7.4. Vsevolod assembled an octahedron from eight identical triangles (depicted in the image), after which he painted each of the twelve segments composing it in red, blue, or green. It turned out that at each of the six vertices of the octahedron, segments of each color arrive. How many segments could have been painted green in total? (Find all the options and show that there are no others.)

![](https://cdn.mathpix.com/cropped/2024_05_06_8a7313c77a58492b8e01g-2.jpg?height=840&width=851&top_left_y=965&top_left_x=611)
Solution: Consider one of the colors. According to the condition, an edge of this color leaves each vertex. The total number of such exits is at least 6, and each edge leaves exactly two vertices. Therefore, the total number of edges of each color is at least 3. From this, it also follows that the number of edges of each color is no more than 6, otherwise there would not be enough edges of other colors.

As can be seen from the diagram, the edges can be colored such that exactly one edge of each color leaves each vertex. The total number of colored edges will be 9, leaving 3 more edges. These 3 edges can be colored in any way. Depending on this, the number of green edges will vary from 3 to 6.

Criteria: Only the answer - 0 points.

Proved that the number of edges is at least 3 - 2 points.

Proved that the number of edges is no more than 6 - 1 point.

Example for each number of edges - 1 point each.

These points are cumulative.