The problem is about assigning numbers to the faces of a cube such that:

1. The cube's faces are numbered using the digits 3 through 9 (each number is used exactly once).
2. The sum of the numbers on any two opposite faces equals 12.

We are tasked to find how many distinct valid arrangements of the cube can be created under these constraints.

Step 1: Understanding opposite faces

For a cube, opposite faces must sum to 12. This condition uniquely pairs the numbers as follows:

• $3 + 9 = 12$
• $4 + 8 = 12$
• $5 + 7 = 12$

Thus, the three pairs of opposite faces are:

1. $(3, 9)$,
2. $(4, 8)$,
3. $(5, 7)$.

Step 2: Determining arrangements

For a cube, we can arrange the three pairs of opposite faces in different orientations. The possible arrangements are calculated based on the rotational symmetries of the cube:

1. Fixing one pair (e.g., $(3, 9)$): Once one pair is fixed, the remaining two pairs can be arranged in $2! = 2$ ways (permuting $(4, 8)$ and $(5, 7)$).
2. Rotational symmetry of the cube: The cube has $24$ rotational symmetries, but fixing one pair (e.g., $(3, 9)$) reduces this to $4$ possible unique orientations for that pair.

So, the total number of unique arrangements is:

$3! \times 4 = 24$

Final Answer:

The number of distinct valid cube arrangements is 24.

Let me know if you want me to expand on this explanation further or calculate similar cube-related problems!

Follow-up Questions:

1. How do rotational symmetries affect the arrangement of numbers on a cube?
2. What would happen if more numbers were allowed (e.g., digits 1–9)?
3. How do opposite-face constraints generalize to other sums (e.g., opposite faces sum to 15)?
4. What is the mathematical reasoning behind $3!$ permutations for the pairs?
5. How can this concept be applied to coloring problems on cubes?

Tip:

When working with problems involving rotational symmetry, always account for constraints that reduce the total number of unique arrangements, such as fixing one face or one pair.