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Algorithm to solve this task

Hi everyone! I'm wandering if there is some actual algorithm to solve this task without brute-forcing. I mean, sure, if it works it ain't stupid, but still. I was trying to find a proper solution, but the only thing I stumble upon is just removing the ring with the biggest number of connections, which was proven to be wrong.

Created: May 17, 2016, 9:53 p.m.
Updated: May 18, 2016, 6:57 a.m.
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TheDVolk007