For which $n$ can a regular $n$-gon be placed on a sheet of lined paper so that all its vertices lie on the lines?

(Lines are parallel straight lines, located at equal distances from each other.)
Suppose a regular $n$-gon can be placed on lined paper. Let $O$ be the center of this polygon. Rotate all lines by an angle of ${ }^{360} \%$ around point $O$ and draw on the same diagram both the original and the new (resulting from the rotation) parallel lines. Now each vertex of the $n$-gon lies at the intersection of two grid lines, meaning the $n$-gon can be placed such that all its vertices fall on the nodes of some lattice. However, as proven in the article by A.A. Egorov "Lattices and Regular Polygons," this is only possible for $n=3,4$ or 6.

Thus, only for these $n$ can a regular $n$-gon be placed on lined paper.

## Answer

For $n=3,4,6$.