Calculating Cubes with One Painted Face

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

Understanding the Cube Division

First, let's figure out how many small cubes fit along each edge of the large cube. The large cube has a side length of 12 cm, and the smaller cubes have a side length of 2 cm.

Number of small cubes along one edge = \( \frac{\text{Side length of large cube}}{\text{Side length of small cube}} \)

Number of small cubes along one edge = \( \frac{12 \text{ cm}}{2 \text{ cm}} = 6 \)

So, there are 6 small cubes along each edge of the original large cube. This means the large cube is cut into a 6x6x6 arrangement of smaller cubes.

The total number of smaller cubes is \( 6 \times 6 \times 6 = 6^3 = 216 \).

Locating Cubes with Only One Painted Face

The smaller cubes that have only one face painted are those that were originally in the center of each face of the large cube. These cubes were only exposed to the paint on one side before the cutting occurred.

Imagine one face of the large cube. It's made up of a grid of \( 6 \times 6 \) small squares, which are the faces of the small cubes.

• The cubes at the 4 corners of this face are also corners of the large cube and have 3 painted faces.
• The cubes along the edges of this face (but not the corners) are edge cubes of the large cube and have 2 painted faces.
• The cubes in the absolute center of this face are the ones we are interested in – they have only 1 painted face.

Calculating Cubes with One Painted Face Per Face

On one face of the large cube, which is a \( 6 \times 6 \) grid of small cube faces, the number of cubes with only one painted face are those not on the edges.

To find the number of non-edge cubes on one face, we can subtract the outer layer of cubes from each dimension of the face grid:

• Number of small cubes along the length of the inner part = \( 6 - 2 \) (subtract 1 from each side) = \( 4 \)
• Number of small cubes along the width of the inner part = \( 6 - 2 \) (subtract 1 from each side) = \( 4 \)

So, the number of small cubes with only one face painted on a single face of the large cube is \( 4 \times 4 = 16 \).

Total Number of Cubes with One Painted Face

The large cube has 6 faces, and each face has 16 small cubes with only one painted face.

Total number of cubes with one painted face = Number of faces \( \times \) Number of one-face painted cubes per face

Total number of cubes with one painted face = \( 6 \times 16 = 96 \).

Therefore, there are 96 smaller cubes that have only one face painted green.

Checking the total: \( 64 + 96 + 48 + 8 = 216 \), which matches the total number of small cubes.

Conclusion

The number of smaller cubes that have only one face painted is 96.

Revision Table - Cube Cutting Problem

Let's quickly review the key aspects of this cube cutting problem to reinforce understanding.

• Large Cube Side: 12 cm
• Small Cube Side: 2 cm
• Cubes per Edge (n): \( \frac{12}{2} = 6 \)
• Total Small Cubes: \( n^3 = 6^3 = 216 \)
• Cubes with 1 Painted Face: Located on the center of each original face.
• Formula for 1 Painted Face: \( 6(n-2)^2 \)
• Calculation: \( 6 \times (6-2)^2 = 6 \times 4^2 = 6 \times 16 = 96 \)

Additional Information - Analyzing Painted Cubes

When a large cube is painted on all faces and cut into smaller cubes of equal size, the smaller cubes can have 0, 1, 2, or 3 painted faces. The number of cubes for each category depends on their position in the original large cube and the number of small cubes along each edge (denoted by 'n').

• 3 Painted Faces: These are the cubes located at the corners of the original large cube. A cube has 8 corners, so there are always 8 such small cubes, provided \( n \ge 2 \).
• 2 Painted Faces: These cubes are located along the edges of the original large cube, but not at the corners. There are 12 edges, and each edge contributes \( (n-2) \) such cubes. So, the formula is \( 12(n-2) \), provided \( n \ge 2 \).
• 1 Painted Face: As calculated in this problem, these cubes are located in the center of each face of the original large cube. There are 6 faces, and each face contributes \( (n-2)^2 \) such cubes. The formula is \( 6(n-2)^2 \), provided \( n \ge 2 \).
• 0 Painted Faces: These are the cubes located completely inside the original large cube, not exposed to any surface. They form a smaller cube of size \( (n-2) \times (n-2) \times (n-2) \) inside. The formula is \( (n-2)^3 \), provided \( n \ge 2 \).

Understanding these categories and formulas helps solve similar problems involving painted cubes.