NIMCET 2020 — Reasoning PYQ

Step 1: Understand the geometry of a cube

A standard cube has a total of $6$ faces. These faces exist in pairs of opposite sides. Specifically:

• Total number of faces = $6$

• Number of pairs of opposite faces = $3$

Step 2: Understand the adjacency rule

The problem states that no two adjacent sides can have the same color.

• On a cube, each face is adjacent to exactly $4$ other faces.

• The only face that is not adjacent to a given face is its opposite face.

Step 3: Calculate the minimum number of colors

To minimize the number of colors used, we can paint opposite faces with the same color, because opposite faces do not share any common edges (they are not adjacent).

Let the three pairs of opposite faces be:

1. $\text{Top}$ and $\text{Bottom}$ $\to$ Color 1

2. $\text{Front}$ and $\text{Back}$ $\to$ Color 2

3. $\text{Left}$ and $\text{Right}$ $\to$ Color 3

Therefore, the mathematical relation for the minimum number of colors $N$ required can be expressed as:

$$N=\frac{\text{Total faces}}{\text{Faces per non-adjacent set}}$$

$$N=\frac{6}{2}=3$$

Thus, a minimum of $3$ colors are required to paint the cube under the given conditions.

Correct Option: A