Directions (76-80) :Study the following information carefully and answer the questions given below:

Nine boxes are placed one above the other. The bottommost shelf is numbered as 1 and just above it is numbered as 2 and so on till the topmost shelf is numbered as 9.

Box C is placed at the position which is a multiple of 3. Three boxes are placed between Box C and Box H. There are as many boxes placed below Box H as above Box L. Box G is placed just above Box E but not placed just below Box C. Box D is placed below Box E. More than two boxes are placed between Box M and Box B. Box F is placed above Box M.

The number of boxes placed between Box D and Box M is half of the number of boxes placed above Box _____.

Correct Answer: Option (Box E)

Step-by-Step Explanation and Derivation:

Let us step-by-step determine the exact arrangement of the nine boxes on shelves numbered 1 (bottommost) to 9 (topmost).

1. Analyzing Box C and Box H:

Box C is placed at a shelf position which is a multiple of 3. Thus, Box C can be on shelf 3, 6, or 9.

There are three boxes placed between Box C and Box H.

• If Box C is on shelf 9: Box H must be on shelf $9-4=5$.

• If Box C is on shelf 6: Box H could be on shelf $6+4=10$ (invalid) or shelf $6-4=2$.

• If Box C is on shelf 3: Box H must be on shelf $3+4=7$.

2. Analyzing Box L:

There are as many boxes placed below Box H as above Box L. This means:

(Position of H - 1) = (9 - Position of L)

Position of L = 10 - Position of H.

Let us test the cases:

• Case 1: C is on shelf 6, H is on shelf 2.

Then L is on shelf $10-2=8$.

• Case 2: C is on shelf 9, H is on shelf 5.

Then L is on shelf $10-5=5$ (Conflict, H and L cannot be on the same shelf).

• Case 3: C is on shelf 3, H is on shelf 7.

Then L is on shelf $10-7=3$ (Conflict, C and L cannot be on the same shelf).

Thus, only Case 1 is valid!

Currently fixed positions:

Shelf 9: _

Shelf 8: Box L

Shelf 7: _

Shelf 6: Box C

Shelf 5: _

Shelf 4: _

Shelf 3: _

Shelf 2: Box H

Shelf 1: _

3. Placing Box G, Box E, and Box D:

Box G is placed just above Box E (i.e., GE adjacent together). Box G is not placed just below Box C, so G cannot be on shelf 5.

Box D is placed below Box E.

Available consecutive empty spots for G and E are:

• Shelves 4 & 3 (G on 4, E on 3): Then Box D must be below E, so D is on shelf 1.

• Shelves 9 & 7 are separated, so no other consecutive pair works above shelf 5.

Therefore, we have:

• Shelf 4: Box G

• Shelf 3: Box E

• Shelf 1: Box D

Updating the shelf order:

Shelf 9: _

Shelf 8: Box L

Shelf 7: _

Shelf 6: Box C

Shelf 5: _

Shelf 4: Box G

Shelf 3: Box E

Shelf 2: Box H

Shelf 1: Box D

4. Placing Box F, Box M, and Box B:

The remaining empty shelves are 9, 7, and 5.

Box F is placed above Box M.

More than two boxes are placed between Box M and Box B (i.e., at least 3 boxes between them).

If Box B is on shelf 9 and Box M is on shelf 5, the number of boxes between them is 3 (shelves 8, 7, 6), which satisfies "more than two".

Then Box F takes the remaining shelf 7. Since F (shelf 7) is above M (shelf 5), this completely satisfies all conditions!

Final Shelf Arrangement:

Shelf 9: Box B

Shelf 8: Box L

Shelf 7: Box F

Shelf 6: Box C

Shelf 5: Box M

Shelf 4: Box G

Shelf 3: Box E

Shelf 2: Box H

Shelf 1: Box D

5. Answering the Question:

Number of boxes placed between Box D (Shelf 1) and Box M (Shelf 5):

The boxes between them are on shelves 2, 3, and 4 (Boxes H, E, G), so there are 3 boxes between Box D and Box M.

According to the statement: "3 is half of the number of boxes placed above Box _____."

Thus, the number of boxes placed above the blank box must be $3\times 2=6$.

The box which has exactly 6 boxes above it (placed on shelf 3) is Box E (boxes on shelves 9, 8, 7, 6, 5, 4 are above it).

Hence, the blank must be filled with Box E.