Unpainted Small Cubes within a Painted Large Cube: A Mathematical Inquiry

In this article, we will explore a mathematical problem involving a large cube made up of 27 smaller cubes. The large cube is painted on the outside, and our goal is to determine how many smaller cubes would have none of their sides painted after the large cube is taken apart.

Problem Statement and Initial Observations

A large cube is made up of 27 small cubes, which is conveniently expressed as 27 33. Therefore, the large cube is a 3 × 3 × 3 cube. When the cube is painted on the outside, only the small cubes on the outer layer will have paint on their sides. The small cubes that are completely unpainted are those that are not exposed to the outside.

Analysis and Solution

Let's start by identifying the structure of the large cube:

The large cube is 3 × 3 × 3.

The small cubes that are completely surrounded and thus not painted are located in the center of the cube. Here is a step-by-step breakdown:

Step 1: Identify the Structure

The large cube is a 3 × 3 × 3 cube, which can be visualized as follows:

(X, X, X) (_, X, X) (X, X, X) (X, X, X) (X, X, X) (X, X, X) (_, X, X) (X, X, X) (X, X, X)

Where X represents a small cube that is exposed to the outside and _ represents a small cube that is completely surrounded and thus not painted.

Step 2: Determine the Painted Cubes

The outer layer of the 3 × 3 × 3 cube consists of:

8 corner cubes (each with 3 exposed faces) 12 edge cubes not counting the corners (each with 2 exposed faces) 6 face center cubes (each with 1 exposed face)

The remaining small cube in the center of the 3 × 3 × 3 structure is the only one that is completely surrounded and thus not painted.

Step 3: Conclusion

There is 1 small cube that has none of its sides painted.

Alternative Cutting Methods

Let's explore some alternative cutting methods to further investigate the number of unpainted small cubes:

8-Cut Method

First, we can cut the large cube through the center along all 3 axes. This will produce 8 smaller cubes of the same size, which we will call the 8-cut. In this case, the number of unpainted cubes is still 1, as the central small cubes are untouched.

20-Cut Method

Alternatively, we can cut the large cube into 27 equal cubes of edge 1/3, then combine 8 of them to form a cube of edge 2/3. This will result in 20 pieces, with 19 of them having an edge of 1/3 and 1 having an edge of 2/3, which we will call the 20-cut. Here are the possible outcomes of this cutting method:

1 2/3 edge cube 7 1/3 edge cubes 20 total pieces 27 smaller cubes

This 20-cut method can produce a variety of outcomes depending on the arrangement and cutting method used.

Analysis of Various Cuts

8-Cut to 20-Cut Combination

If we first perform the 20-cut and then the 8-cut to one of the 1/2 edge cubes, the possible configurations are:

1 2/3 edge cube with no exposed faces (1 hidden cube) 4 tiny cubes with edge 1/6 are embedded (4 hidden cubes) 6 tiny cubes inside (6 hidden cubes) 7 tiny cubes inside (7 hidden cubes)

By rearranging the cubes, we can achieve different numbers of hidden cubes. For example:

Move tiny cubes to the ends and bring the 2 small cubes together to achieve 0 hidden cubes Exchange the positions of 9 tiny cubes with their touching 1/2 edge cube to achieve 0, 1, or 3 hidden cubes respectively

20-Cut to 8-Cut Combination

Regardless of the cutting methods, the core insight remains that the only cube that is completely surrounded and thus not painted is the one in the center of the 3 × 3 × 3 arrangement. This problem demonstrates the unique properties of the 3 × 3 × 3 cube and the way in which small cubes can be arranged to explore different numbers of unpainted cubes.