Question 9
You are given a regular $n$-gon. Each vertex is marked $+1$ or $-1$. A move consists of changing the sign of all the vertices which form a regular $k$-gon for some $1 \lt k \leq n$. [A regular $2$-gon means two vertices which have the center of the $n$-gon as their midpoint.] For example, if we label the vertices of a regular $6$-gon $1, 2, 3, 4, 5, 6$, then you can change the sign of $\{1, 4\}$, $\{2, 5\}$, $\{3, 6\}$, $\{1, 3, 5\}$, $\{2, 4, 6\}$ or $\{1, 2, 3, 4, 5, 6\}$. Show that for (1) $n = 15$, (2) $n = 30$, (3) any $n \gt 2$, we can find some initial marking which cannot be changed to all $+1$ by any series of moves. Let $f(n)$ be the largest number of markings, so that no one can be obtained from any other by any series of moves. Show that $f(200) = 2^{80}$.