Painting and Partitioning Cubes: How Many Smaller Cubes Have Painted Sides?

When a cube with a side length of 8 cm is painted red on all sides and then cut into smaller cubes, each with a side length of 2 cm, how many of these smaller cubes will have one or more sides painted red? This article explores the solution to this intriguing problem and introduces several key concepts in geometric geometry.

Understanding the Geometry and Volume

Let's begin by breaking down the problem step by step. We start with a larger cube that has a side length of 8 cm. Calculating its volume and using it to determine the number of smaller cubes is the first step.

The volume of the larger cube, denoted as (V_{text{large}}), is:

V_{text{large}} 8 times 8 times 8 512 text{ cm}^3

Each smaller cube has a side length of 2 cm, giving it a volume of:

V_{text{small}} 2 times 2 times 2 8 text{ cm}^3

The number of smaller cubes obtained from cutting the larger cube is:

text{Number of smaller cubes} frac{V_{text{large}}}{V_{text{small}}} frac{512}{8} 64

Identifying Cubes with Painted Sides

The newly formed smaller cubes are arranged in a 4 by 4 by 4 grid, as (8 div 2 4). This structure allows us to count the number of smaller cubes with one or more painted sides by first identifying those without any painted sides, often referred to as inner cubes.

Unpainted Inner Cubes

The interior of the larger cube forms a smaller cube with side lengths reduced by 2 cm on each side (since 1 cm is removed from each face when cutting). Therefore, the side length of the inner cube is:

4 - 2 2 text{ cm}

The volume of this inner cube is:

V_{text{inner}} 2 times 2 times 2 8 text{ cm}^3

The number of inner cubes is:

text{Number of inner cubes} frac{8}{8} 1

Calculating Painted Cubes

To find the number of smaller cubes with at least one side painted, we subtract the number of unpainted cubes from the total number:

text{Number of painted cubes} 64 - 1 63

Thus, 63 of the smaller cubes have one or more sides painted red.

Alternative Method: Face Painting Analysis

Another way to solve this problem involves examining the painted faces. When the larger cube is painted, each face has:

24 cubes with one face painted (4 along each edge, but excluding the corners and the outer edge). 24 cubes with two faces painted (located at the edges, excluding corners). 8 cubes with three faces painted (the corner cubes).

Summing these gives a total of 56 smaller cubes with at least one side painted red.

The Importance of Surface Area in Cubes

The total surface area of the larger cube, after it is painted, covers 48 square centimeters per face, or 4224 square centimeters total (8 times 8 times 6). Understanding the surface area helps in visualizing the painted regions more precisely.

Using the geometric principles discussed here can help in solving a variety of similar problems involving partitioning and painting of cubes. This concept is central to fields such as geometric geometry and has applications in engineering and architecture, particularly in understanding structural integrity and surface treatment of materials.

By exploring these principles, one not only solves mathematical puzzles but also gains valuable insights into the practical applications of geometric concepts.