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Mail-from: From ponder@austin.ibm.com Thu Jul 31 17:16:55 1997
Date: Thu, 31 Jul 1997 16:13:34 -0500
From: ponder@austin.ibm.com (Ponder)
Message-Id: <9707312113.AA33486@roosevelt.austin.ibm.com>
To: Cube-Lovers@ai.mit.edu
Subject: puzzle to be simulated
Cc: ponder@austin.ibm.com


I've seen a number of Rubik's Cube simulations on the web,
and was wondering if any of you would be interested in
implementing the following puzzle that I call "the hell-hole".
It has 16 faces that each work like the faces of a rubik's
cube, but:

	1] The faces are layed out on a 4x4 grid. Each
	   "corner" joins four surrounding faces instead
	   of three.

	2] The grid is rolled into a cylinder and then
	   joined at both ends to forma torus.

However, the torus is given a "twist" when you join the two
ends together, as follows:

	  1 2 3 4
	  _ _ _ _
	a|_|_|_|_|b
	b|_|_|_|_|c
	c|_|_|_|_|d
	d|_|_|_|_|a
	  1 2 3 4

First join the 1-2-3-4 sides together to form the cylinder,
then the a-b-c-d ends together to get the torus. Each of the
squares is a 3x3 face like a Rubik's Cube.

The combinatorics get a *lot* messier because of the twist.
Without it, you can't "flip" the edge pieces. With it, you
can, but only by moving the edge-piece in a full-circle around
the torus.

No way to build it, either, since the pieces would need to
flex between convex and concave. It could be simulated on a
computer, however. I have a paper coming out in the Journal of
Recreational Mathematics on how to solve these kinds of things,
and it is pretty messy.

Thanks,

	Carl Ponder
	ponder@austin.ibm.com