Message #145

From: David Vanderschel <DvdS@Austin.RR.com>
Subject: Re: Cubes, beer and Beethoven
Date: Thu, 05 May 2005 14:55:49 -0500

>In the 3D version there are two independent parity
>problems if you follow the ‘ultimate solution’: the
>last pair of edges is inverted 50% of the time,
>resolvable by reconstructing the faces and the last
>two corners are switched 50% of the time, resolvable
>by reconstructing the edges.

The relevant ‘parity’ considerations for 3^3 are a bit
more complicated than that. See the following article
on the subject, which I wrote in 1980!: http://tinyurl.com/2zy4o

>I guessed there might be three parity problems in the
>4^4:

Since you speak of "centers", I am wondering if you
meant "3^4". Assuming that, then there are some known
relationships which must hold. Eric Balandraud’s
article on the MC4D web site, "Calculating the
Permutations of 4D Magic Cubes", includes (somewhat
implicitly) some insight into such ‘parities’. His
observations are correct, but not very obvious. ;-)
Link to Eric’s page:
http://www.superliminal.com/cube/permutations.html

>Was I lucky?

Probably.

>Would a layer by layer solution, as Marc Guegueniat
>has found, remove the parity issues?

No.

Regards,
David V.