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 Post subject: Solving every even permutation with a single commutator.Posted: Wed Apr 30, 2014 12:27 am

Joined: Fri Nov 05, 2010 2:20 am
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http://www.speedsolving.com/forum/showt ... -challenge

Saw this on the speedsolving forum. I don't know much about commutators, but this looks like a nice challenge. In fact, someone may have solved it and wanted to make it his PHD publication.

What do you think?

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 Post subject: Re: Solving every even permutation with a single commutator.Posted: Wed Apr 30, 2014 3:01 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
rubikcollector123 wrote:
Saw this on the speedsolving forum. I don't know much about commutators, but this looks like a nice challenge. In fact, someone may have solved it and wanted to make it his PHD publication.

What do you think?

It's trivial to prove impossible by counter-example unless further restrictions are placed on what is meant by "even permutation". For example, on a Helicopter cube the sequence [RU FU]x3 performs an even permutation of the whole puzzle (even though it's odd for each orbit) and it can't be resolved via a commutator or any sequence of only commutators.

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 Post subject: Re: Solving every even permutation with a single commutator.Posted: Thu May 01, 2014 2:40 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
I had a chance to think about this more. The Helicopter Cube orbits nicely demonstrate the idea but twist / orientation of pieces can be used too.

For example, if you 3-cycle three corners of a Skewb in the same orbit then it twists an corner in the other orbit. This is an even permutation of pieces. It's even an even permutation of stickers.

A Super Megaminx has a similar property where the twist of a center prevents the puzzle from being solved with commutators alone.

Lastly, in the above examples you could argue that twists in stickered pieces shouldn't be excluded from the challenge but in a Master Pyraminx you can 3-cycle centers and no stickered piece on the puzzle is twisted (although the core is twisted).

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