Understanding the Painted Cube Problem

This problem involves a large cube that is painted and then cut into many smaller identical cubes. We need to determine how many of these smaller cubes have paint on exactly one of their faces.

Let the side length of the large cube be \(L\) and the side length of the smaller cubes be \(s\).

Given:

• Side of large cube, \(L = 49\) cm
• Side of small cube, \(s = 7\) cm

First, we need to find out how many smaller cubes fit along one edge of the large cube. This number, let's call it \(n\), is given by:

\[ n = \frac{L}{s} \]

Substituting the given values:

\[ n = \frac{49 \, \text{cm}}{7 \, \text{cm}} = 7 \]

So, the large cube is cut into \(7 \times 7 \times 7\) smaller cubes.

The total number of smaller cubes is \(n^3 = 7^3 = 343\).

Locating Cubes with One Painted Face

When a large cube is painted on all its faces and then cut, the smaller cubes will have different numbers of painted faces depending on their original position in the large cube:

• Corner cubes: Located at the 8 corners of the large cube. They have 3 painted faces.
• Edge cubes: Located along the edges of the large cube (but not at the corners). They have 2 painted faces.
• Face cubes: Located on the faces of the large cube (but not on the edges or corners). They have 1 painted face.
• Inner cubes: Located completely inside the large cube. They have 0 painted faces.

We are interested in the smaller cubes that have only one face painted. These are the cubes that were originally on the faces of the large cube, away from the edges and corners.

Calculating Cubes with Only One Painted Face

Consider one face of the original large cube. This face is a square grid of \(n \times n\) smaller cube faces. In our case, it's a \(7 \times 7\) grid.

The smaller cubes on this face that have only one painted face are those that are not on the edges of this face grid. On a \(7 \times 7\) grid, the cubes along the edges are those in the first and last rows, and the first and last columns.

The number of cubes along one edge is \(n\). The number of cubes along each edge that are *not* corners is \((n-2)\). There are 12 edges on a cube, so there are \(12 \times (n-2)\) edge cubes (2 painted faces).

The cubes on the face that have only one painted face form an inner square on that face. The dimensions of this inner square are \((n-2) \times (n-2)\).

For our case, \(n=7\), so the inner square on one face contains \((7-2) \times (7-2) = 5 \times 5 = 25\) smaller cubes.

Each of these 25 cubes is located on one specific face of the large cube and has only that face painted.

A cube has 6 faces. Since each face contributes \((n-2)^2\) cubes with one painted face, the total number of smaller cubes with only one painted face is:

\[ \text{Number of one-faced cubes} = 6 \times (n-2)^2 \]

Substituting \(n=7\):

\[ \text{Number of one-faced cubes} = 6 \times (7-2)^2 \]

\[ \text{Number of one-faced cubes} = 6 \times (5)^2 \]

\[ \text{Number of one-faced cubes} = 6 \times 25 \]

\[ \text{Number of one-faced cubes} = 150 \]

Therefore, there are 150 smaller cubes having only one face painted.

Summary of Cube Types by Painted Faces

We calculated the number of small cubes with one painted face. Let's quickly check the counts for other types to ensure the total adds up correctly:

• 3-faced cubes (corners): 8
• 2-faced cubes (edges): 60
• 1-faced cubes (faces): 150
• 0-faced cubes (inner): 125

Total cubes = \(8 + 60 + 150 + 125 = 343\). This matches the total number of small cubes \(7^3 = 343\).

The number of smaller cubes with only one face painted is 150.

Revision Table: Cube Cutting Formulas

These formulas are valid when the large cube is painted on all 6 faces and \(n > 2\).

Additional Information on Painted Cube Problems

Painted cube problems are common in spatial reasoning and quantitative aptitude tests. They require you to visualize how a 3D object is divided and how properties (like paint) are distributed. The key is to understand the different locations within the large cube (corners, edges, faces, interior) and how these relate to the number of exposed faces before cutting.

For variations of this problem, consider:

• What if the cube is painted on only 5 faces? Or 4? Or 3 (adjacent or opposite)?
• What if the large shape is not a cube but a cuboid?
• What if the cuts are not uniform?

In cases where not all faces are painted, you need to analyze which specific small cubes (based on their original position) would receive paint. For example, if only 5 faces are painted, the cubes on the unpainted face will have 0 painted faces instead of 1 (if they were face cubes), 1 instead of 2 (if they were edge cubes connected to the unpainted face), and 2 instead of 3 (if they were corner cubes connected to the unpainted face). Cubes on edges or faces not connected to the unpainted face would retain their original number of painted faces.

Mastering the standard cube-cutting formulas for all faces painted provides a strong foundation for tackling these more complex variations.