Message #1011

From: Andrey <andreyastrelin@yahoo.com>
Subject: Re: [MC4D] Magic Tiles
Date: Fri, 16 Jul 2010 04:14:23 -0000

Melinda,
>
> Your method of solving the {7,3} (AKA the Klein quartic
> <http://math.ucr.edu/home/baez/klein.html>) is not guaranteed to work.
> Nelson Garcia was the first to point out the possible "two bottoms"
> <http://games.groups.yahoo.com/group/4D_Cubing/message/837> problem that
> can result which lead to a nice discussion of topology and genus. In
> short, the top-down method is only guaranteed to work on surfaces
> topologically equivalent to the sphere (I.E. genus == 0). All of the
> puzzles that we’ve dealt with up till then had this property, but the
> Klein quartic has genus == 3.

No, different topology is not a problem for this method. As I mentioned, I keep the set of "non-solved centers" connected, so when the next selected face is connected with two different parts of "perimeter" of solved centers area, then I check that it’s not splitting area but only reduces its genus. The problem may occur when the puzzle has "last corner problem" (possibility of single twisted 3C), but I don’t know is it possible or not.

Andrey