Solving the Painted Cubes Puzzle: A Geometric Challenge

When dealing with solids of various geometric shapes, such as cubes, applying mathematical principles often reveals fascinating patterns. Consider a cube with a side length of 5 cm that is entirely painted. Upon cutting this cube into 1 cm cubes, how many of these smaller cubes will have exactly one of their sides painted? This problem combines 3D geometry with a touch of logical deduction, making it not only a fun challenge but also an excellent learning tool in teaching geometric thinking and problem-solving skills.

Introduction to Painted Cubes

The concept of painting a solid cube and dissecting it into smaller cubes has been the subject of many mathematical puzzles. In this article, we explore a specific case where a 5 cm cube is painted entirely and then sliced into 1 cm cubes. Our focus is on determining how many of these smaller cubes will have exactly one side painted.

Understanding the Solution

To solve this puzzle, we begin by considering that the original cube has 6 faces. Each face of the cube, when divided into smaller 1 cm cubes, will have a 5 by 5 grid, which equals 25 cubes per face. If we focus only on the cubes that lie on the surface but not on the edges or corners, we can deduce the number of cubes with exactly one side painted.

Step-by-Step Calculation

1. Total Number of 1 cm Cubes: The total number of 1 cm cubes that result from slicing the 5 cm cube is 125 (since (5 times 5 times 5 125)). 2. Cubes with Exposed Faces: If we examine any single face of the cube, there will be a 3 by 3 grid of cubes in the middle that will have exactly one side painted. This middle section contains 9 cubes. Since there are 6 faces on the cube, we need to multiply this by 6 to get the total number of cubes with exactly one side painted, resulting in 54 cubes (since (9 times 6 54)).

Deeper Insight into the Puzzle

Another approach to solving this puzzle involves using a formula. Let's derive this formula:

General Formula for One-Sided Painted Cubes

The general formula for the number of smaller cubes with exactly one side painted in a cube of side length (n) is: [6(n - 2)^2]

For a cube of side length 5 cm, we can substitute (n 5) into the formula:

[6 times (5 - 2)^2 6 times 3^2 6 times 9 54]

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Conclusion

Solving geometric puzzles like the painted cubes problem not only sharpens mathematical skills but also enhances logical reasoning. By understanding the solution method and applying the appropriate formulas, one can efficiently tackle similar problems. Whether you are a student, a teacher, or someone who enjoys mathematical puzzles, the painted cube challenge is a great way to engage with geometric concepts.