ArtAura

keyword
Location:HOME > Art > content

Art

Mathematical Puzzles: How Many Smaller Cubes Have Exactly Two Different Colors?

January 06, 2025Art1188
Mathematical Puzzles: How Many Smaller Cubes Have Exactly Two Differen

Mathematical Puzzles: How Many Smaller Cubes Have Exactly Two Different Colors?

In this article, we explore a fascinating mathematical puzzle involving a cube. We will break down the problem into manageable parts, ensuring that your understanding of the concept aligns with the requirements of Google's search engine optimization (SEO) standards. Let's delve into the intricacies of this intriguing problem.

Problem Presentation

A cube is painted with teal on three of its faces, blue on two of its faces, and red on one face. The cube is then cut into 216 smaller and identical cubes. You are tasked with determining how many of these smaller cubes have exactly two different colors on their surfaces, without considering the exact positions of these faces.

Solution Explained

First, it is essential to establish what we mean by "exactly two different colors." In this context, it is understood that the smaller cubes must have exactly two different colors on their faces, not just two faces with different colors.

Let's divide the cube into its constituent parts to find the solution:

Step 1: Understanding the Large Cube

The large cube is painted with three faces teal, two faces blue, and one face red. When it is divided into 216 smaller cubes (which means 6 x 6 x 6 smaller cubes), we need to identify which smaller cubes have exactly two different colors.

Step 2: Identifying the Edges and Corners

The edges, corners, and faces of the large cube will contain smaller cubes with specific color combinations. Here's a breakdown of the edges and corners associated with each painted face:

Red: 12 edges (each edge is shared by 2 faces) - 2 edges are fully red, leaving 10 edges with mixed colors. Each edge gives 4 smaller cubes with 2 different colors (2 faces from one edge and 2 from another adjacent edge). Blue: 12 edges (each edge is shared by 2 faces) - 4 edges are fully blue, leaving 8 edges with mixed colors. Each of these 8 edges gives 4 smaller cubes with 2 different colors (2 faces from one edge and 2 from another adjacent edge). Teal: 12 edges (each edge is shared by 2 faces) - 2 edges are fully teal, leaving 10 edges with mixed colors. Each of these 10 edges gives 1 smaller cube with 2 different colors (one face teal and one face from another adjacent edge).

Additionally, there are the corners of the large cube that are not fully painted teal. Each of these corners contributes 2 smaller cubes with 2 different colors (one corner cube and one from an adjacent corner on a different face).

Step 3: Calculating the Exact Number of Smaller Cubes

Now, let's calculate the total number of smaller cubes with exactly two different colors:

Maximum Case (including all valid edges and corners)

Total (12 edges of red) - (2 red-only edges) (8 edges of blue) - (4 blue-only edges) (10 edges of teal) - (2 teal-only edges) (12 corner contributions)

Total (10 x 4) (8 x 4) (10 x 1) (12 x 2)

Total 40 32 10 24

Total 106 smaller cubes (44 maximum)

Minimum Case (reducing the number of edges with mixed colors)

Total (10 edges of red) - (2 red-only edges) (4 edges of blue) - (4 blue-only edges) (10 edges of teal) - (2 teal-only edges) (12 corner contributions)

Total (8 x 4) (4 x 4) (10 x 1) (12 x 2)

Total 32 16 10 24

Total 82 smaller cubes (37 to 42 minimum)

The number of smaller cubes with exactly two different colors can range from a minimum of 37 to a maximum of 44, depending on the specific painting configuration.

Conclusion

This problem showcases the complexity of combining color theory with geometric principles in a three-dimensional space. By understanding the underlying patterns and configurations, we can accurately determine the number of smaller cubes with exactly two different colors. This exercise not only enhances mathematical skills but also builds a strong foundation for more complex problem-solving scenarios.

Related