# 5. Option 1

Unit cubes were used to build a large cube. 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 4 neighbors is 132. Find the number of cubes that have exactly 5 neighbors.
Answer: 726.

Solution: Cubes that have exactly 4 neighbors touch exactly one edge of the large cube. There are 12 edges in total, so 11 such cubes touch each edge. Therefore, the number of cubes that are strictly inside each face is $11 \cdot 11 = 121$. Since there are 6 faces, the answer is: $121 \cdot 6 = 726$.

## Variant 2

Unit cubes were used to build a large cube. Two cubes are called neighbors if they touch by a face. Thus, one cube can have up to 6 neighbors. It is known that the number of cubes with exactly 4 neighbors is 84. Find the number of cubes with exactly 5 neighbors.

Answer: 294.

## Variant 3

Unit cubes were used to build a large cube. Two cubes are called neighbors if they touch by a face. Thus, one cube can have up to 6 neighbors. It is known that the number of cubes with exactly 4 neighbors is 156. Find the number of cubes with exactly 5 neighbors.

Answer: 1014.

## Variant 4

Unit cubes were used to build a large cube. Two cubes are called neighbors if they touch by a face. Thus, one cube can have up to 6 neighbors. It is known that the number of cubes with exactly 4 neighbors is 108. Find the number of cubes with exactly 5 neighbors.

Answer: 486.

## Variant 5

Unit cubes were used to build a large cube. Two cubes are called neighbors if they touch by a face. Thus, one cube can have up to 6 neighbors. It is known that the number of cubes with exactly 4 neighbors is 180. Find the number of cubes with exactly 5 neighbors.

Answer: 1350.