Painted Faces on Sliced Cubes

Problem Overview

Consider a cube with a side length of 8 cm that is painted on all its faces. If this cube is cut into 1 cm cubes, how many of these smaller cubes will have exactly one of their faces painted?

Solution Steps

The total number of 1 cm cubes that can be obtained from the larger 8 cm cube is calculated as follows:

Total Number of Cubes

Formula

For a 5 cm cube (since we are working with cubes that have sides of 1 cm:

No. of 1 cm cubes 5 x 5 x 5 125

Explanation

This means there are 125 smaller cubes in the larger cube.

Excluding Painted Cubes Along Edges and Corners

The smaller cubes that lie along the edges and corners of the larger cube have more than one face painted. We need to subtract these from our count of cubes with exactly one face painted.

Excluding Cubes Along the Edges

No. of cubes on each face 8 x 8 64. Since there are 6 faces, the total number is 64 x 6 384. However, the cubes along the edges have been counted more than once.

Cubes along the edges that have more than one face painted are:

No. of cubes along the edges 8 x 4 - 4 x 4 28 (4 cubes at each corner have been counted twice).

Final Calculation

No. of cubes with exactly one face painted 64 - 28 36 per face. For all faces, this is 36 x 6 216.

Geometric Probability for Larger Cubes

5 cm Cube Example

For a 5 cm cube sliced into 1 cm cubes, the number of cubes with exactly one face painted can be calculated using the following formula:

Formula

No. of cubes with one face painted 6 x (n-2)2

Where n is the side length of the cube in cm.

Here, n 5:

No. of cubes with one face painted 6 x (5-2)2 6 x 9 54

General Approach

For any cube of side length n cm, the number of 1 cm cubes with exactly one face painted can be calculated using the formula mentioned. This is a valuable geometric consideration for understanding the distribution of painted faces in sliced cubes.

Conclusion

The problem of determining how many of the 1 cm cubes have exactly one face painted is a fascinating example of applying geometric principles to real-world scenarios. Understanding these calculations can provide insights into more complex geometrical and physical problems in mathematics and engineering.