Two adjacent faces of a solid cube are painted with black colour. The faces opposite to the black faces is painted with blue colour while the remaining faces are painted with yellow colour. After painting, this cube has been divided into 125 equal cubes. How many cubes have no faces painted?

Question

Prepp Team · Accepted Answer

Understanding the Cube Painting and Division Problem

The problem describes a large solid cube that has been painted on its faces with three different colours: black, blue, and yellow. After painting, the cube is cut into many smaller, equal-sized cubes. We need to find out how many of these small cubes have none of their faces painted.

First, let's understand the dimensions of the large cube after it's divided. The problem states the cube is divided into 125 equal smaller cubes. When a large cube of side length N (in terms of small cube units) is divided into smaller equal cubes, the total number of smaller cubes is $N^3$.

Given the total number of small cubes is 125, we have:

$$N^3 = 125$$

To find N, we take the cube root of 125:

$$N = \sqrt[3]{125}$$ $$N = 5$$

This means the original large cube was divided into a 5x5x5 grid of smaller cubes. Each edge of the large cube is made up of 5 small cubes.

Analyzing the Painted Faces

The problem specifies the painting pattern:

Two adjacent faces are painted black.
The faces opposite to the black faces are painted blue.
The remaining faces are painted yellow.

A cube has 6 faces. Let's visualize this:

Imagine the cube resting on a table. Let the top face and one side face (say, the front face) be black. These are adjacent faces.
The face opposite the top face is the bottom face. This will be blue.
The face opposite the front face is the back face. This will also be blue.
The remaining two faces are the other two side faces (left and right). These will be yellow.

So, we have:

2 adjacent black faces
2 adjacent blue faces (opposite the black ones)
2 opposite yellow faces (the remaining ones)

This painting pattern covers all 6 faces of the large cube.

Counting Cubes with No Faces Painted

We are looking for the number of small cubes that have none of their faces exposed on the outside of the original large cube. These are the cubes hidden completely inside the large cube.

Consider a cube divided into N x N x N smaller cubes. The small cubes that are unpainted are those that do not touch any of the original faces of the large cube.

If the large cube has dimensions N x N x N, the inner core of cubes that are not exposed to any face will have dimensions (N-2) x (N-2) x (N-2).

In our case, N = 5. So, the dimensions of the inner unpainted cube core are (5-2) x (5-2) x (5-2).

This gives us:

Length of unpainted core: $5 - 2 = 3$ small cubes
Width of unpainted core: $5 - 2 = 3$ small cubes
Height of unpainted core: $5 - 2 = 3$ small cubes

The number of cubes with no faces painted is the product of these dimensions:

Number of unpainted cubes = $(N-2) \times (N-2) \times (N-2)$

Number of unpainted cubes = $(5-2) \times (5-2) \times (5-2)$

Number of unpainted cubes = $3 \times 3 \times 3$

Number of unpainted cubes = $27$

These 27 cubes form a smaller cube inside the original 5x5x5 cube, surrounded by the painted layers.

Summary of Cube Types

For a N x N x N cube division, the number of cubes with different numbers of painted faces can be categorized:

Type of Cube	Location on Large Cube	Formula for N>2	Number for N=5
3 faces painted	At the corners	8	8
2 faces painted	Along the edges (excluding corners)	$12 \times (N-2)$	$12 \times (5-2) = 12 \times 3 = 36$
1 face painted	On the faces (excluding edges and corners)	$6 \times (N-2)^2$	$6 \times (5-2)^2 = 6 \times 3^2 = 6 \times 9 = 54$
0 faces painted	Inside the cube	$(N-2)^3$	$(5-2)^3 = 3^3 = 27$