# 7. Variant 1

Unit cubes were used to assemble a large parallelepiped with sides greater than 4. 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 836. Find the number of cubes that have no more than four neighbors.
Answer: 144.

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 2 and their product equals the product of four prime numbers $2,2,11$ and 19, we have that the numbers $a-2$, $b-2$ and $c-2$ are 4, 11, and 19 in some order. The number of cubes with exactly 4 neighbors is $4 \cdot(a-2+b-2+c-2)$ (since each such cube is adjacent to exactly one edge). Thus, we get the answer: $4 \cdot(4+11+19)+8=144$.

## Variant 2

A large parallelepiped was assembled from unit cubes with sides greater than 4. Two cubes are called neighboring if they touch by faces. Thus, a cube can have up to 6 neighbors. It is known that the number of cubes with exactly 6 neighbors is 1564. Find the number of cubes with no more than four neighbors.

Answer: 184.

## Variant 3

A large parallelepiped was assembled from unit cubes with sides greater than 4. Two cubes are called neighboring if they touch by faces. Thus, a cube can have up to 6 neighbors. It is known that the number of cubes with exactly 6 neighbors is 812. Find the number of cubes with no more than four neighbors.

Answer: 168.

## Variant 4

A large parallelepiped was assembled from unit cubes with sides greater than 4. Two cubes are called neighboring if they touch by faces. Thus, a cube can have up to 6 neighbors. It is known that the number of cubes with exactly 6 neighbors is 1012. Find the number of cubes with no more than four neighbors.

Answer: 160.

## Variant 5

A large parallelepiped was assembled from unit cubes with sides greater than 4. Two cubes are called neighboring if they touch by faces. Thus, a cube can have up to 6 neighbors. It is known that the number of cubes with exactly 6 neighbors is 868. Find the number of cubes with no more than four neighbors.

Answer: 176.