Nine trolls are placed in the cells of a three-by-three square.
The trolls in neighboring cells shake hands with each other.
Later they re-arrange themselves in the square and the neighbors greet each other once more.
Then they repeat it again for the 3rd time.

Prove (or provide a counterexample) that there is at least one pair of trolls who didn’t greet each other.

Based on a problem in Russian "Kvantik",2012

(In reply to re: computer solution by Charlie)

BTW, an example of the farthest it was able to get without duplicating a handshake is:

Posted by Charlie on 2015-07-23 12:22:03