Nine boxes named P, Q, R, S, T, U, V, W, and X are stacked one above the other in an unknown order. The following conditions apply:

- Exactly five boxes are placed between P and R.
- T is immediately above R.
- Exactly three boxes are placed between T and S.
- The number of boxes between P and S is the same as the number of boxes between Q and T.
- U is placed below Q but is not at the bottom.
- More than four boxes are placed between T and U.
- Exactly one box is placed between U and V.
- Box X is placed above box W.

Based on these conditions, what is the position of box W from the bottom?

Question

Nine boxes named P, Q, R, S, T, U, V, W, and X are stacked one above the other in an unknown order. The following conditions apply:

- Exactly five boxes are placed between P and R.
- T is immediately above R.
- Exactly three boxes are placed between T and S.
- The number of boxes between P and S is the same as the number of boxes between Q and T.
- U is placed below Q but is not at the bottom.
- More than four boxes are placed between T and U.
- Exactly one box is placed between U and V.
- Box X is placed above box W.

Based on these conditions, what is the position of box W from the bottom?

Accepted Answer

**Solution:**

1. There are 9 boxes: P, Q, R, S, T, U, V, W, X.

2. Since exactly 5 boxes are between P and R, the positions of P and R differ by 6 (because 5 boxes are in between).

3. T is immediately above R, so if R is at position *n*, T is at *n-1*.

4. There are 3 boxes between T and S.

5. The number of boxes between P and S equals the number of boxes between Q and T.

6. U is below Q but not at the bottom.

7. More than 4 boxes are between T and U.

8. Exactly one box is between U and V.

9. X is above W.

**Step-by-step arrangement:**

- Let’s number positions from top (1) to bottom (9).

- Since P and R have 5 boxes between them, possible pairs are (1,7), (2,8), or (3,9).

- T is immediately above R, so if R is at 7, T is at 6; if R is at 8, T is at 7; if R is at 9, T is at 8.

- More than 4 boxes between T and U means the difference in their positions is at least 6.

- Try P at 1 and R at 7:
  - T at 6
  - Boxes between T(6) and S = 3, so S can be at 2 or 10 (not possible), so S at 2.
  - Boxes between P(1) and S(2) = 0, so boxes between Q and T(6) must be 0, so Q is at 5 or 7 (7 is R), so Q at 5.
  - U is below Q(5), not at bottom(9), so U can be 6,7,8.
  - More than 4 boxes between T(6) and U means difference ≥6, impossible here.

- Try P at 2 and R at 8:
  - T at 7
  - Boxes between T(7) and S = 3, so S at 3 or 11 (not possible), so S at 3.
  - Boxes between P(2) and S(3) = 0, so boxes between Q and T(7) = 0, so Q at 6 or 8 (8 is R), so Q at 6.
  - U is below Q(6), not at bottom(9), so U can be 7 or 8.
  - More than 4 boxes between T(7) and U means difference ≥6, impossible here.

- Try P at 3 and R at 9:
  - T at 8
  - Boxes between T(8) and S = 3, so S at 4 or 12 (not possible), so S at 4.
  - Boxes between P(3) and S(4) = 0, so boxes between Q and T(8) = 0, so Q at 7 or 9 (9 is R), so Q at 7.
  - U is below Q(7), not at bottom(9), so U at 8.
  - More than 4 boxes between T(8) and U(8) is impossible, so this fails.

- Reconsider the assumption that boxes between P and S = boxes between Q and T = 0.

- If boxes between P and S = 1, then P and S differ by 2.

- Try P at 1, S at 3 (1 box between), then boxes between Q and T = 1.

- T is immediately above R, and 5 boxes between P and R.

- If P at 1, R at 7 (5 boxes between), T at 6.

- Q and T differ by 2 (1 box between), so Q can be at 4 or 8.

- U is below Q but not at bottom.

- More than 4 boxes between T(6) and U means U at position 1 or below 1 (not possible), so U must be below 6 by at least 5 positions, so U at 9.

- One box between U(9) and V means V at 7 or 11 (11 not possible), so V at 7.

- But 7 is R, so V cannot be 7.

- Try Q at 8:
  - U below Q(8), not at bottom(9), so U at 9.
  - One box between U(9) and V means V at 7.

- X is above W.

- Remaining positions for W and X are 2 and 5.

- X above W means X at 2, W at 5.

**Final order from top (1) to bottom (9):**

1: P
2: X
3: S
4: Q
5: W
6: T
7: R
8: V
9: U

Therefore, W is at position 5 from the top, which is 4th from the bottom (since total 9 boxes).

**Answer:** Fourth from the bottom

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