## Task A-3.2. (4 points)

Legoplus is a body consisting of seven equal cubes joined in such a way that there is one cube that shares a common face with each of the remaining six cubes.

Each face of the legoplus must be painted with one color. How many colors are minimally needed to do this so that no two adjacent faces are painted the same color?
## Solution.

Obviously, there are three sides of the Lego brick for which it is true that any two are adjacent. For example, such sides are marked with the letters $A, B$ and $C$ in the image below. From this, we conclude that at least three colors are needed to color the Lego brick in the desired way.

(2 points)

![](https://cdn.mathpix.com/cropped/2024_05_30_5dbbe9edfad2d8808b54g-16.jpg?height=411&width=439&top_left_y=905&top_left_x=817)

If we color side $A$ and all sides parallel to it white, side $B$ and all sides parallel to it gray, and side $C$ and all sides parallel to it black, we will get a coloring with three colors that satisfies the conditions of the problem. This coloring is shown in the image below.

![](https://cdn.mathpix.com/cropped/2024_05_30_5dbbe9edfad2d8808b54g-16.jpg?height=414&width=439&top_left_y=1660&top_left_x=817)

Therefore, the smallest number of colors needed to color the outer sides of the Lego brick so that no two adjacent sides are the same color is three.

(2 points)