Thirteen boxes are kept in two stacks (Stack 1 and Stack 2) with seven shelves each. Stack 1 is to the west of Stack 2. Each shelf can hold one box, and one shelf is vacant in one of the stacks. Various conditions about the relative positions of boxes M, L, O, X, U, T, Q, R, N, C, S, A, and numbered boxes (11, 17, 21, 27, 29, 33, 38, 39, 51, 58) are given.

**Question:** How many boxes are kept above box M in the same stack?

Question

Thirteen boxes are kept in two stacks (Stack 1 and Stack 2) with seven shelves each. Stack 1 is to the west of Stack 2. Each shelf can hold one box, and one shelf is vacant in one of the stacks. Various conditions about the relative positions of boxes M, L, O, X, U, T, Q, R, N, C, S, A, and numbered boxes (11, 17, 21, 27, 29, 33, 38, 39, 51, 58) are given.

**Question:** How many boxes are kept above box M in the same stack?

Accepted Answer

**Solution:**

1. **Understanding the problem:**
   - There are two stacks (Stack 1 and Stack 2), each with 7 shelves.
   - Total boxes = 13, so one shelf is vacant.
   - Stack 1 is west of Stack 2.
   - Boxes are placed such that each shelf holds one box.
   - Various conditions relate the positions of boxes and numbers.

2. **Key points from the conditions:**
   - Box M is two shelves above the box numbered 51.
   - More than one odd number of boxes are below the box numbered 51.
   - Box M and box 21 are not on the topmost shelf.
   - Box M and Box L are on adjacent shelves but in different stacks.
   - Box O is below M and above box 11.
   - Box M and Box O are not in the same stack.
   - Box M and box 27 are neither adjacent nor on the same shelf.
   - The box numbered 58 is just above Box U, both below M.
   - Box U is below Box L but not on the bottom shelf.
   - Other conditions restrict positions of boxes X, T, Q, R, N, C, S, A.

3. **Deductions:**
   - Since Box M is two shelves above box 51, and more than one odd number of boxes are below 51, 51 cannot be near the bottom.
   - Box M and Box L are adjacent shelves but in different stacks, so if M is in Stack 1 at shelf *i*, L is in Stack 2 at shelf *i+1* or *i-1*.
   - Box O is between M and box 11, so O is below M and above 11.
   - Box M and Box O are in different stacks.
   - Box U and box 58 are below M.
   - Box U is below L but not on bottom shelf.

4. **Arranging boxes in stacks:**
   - Assign Box M to Stack 1 at shelf 4.
   - Then box 51 is at shelf 2 in Stack 1 (since M is two shelves above 51).
   - Boxes below 51 in Stack 1 are shelves 1 (bottom) and possibly vacant shelf.
   - Box L is adjacent to M but in Stack 2, so L is at shelf 3 or 5 in Stack 2.
   - Box O is below M and above 11, so O is at shelf 3 in Stack 2, 11 at shelf 1 or 2 in Stack 2.
   - Box U and 58 are below M in Stack 1, so U at shelf 1 or 2, 58 just above U.

5. **Counting boxes above M in the same stack:**
   - M is at shelf 4 in Stack 1.
   - Shelves above M in Stack 1 are shelves 5, 6, and 7.
   - Since total boxes are 13 and one shelf is vacant, and given the constraints, shelves 5 and 6 in Stack 1 are occupied.
   - Shelf 7 is likely vacant or occupied depending on arrangement.
   - From conditions, topmost shelf is not vacant.
   - Therefore, shelves 5, 6, and 7 are occupied in Stack 1.

6. **Conclusion:**
   - Number of boxes above M in the same stack = 2 (shelves 5 and 6).

**Answer:** (a) Two

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