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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.

Salvador Dali
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  • Let the grasshoppers sit on the vertexes of a triangle then the questions says that (i) every second each grasshopper hops to any other vertex OR (ii) in one second one grasshopper hops the second second next grasshopper hops and then third grasshopper hops in third second. Please clear me that question is saying (i) or it is saying (ii). If it is (ii) then please give an example to make it more clear. – Singh May 21 '15 at 21:45

2 Answers2

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

Adayah
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  • what does $g_2 - g_1$ stands for? Is this an attempt to write coordinates of grasshopper1 jumped over grasshopper2? – Salvador Dali May 21 '15 at 08:40
  • No, it is a vector with initial point $g_1$ and terminal point $g_2$. The sign of determinant basically tells if the iteration through $g_1, g_2, g_3$ is clockwise or counterclockwise. – Adayah May 21 '15 at 09:21
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(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.

Christian Blatter
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