three grasshoppers jumping on a plane.

Question

The problem is dead simple:

Three grasshoppers sit on a plane not in a line. Every second just one of the grasshoppers hops symmetrically over one of the others. Can they return to the initial positions after n seconds?

The very tempting answer that it is not possible if the n is odd and possible if the n is even is correct. Also it is not hard to show that if n is even it is correct (you can just use one grasshopper and jump back and forward). But I fail to get a prove for n is odd.

I understand that I have to look for some sort of invariant and show that when n is odd there is a contradiction, but I fail to find the invariant.

Answer 1

Hint: if positions of grasshoppers are $g_1, g_2, g_3$, check out what happens to $\det( g_2 - g_1, g_3 - g_1 )$.

Answer 2

(This is @Adayah 's answer told in other words.)

Let $g_i$ $(1\leq i\leq3)$ be the positions of the three grasshoppers. The triangle $\triangle$ formed by the $g_i$ changes its orientation at each jump. When the $g_i$ are arranged counterclockwise along $\partial\triangle$ at the start then they will be arranged clockwise after an odd number of jumps.