How Many Smaller Cubes Have No Painted Faces After Cutting a Larger Cube?

In this article, we will explore a geometric puzzle involving a cube and its painted faces. We start with a cube of 3 cm on each side that is painted red on all of its faces. The cube is then divided into smaller cubes, each with a side length of 1 cm. The goal is to figure out how many of these smaller cubes do not have any faces painted. We will walk through the problem step-by-step to ensure clarity.

Problem Analysis

Our initial cube has a side length of 3 cm. Let's begin by calculating the volume of the original cube:

Volume of the original cube: ( 3 text{ cm} times 3 text{ cm} times 3 text{ cm} 27 text{ cubic centimeters} )

When the original cube is divided into smaller cubes, each with a side length of 1 cm, the total number of these smaller cubes is:

Total number of smaller cubes: ( 3 times 3 times 3 27 )

Cube Division and Identifying Unpainted Cubes

Next, we need to determine which of these smaller cubes do not have any faces painted. To achieve this, we consider the structure of the original cube.

Inner Cube Analysis

The unpainted cubes will be located in the inner part of the original cube away from the outer layer. If we visualize the inner cube, it will be a smaller cube with side lengths reduced by the thickness of the outer layer, which is 1 cm on each side. Therefore, the inner cube has dimensions of 1 cm on each side.

Volume of the inner cube: ( 1 text{ cm} times 1 text{ cm} times 1 text{ cm} 1 text{ cubic centimeter} )

This means there is only one cube (the central one) that is completely unpainted.

Verifying with Options

Given the options provided, we need to check if the calculations align with the options:

a. 9 b. 3 c. 6 d. 12

Based on our analysis, the correct answer is that only 1 cube is completely unpainted. None of the given options A, B, C, or D match this result.

Misconceptions and Clarifications

It's important to note that other cubes may have different numbers of painted faces. For example:

The original cube has 8 vertices, and 8 cubes of 1 cm sides will have red paint on three faces. There will be 4 cubes per face (8 cubes in total) with three sides painted. There is only one cube in the center that is completely unpainted.

If you have any further questions or need clarification, feel free to ask!

Key Takeaways

The volume of the original cube with side length 3 cm is 27 cubic centimeters. The total number of smaller cubes when divided is 27. The only cube that is completely unpainted is the one in the center of the original cube.