# 7. Option 1.

Unit cubes were used to assemble a large parallelepiped with sides greater than 3. Two cubes will be called adjacent if they touch by faces. Thus, one cube can have up to 6 neighbors. It is known that the number of cubes that have exactly 6 neighbors is 429. Find the number of cubes that have exactly 4 neighbors.
Answer: 108.

Solution: Let $a, b$ and $c$ be the lengths of the sides of the large parallelepiped. Then, the number of cubes with exactly 6 neighbors is: $(a-2)(b-2)(c-2)$. Since each of the factors $a-2, b-2$, and $c-2$ is greater than 1 and their product equals the product of the three prime numbers 3, 11, and 13, these numbers must be 3, 11, and 13 in some order. The number of cubes with exactly 4 neighbors is $4(a-2+b-2+c-2)$ (since each such cube is adjacent to exactly one edge). Thus, we get the answer: $4(3+11+13)=108$.