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From: Dan Hoey <Hoey@aic.nrl.navy.mil>
To: Cube-Lovers@ai.mit.edu, ponder@austin.ibm.com
In-Reply-To: <9707312113.AA33486@roosevelt.austin.ibm.com>
Subject: Re: puzzle to be simulated

Carl Ponder describes his "hell-hole" puzzle that (if I undestand
correctly) has 36 facets laid out like

   m : n n n : p p p : r r r : s s s : a
   ..:.......:.......:.......:.......:..
   s : a a a : b b b : c c c : d d d : e
   s : a a a : b b b : c c c : d d d : e
   s : a a a : b b b : c c c : d d d : e
   ..:.......:.......:.......:.......:..
   d : e e e : f f f : g g g : h h h : j
   d : e e e : f f f : g g g : h h h : j
   d : e e e : f f f : g g g : h h h : j
   ..:.......:.......:.......:.......:..
   h : j j j : k k k : l l l : m m m : n
   h : j j j : k k k : l l l : m m m : n
   h : j j j : k k k : l l l : m m m : n
   ..:.......:.......:.......:.......:..
   m : n n n : p p p : r r r : s s s : a
   m : n n n : p p p : r r r : s s s : a
   m : n n n : p p p : r r r : s s s : a
   ..:.......:.......:.......:.......:..
   s : a a a : b b b : c c c : d d d : e

with opposite boundaries identified so that the letters match up.  A
turn rotates 25 facets--one of the 3x3 "faces" marked with dots--and
the sixteen neighboring facets from the neighboring faces.

> 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.

If I've got the puzzle right, you could get edge flippability just by
using a 3x3 array of faces instead of 4x4.  Or perhaps 3x5.  Another
nice idea that uses a "square" torus is

   k : l l l : m m m : n n n : d
   ..:.......:.......:.......:..
   h : a a a : b b b : c c c : j
   h : a a a : b b b : c c c : j
   h : a a a : b b b : c c c : j j j : k k k : l
   ..:.......:.......:.......:.......:.......:..
   n : d d d : e e e : f f f : g g g : h h h : a
   n : d d d : e e e : f f f : g g g : h h h : a
   n : d d d : e e e : f f f : g g g : h h h : a
   ..:.......:.......:.......:.......:.......:..
   c : j j j : k k k : l l l : m m m : n n n : d
   c : j j j : k k k : l l l : m m m : n n n : d
   c : j j j : k k k : l l l : m m m : n n n : d
   ..:.......:.......:.......:.......:.......:..
   f : g g g : h h h : a a a : b b b : c c c : j

This corresponds to a square of side sqrt(13) with opposite edges
identified, cut on the bias into 13 square faces.
    ............................
    :.'   a  .' `.. m .'     `.:
    : `..   .'     `..'   n  .':
    :    `..'   b  .' `..   .' :
    : d  .' `..   .'     `..' d:
    :   .'     `..'   c  .' `..:
    :`..'   e  .' `..   .'     :
    :.' `..   .'     `..'   j  :
    :'     `..'   f  .' `..   .:
    :   k  .' `..   .'     `..':
    :..   .'     `..'   g  .' `:
    :  `..'   l  .' `..   .'   :
    :h .' `..   .'     `..'   h:
    : .'  a  `..'   m   .'`..  :
    :.'.......'b`.......'.n..`.:

after which each face is cut up into nine facets.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil