An invisible ghost jumping on a regular hexagon Given a regular hexagon and an invisible ghost at one of the vertices of the hexagon (we don’t know which). We have a special gun, that can kill ghosts. In a step we are able to shoot the gun twice (i.e. choose two vertices and see if the ghost is there). After every step, the gost moves to an adjacent vertex. What is the minimum number of moves required to kill the ghost?
I have an example with 6 steps. I am sure this is not the minimum. My friend has an example for 4. So what is the minimum? And can you generalize for a regular $n$-gon? Thanks!

Brute force exhaustion of possible strategies gives two solutions requiring four turns:


*

*Shoot at $(1,3)$ then $(4,6)$ then $(2,4)$ then $(1,5)$

*Shoot at $(1,3)$ then $(4,6)$ then $(4,6)$ then $(1,3)$


along with reflections and rotations of these basic solutions. There are none requiring three turns.
To see that the second solution works, we can use the following method to analyze the locations where there could possibly be a live ghost:


*

*The ghost is among $1,2,3,4,5,6$. After shooting $(1,3)$, the ghost is among $2,4,5,6$.

*The ghost is among $1,3,4,5,6$. After shooting $(4,6)$, the ghost is among $1,3,5$.

*The ghost is among $2,4,6$. After shooting $(4,6)$, the ghost is at $2$.

*The ghost is among $1,3$. After shooting $(1,3)$, it can't be anywhere.

