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Nine trolls (Posted on 2015-07-23) Difficulty: 2 of 5
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

No Solution Yet Submitted by Ady TZIDON    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: computer solution | Comment 5 of 9 |
(In reply to computer solution by Daniel)

Daniel, 

It sounds as if you're saying that even after only one shuffle, there has to be a repeated handshake from the initial position. However, this is not the case:

123
456
789

159
834
276

Note that diagonals do not count as handshakes; otherwise we'd have more than 12 handshakes per stage of the game and more than 36 for the set of 3 grids.

  Posted by Charlie on 2015-07-23 11:37:25
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