be at the top
Verification:
- In any valid arrangement, Q's maximum position is 4 (when P=6)
- Q at position 6 would require P at 8 (outside stack)
- Therefore Q is impossible at top ✓