Kalinin D.A.

Find the maximum number of colors in which the edges of a cube can be painted (each edge in one color) so that for each pair of colors there are two adjacent edges painted in these colors. Adjacent are considered edges that have a common vertex.

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There are several ways to color the edges of a cube in six colors while satisfying the condition of the problem. Here is one of them:

![](https://cdn.mathpix.com/cropped/2024_05_06_26f3e6a31f0097f83768g-13.jpg?height=620&width=694&top_left_y=1165&top_left_x=682)

We will show that more than six colors are not possible.

Suppose we have colored the edges of the cube in seven or more colors. Since the cube has a total of 12 edges, there must be a color, for example, white, in which only one edge is painted. For each edge of the cube, there are exactly four adjacent edges. Therefore, the white color can be paired with no more than four other colors, which means there cannot be more than five different colors in total. This is a contradiction.

## Answer

6 colors.