Counting Cubes with Only One Face Painted in a Rectangular Parallelepiped

Introduction

In this article, we will explore a problem involving a rectangular parallelepiped that has been painted and then cut into 1 cm3 unit cubes. Specifically, we will determine how many of these unit cubes have exactly one face painted. We will present a step-by-step approach to solving this problem, and provide a general formula for any given dimensions.

Problem Description

A rectangular parallelepiped with dimensions 4 cm x 6 cm x 5 cm is painted all over and then cut into 1 cm cubes. The cubes that we are interested in are those that have only one of their six faces painted. This problem is a classic application of combinatorial geometry.

Understanding the Geometry and Analyzing the Problem

To solve the problem, we need to analyze the structure of the cube and its painted faces. The painted faces of the original parallelepiped must be the same as the faces of the unit cubes with exactly one painted face. Let's break down this analysis step by step.

Step-by-Step Solution

Step 1: Identifying Cubes with Only One Face Painted

Consider a cube of dimensions 4 cm x 6 cm x 5 cm. We can divide the parallelepiped into three slices: the top, the bottom, and the middle slice. Each of these slices has 4 cm x 6 cm 24 unit cubes, and we need to identify which of these have exactly one face painted.

Bottom Slice: The bottom slice has 24 unit cubes. The outermost cubes (those on the edges and corners) are not counted, as they have more than one face painted. The innermost cubes with only one face painted are: 1 x 6 x 5 30 - 4 corner cubes (each counted twice), so we subtract 8 to avoid double counting: 30 - 8 22. Top Slice: The top slice is similar to the bottom slice, so it also has 22 unit cubes with only one face painted. Medium Slice: The middle slice is more complex. It consists of 2 x 6 x 5 60 unit cubes, which include 4 corner cubes and 24 inner cubes. We need to subtract the corner cubes and the inner cubes (those with no painted faces) to find the cubes with only one face painted: 60 total - 4 corner cubes - 24 inner cubes 28 unit cubes with only one face painted.

Total Calculation:

22 (bottom) 22 (top) 28 (middle) 52 unit cubes.

General Formula

For a rectangular parallelepiped with dimensions a cm x b cm x c cm, the number of cubes with only one face painted can be calculated using the formula:

Number of single-side-painted cubes 2[(a - 2) x (b - 2) (b - 2) x (c - 2) (c - 2) x (a - 2)]

Applying this formula to our specific example:

Number of single-side-painted cubes 2[(4 - 2) x (6 - 2) (6 - 2) x (5 - 2) (5 - 2) x (4 - 2)]

Number of single-side-painted cubes 2[2 x 4 4 x 3 3 x 2] 2[8 12 6] 2 x 26 52

Conclusion

In conclusion, we have demonstrated a method to solve the problem of counting unit cubes with only one face painted in a rectangular parallelepiped. The specific example provided shows that 52 unit cubes have exactly one face painted. A general formula has been derived for any dimensions of the rectangular parallelepiped.