Vasilev N.B.

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?

(The lines are parallel and are spaced 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 $360^{\circ} \%$ around point $O$ and draw both the original and the new (resulting from the rotation) parallel lines on the same diagram. 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, a regular $n$-gon can only be placed on lined paper for these values of $n$.

## Answer

For $n=3,4,6$.