We colored the faces of a regular dodecahedron with red, blue, yellow, and green such that any two adjacent faces have different colors. How many edges does the dodecahedron have where one of the two adjacent faces is blue and the other is green?
A dodecahedron has 12 faces, each of which is adjacent to exactly 5 others. The faces are labeled from $A$ to $L$ such that two faces of the dodecahedron are adjacent if and only if the corresponding regions on the first diagram are adjacent (the first diagram can be obtained by projecting the dodecahedron from a point "above" the $A$-labeled face onto the plane of the $L$-labeled face).

Suppose $A$ is red. Then $B, C, D, E$, and $F$ cannot be red. If $L$ is red, then $G, H, I, J$, and $K$ cannot be red. If $L$ is not red, then at most two of $G, H, I, J$, and $K$ can be red. Therefore, the number of faces that are the same color as $A$, including $A$ itself, is at most three. By the symmetries of the dodecahedron, this can be said about every face. Thus, each of the four colors appears on at most three faces; however, we have 12 faces in total, so each of the four colors must appear on exactly three faces.

Write down the three colors that can meet at each vertex. Since faces of the same color do not share a vertex and there are three faces of each color, each color is written down at $5 \cdot 3 = 15$ vertices, and thus $20 - 15 = 5$ vertices do not have any color written down. An edge connecting a blue and a green face must start from exactly one of the vertices where either the red or the yellow color is not present. This amounts to $2 \cdot 5 = 10$ vertices, but since each edge has two endpoints, there are exactly 5 edges where one face is blue and the other is green. Based on the work of Gábor Lippner (Fazekas M. Fóv. Gyak. Gimn., 12th grade).

Note. The dodecahedron can be colored in several ways that meet the conditions - one possible coloring is shown on the back cover - and for every valid coloring, it is true that for any two colors, there are exactly 5 edges of the dodecahedron where the two adjacent faces are of the given two colors.

![](https://cdn.mathpix.com/cropped/2024_05_02_a92da8a7c48288df6f11g-1.jpg?height=550&width=548&top_left_y=934&top_left_x=775)