Understanding the Geometry of Painted Cubes in a Rectangular Parallelepiped

In a recent geometry challenge, a rectangular parallelepiped of dimensions 4 cm, 6 cm, and 5 cm was painted red and then cut into smaller cubes, each with a side length of 1 cm. The intriguing question was how many of these cubes have only one face painted. Let's delve into the details and explore the mathematical solution step-by-step.

Step-by-Step Solution

First, visualize the parallelepiped as a 3D structure with dimensions 4 cm x 6 cm x 5 cm. Each small cube has a side length of 1 cm. There are three sets of similar faces:

Let's start with the first set of 4x5 faces. The number of cubes with one face painted on each of these faces is given by the product of two adjacent sides minus the corners and edges. Specifically, each 4x5 face has:

2 x 3 6 cubes with one side painted red.

Moving to the second set of 5x6 faces, each 5x6 face has:

3 x 4 12 cubes with one side painted red.

For the third set of 6x4 faces, each 6x4 face has:

4 x 2 8 cubes with one side painted red.

Adding these up, we get:

6 12 8 26 cubes.

Since there are two sets of similar faces, we multiply this by 2:

2 x 26 52 cubes.

Alternative Solution

To further simplify the process, consider the total number of unit cubes forming the parallelepiped. The total number is:

4 x 6 x 5 120 unit cubes.

Now, let's identify the cubes with only one face painted. The cube's faces without edges or corners that are painted are:

1 x 6 x 5 30 cubes on the bottom slice, 1 x 6 x 5 30 cubes on the top slice, and 2 x 6 x 5 60 cubes in the middle slice, excluding the corner pillars and the interior cubes.

The corner pillars in the middle slice are:

4 2 6 cubes (4 on the corners and 2 in the middle edges).

The interior cubes in the middle slice are:

2 x 4 x 3 24 cubes.

Therefore, the number of cubes with only one face painted in the middle slice is:

60 - 6 - 24 30 cubes plus the 60 from the bottom and top slices, giving us:

30 30 30 90 - 38 52 cubes.

General Formula

For a general case of dimensions a, b, and c, the number of cubes with only one face painted can be determined using the following formula:

2[a - 2b - 2 b - 2c - 2 c - 2a - 2]

Plugging the values of 4 cm, 6 cm, and 5 cm into this formula:

2[4 - 2(6) - 2 6 - 2(5) - 2 5 - 2(4) - 2] 2[4 - 12 - 2 6 - 10 - 2 5 - 8 - 2] 2[-22] 52

Conclusion

In conclusion, the number of cubes that have only one face painted is:

52 cubes.

This detailed step-by-step process not only provides a clear understanding of the problem but also illustrates the power of geometric reasoning and the elegance of mathematical formulas in solving complex problems.