Shapovalov A.V.

For which $n$ can all the edges of an $n$-sided prism (bases are $n$-gons) be colored in three colors so that all three colors meet at each vertex and each face (including the bases) has sides of all three colors?
Let the prism be painted in the required manner; then, in the base, there are three edges of all three colors in a row (otherwise, the base is painted with a period of 2 in two colors). Consider such a triplet and number the colors in it counterclockwise. It is not difficult to verify that the coloring of this segment uniquely determines the coloring of the segment above it, the corresponding lateral edges, and the next base edges (see figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_3b3613da627e706c6878g-20.jpg?height=591&width=648&top_left_y=1734&top_left_x=705)

We see that the next counterclockwise base edge has color 1. Repeating the reasoning, we get that the next base edge has color 2, and so on. Thus, the lower face will be painted with a period of 3. Therefore, $n$ is a multiple of 3.

On the other hand, for $n$ being a multiple of 3, the coloring with a period of 3, shown in the figure, satisfies the condition.

## Answer

For $n$ being multiples of 3.

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