Six colored cube | Puzzles for Interview - Interview Preparation PDF Download

Introduction

If you have a six-faced cube and six distinct colors, you might be wondering how many unique ways you can paint the cube so that no two faces have the same color. Let's break it down step by step.

Solution

Step 1: Fix the Color of the Top Face

To avoid repetition, let's start by fixing the color of the top face. This means that the top face will always be the same color in every possible arrangement.

Step 2: Paint the Bottom Face

Once the top face is fixed, we have five colors left to choose from for the bottom face. This means that the bottom face can be painted in 5 ways.

Step 3: Arrange the Remaining Four Colors

Now that the top and bottom faces are painted, we have four colors left to choose from for the remaining four faces. These four colors can be arranged in a circular pattern around the cube.

To calculate the number of distinct circular arrangements for n distinct objects, we use the formula (n-1)! For our four colors, this means there are (4-1)! = 3! = 6 distinct circular arrangements.

Step 4: Multiply the Possibilities

To find the total number of unique ways to paint the cube, we simply multiply the number of options for the bottom face (5) by the number of circular arrangements for the remaining four colors (6). This gives us a total of 5*6 = 30 unique ways to paint the cube with six distinct colors.

Conclusion

If you have a six-faced cube and six distinct colors, there are 30 unique ways to paint the cube so that no two faces have the same color. By following the steps above, you can easily calculate the possibilities and create your own unique design.