Cube Coloring Problem

Question: How many minimum numbers of colours will be required to paint all the sides of a cube without the adjacent sides having the same colours?

Answer: (A) 3

Explanation:

To paint all the sides of a cube such that no adjacent sides have the same color, a minimum of 3 colors are required. Here's how:

1. Color 1: Paint the top face of the cube with Color 1.
2. Color 2: Paint an adjacent side (e.g., the front face) with Color 2.
3. Color 3: Paint another adjacent side (e.g., the right face) with Color 3.
   This side is adjacent to both the top (Color 1) and the front (Color 2), so it needs a new color.
4. Remaining Faces:
   • The bottom face is opposite the top face (Color 1), so it can also be painted with Color 1.
   • The back face is opposite the front face (Color 2), so it can also be painted with Color 2.
   • The left face is opposite the right face (Color 3), so it can also be painted with Color 3.

In this arrangement, no two adjacent faces share the same color.
Therefore, 3 colors are the minimum required.