Cubes with no paint in a 4 × 4 × 4 cube cut?

The number of cubes with no paint in a 4 × 4 × 4 cube cut is 8. To understand why this is the case, consider the structure of the cube. A 4 × 4 × 4 cube consists of 64 smaller 1 × 1 × 1 cubes arranged in three dimensions. When the outer layer of the cube is painted, only the cubes that are not on the exterior will remain unpainted.

In a 4 × 4 × 4 cube, the outer layer consists of all the cubes that have at least one face exposed. This includes the cubes on all six faces of the larger cube. To find the cubes that remain unpainted, we need to focus on the inner layer of the cube. The inner cube is formed by excluding the outer layer of cubes, which essentially reduces the dimensions of the cube from 4 × 4 × 4 to 2 × 2 × 2.

Calculating the number of cubes in this inner section, we find that a 2 × 2 × 2 cube contains exactly 8 smaller 1 × 1 × 1 cubes. Therefore, these 8 cubes are completely surrounded by other cubes and do not have any paint on them. Understanding this distinction is crucial, especially in competitive exams where spatial reasoning and visualization play a significant role.

In summary, while the outer layer of the 4 × 4 × 4 cube has cubes with paint, the inner section consists solely of the 8 cubes that remain unpainted. This concept is fundamental in problems involving painted surfaces on cubes, making it an important topic for exam preparation.