Message #2223

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] "Three strange {8,4} colorings"
Date: Fri, 01 Jun 2012 10:22:22 -0500

I can’t help in the solution department, but here is some basic info about
the topologies :)

*{8,4} 5-Color*
Faces 5
Edges 8
Vertices 4
Euler Characteristic 1

*{8,4} 9-Color*
Faces 9
Edges 16
Vertices 8
Euler Characteristic 1

*{8,4} 10-Color*
Faces 10
Edges 16
Vertices 8
Euler Characteristic 2

So the first two have the topology of the projective plane
(non-orientable), and the last of the sphere.

Anyone want to figure out counts and kinds (henagons, digons, etc.) of the
particular faces? The 10C should have a nice, planar graph representation.
Roice

On Fri, Jun 1, 2012 at 5:23 AM, Melinda Green <melinda@superliminal.com>wrote:

> That was about all that Roice said about these three puzzles and nobody
> seems to have noticed them. That’s understandable because he was
> dropping hundreds of new puzzles on us at the same time and the {8,4}s
> were at the very end. Well I stumbled into them a couple of days ago and
> can say "Mighty strange indeed!" As you know, I’ve been focusing on
> edge-turning puzzles that I can solve intuitively and found the 5-color
> and 10-color versions to be a lot of fun. They start out easy enough and
> finish with enough of a brain stretch to be quite rewarding to solve. I
> had tried and failed with with the 9-color version after a couple of
> half-hearted attempts, but since I had solved the other two I figured it
> was time to make a serious attempt to collect all three, and all I can
> say is "OH………………………….., MY GOD!" Roice mentioned
> that they have some interesting topological properties that would be fun
> to study and I completely agree. The best single word I can find to
> describe tthe 9-color version is "perverse". Rather than try to
> describe what I found, I want to invite Ed and Nan and any other serious
> puzzlers to give these a shot. Then please let us know what you think.
> Do they yield easily to your standard methods? Do the face and vertex
> turning version behave as oddly as the edge-turning? I definitely want
> to learn more about these bad boys!
>
> -Melinda
>