Question 12
Define a regular $n$-pointed star to be a union of $n$ line segments $P_{1}P_{2}$, $P_{2}P_{3}$, $\ldots$, $P_{n}P_{1}$ such that
- the points $P_{1}, P_{2}, \ldots, P_{n}$ are coplanar and no three of them are collinear;
- each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint;
- all of the angles at $P_{1}, P_{2}, \ldots, P_{n}$ are congruent;
- all of the $n$ line segments $P_{1}P_{2}, P_{2}P_{3}, \ldots, P_{n}P_{1}$ are congruent; and
- the path $P_{1}P_{2}\ldots P_{n}P_{1}$ turns counterclockwise at an angle less than $180^{\circ}$ at each vertex.
There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. Find all possible values of $n$ such that there are exactly 29 non-similar regular $n$-pointed stars.
- the points $P_{1}, P_{2}, \ldots, P_{n}$ are coplanar and no three of them are collinear;
- each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint;
- all of the angles at $P_{1}, P_{2}, \ldots, P_{n}$ are congruent;
- all of the $n$ line segments $P_{1}P_{2}, P_{2}P_{3}, \ldots, P_{n}P_{1}$ are congruent; and
- the path $P_{1}P_{2}\ldots P_{n}P_{1}$ turns counterclockwise at an angle less than $180^{\circ}$ at each vertex.
There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. Find all possible values of $n$ such that there are exactly 29 non-similar regular $n$-pointed stars.