Understanding the Painted Cube Problem

This problem involves a large cube that is painted on all its faces and then cut into many smaller cubes of equal size. We need to determine how many of these smaller cubes have paint on a specific number of faces.

The large cube is cut into 27 small cubes. Since $27 = 3 \times 3 \times 3$, this means the large cube was divided into 3 sections along its length, 3 along its width, and 3 along its height. We can represent this as an $n \times n \times n$ division where $n=3$.

Classifying Small Cubes by Painted Faces

When a large painted cube is cut into smaller cubes, the number of painted faces on each small cube depends on its original position within the large cube.

The small cubes can be classified into four types based on how many of their faces are painted:

• Cubes with 3 faces painted: These are the cubes located at the corners of the large cube.
• Cubes with 2 faces painted: These are the cubes located on the edges of the large cube, but not at the corners.
• Cubes with 1 face painted: These are the cubes located on the center of each face of the large cube, not along the edges or at the corners.
• Cubes with 0 faces painted: These are the inner cubes, located completely inside the large cube.

Calculating Cubes Painted on One Face Only

The question asks for the number of small cubes painted on one face only. These are the cubes located on the center part of each face of the original large cube.

For a cube cut into $n \times n \times n$ small cubes, the number of small cubes with one face painted is given by the formula:

\(\text{Number of cubes with 1 face painted} = 6 \times (n-2)^2\)

In this problem, the total number of small cubes is 27, which means $n^3 = 27$. Taking the cube root, we find $n=3$.

Now, we can substitute $n=3$ into the formula for cubes with one face painted:

\(6 \times (3-2)^2\)

\(6 \times (1)^2\)

\(6 \times 1\)

\(6\)

Therefore, there are 6 small cubes that are painted on one face only.

Detailed Breakdown for a \(3 \times 3 \times 3\) Cube

Let's verify the counts for all types of cubes for $n=3$:

• Cubes with 3 faces painted (Corners): There are always 8 corners in a cube. Formula: 8. Count = 8.
• Cubes with 2 faces painted (Edges): These are on the edges, excluding corners. There are 12 edges. Each edge has \(n-2\) such cubes. Formula: \(12 \times (n-2)\). For $n=3$, this is \(12 \times (3-2) = 12 \times 1 = 12\). Count = 12.
• Cubes with 1 face painted (Faces): These are on the center of each face. There are 6 faces. Each face has \((n-2) \times (n-2)\) such cubes. Formula: \(6 \times (n-2)^2\). For $n=3$, this is \(6 \times (3-2)^2 = 6 \times 1^2 = 6 \times 1 = 6\). Count = 6.
• Cubes with 0 faces painted (Inner): These are in the very center of the cube. Formula: \((n-2)^3\). For $n=3$, this is \((3-2)^3 = 1^3 = 1\). Count = 1.

Total number of cubes = $8 + 12 + 6 + 1 = 27$. This matches the total number of small cubes the large cube was cut into.

The number of small cubes painted on one face only is 6.

One of the options provided is 12. As shown above, 12 represents the number of small cubes painted on two faces only (the edge cubes, excluding corners).

Revision Table: Cube Cutting Formulas

Here is a summary of the formulas used for finding the number of small cubes with a specific number of painted faces when a large cube is cut into \(n^3\) smaller cubes:

Additional Information: Cube Cutting Concepts

What is 'n' in cube cutting problems?
'n' represents the number of divisions along each edge of the large cube. If a cube is cut into \(n^3\) small cubes, then there are 'n' small cubes along each edge of the large cube. In this problem, 27 small cubes means \(n^3=27\), so \(n=3\). There are 3 small cubes along each edge of the original large cube.

Why do corners have 3 painted faces?
The 8 corner cubes of the large cube are at the intersection of three faces. When the large cube is painted, these three exposed faces on the corner cubes get painted.

Why do edge cubes have 2 painted faces?
The cubes along the edges (but not at the corners) are at the intersection of two faces of the large cube. When the large cube is painted, these two exposed faces on the edge cubes get painted.

Why do face cubes (center) have 1 painted face?
The cubes in the exact center of each face of the large cube are only exposed on that one face to the outside of the large cube. When the large cube is painted, only this single exposed face on these cubes gets painted.

Why do inner cubes have 0 painted faces?
The cubes entirely within the interior of the large cube have no face exposed to the outside surface of the large cube. Therefore, none of their faces get painted.