Understanding the Painted Cube Problem: Solution and Analysis

The painted cube problem is a classic geometry challenge that involves analyzing the distribution of painted faces on a cube when it is subdivided into smaller cubes. In this article, we will explore the intricacies of the problem and provide a detailed solution step-by-step. We will also discuss some key mathematical concepts and their applications to this problem.

Solution to the Painted Cube Problem

Consider a cube with side lengths of 5 cm, which is painted on all its faces. This cube is then sliced into 1 cm cubes. The goal is to determine how many of these smaller cubes will have exactly one of their faces painted.

Determine the Total Number of 1 cm Cubes

The volume of the original 5 cm cube is calculated as follows:

[5 cm x 5 cm x 5 cm 125 cm3]

Since each 1 cm x 1 cm x 1 cm unit will be a separate cube, we can conclude that there are 125 individual 1 cm cubes in the original 5 cm cube.

Identifying Cubes with Exactly One Face Painted

To solve the problem, we need to count the number of smaller cubes that have exactly one face painted. These cubes are located on the faces of the larger cube but not on the edges or corners.

Counting the Interior Cubes on Each Face

Consider one face of the 5 cm cube. This face is a 5 cm x 5 cm square, consisting of 25 smaller 1 cm x 1 cm squares. However, the cubes on the edges and corners of this face cannot have exactly one face painted.

Excluding Edges and Corners

Each face of the cube has 12 edge cubes (4 on each edge) and 8 corner cubes. Each edge cube shares its painted face with an adjacent face, and each corner cube has three painted faces. Therefore, we need to count only the interior cubes of each face.

The interior of a 5 cm x 5 cm face consists of a 3 cm x 3 cm square, which contains:

[3 cm x 3 cm 9 text{ cubes}]

This is true for each of the 6 faces of the cube. Therefore, the total number of 1 cm cubes with exactly one face painted is:

[6 times 9 54 text{ cubes}]

Alternative Formula for Calculation

There is a general formula to calculate the number of smaller cubes with exactly one face painted for an n x n x n cube:

[6(n - 2)^2]

Using this formula for our 5 cm cube (n 5), we get:

[6(5 - 2)^2 6 times 3^2 6 times 9 54]

Conclusion

In summary, the number of 1 cm cubes that have exactly one of their faces painted in a 5 cm cube, when sliced, is 54. This problem not only tests basic geometry and arithmetic skills but also requires an understanding of cube geometry and spatial reasoning.

Understanding and solving the painted cube problem can be useful in various fields, including geometry, engineering, and computer science. It provides a framework for analyzing and solving similar problems involving the distribution of properties across smaller units.

Further Exploration

To deepen your understanding of this problem, you can explore how the solution changes for different cube sizes. Try calculating the number of 1 cm cubes with exactly one face painted for a 4 cm cube or a 6 cm cube. This exercise will help solidify your skills in solving similar problems and understanding the underlying mathematical concepts.

By working through these examples, you will gain a deeper insight into the geometry of cubes and the distribution of properties across sub-units. This knowledge can be applied to various fields, including design, construction, and even in theoretical mathematics.

Key Terms: painted cube problem, 1 cm cubes, face painting