   Date: Wed, 22 Dec 93 13:58:42 -0500 (EST)   From: Dan Hoey <hoey@aic.nrl.navy.mil >
~~~ ~~~ Subject: The 4^3 and 3^4 Rubik puzzles

[ Cube-Lovers,

There has recently been a discussion on Usenet group rec.puzzles
about some cube topics. There were a few pieces of new information,
such as that you can now get Ishi's 5^3 cubes in a lot of places (I
got mine in Learningsmith's) for about \$35. Here's a message I sent
that's relevant to some Cube-Lovers topics.

By the way, I'm still working through Jerry Bryan's articles on his
brute-force program and his approach to symmetry. I hope to get a
]

eric@gsb002.cs.ualberta.ca (Holleman Eric) wrote:

By the way, I found the Revenge somewhat easier than the Cube, and I
don't think that it was because of my familiarity with the earlier
puzzle.

x87bennett@gw.wmich.edu (Joe) wrote:

From my experience, if you can solve a Rubik's Revenge, you can
solve the Cube very easily. Once you get each of the middle 2 cubes
on each edge to match, and all 4 center cubes on each face to match,
it works exactly like a Rubik's cube.

and alan@saturn.cs.swin.oz.au (Alan Christiansen) wrote:

I have both. I solved both. The 4x4x4 is a superset of the 3x3x3.
ie by fixing all the face centres and then pairing all edges you are
left with a 3x3x3 cube, except that when you have solved this 3x3x3
there may be a single pair of edges flipped. This is impossible
on a real 3x3x3. Fixing this requires a middle layer to be rotated
1/4 revolution and then all the bits put back.

I cant see how it can be [any] easier than a 3x3x3.

I, too, found the 3^3 easier than the 4^3. But I can imagine ways in
which a solver could find the 4^3 easier. Let us first consider a 4^3
with the faces fixed, the edges together, and the correct simulated
edge flip parity. I would solve this as if it were a 3^3, and a lot
of people do. But another solver might find it easier to take
advantage of the extra moves that are not possible on a 3^3. To take
a concrete example, it could be that the solver has a hard time with
flipping edges by pairs, as is needed to solve the 3^3. On the 4^3
you can flip one edge at a time. So the solver would find the 4^3
position easier than the corresponding position on a 3^3. If the
solver finds this so much easier that it overcomes the difficulty of
putting the faces and edges together--or in fact puts the faces and
edges together in the course of solving the corners and the edge
positions--then the 4^3 could be easier. It depends on the solution
procedure.

alan@saturn.cs.swin.oz.au (Alan Christiansen) continues:

ANyway the real reason I am writing this is that I have written
a cube simulator.
It can simulate 3x3x3 4x4x4 5x5x5 .... cubes.
I am working on 4x4x4x4 cube simulation.

This is interesting, as there is more than one way to model the
four-dimensional cube problem. Consider the 3^4 cube. It has eight
hyper-faces, each in the shape of a cube. One model of this puzzle is
that you could turn any face of any hyper-face as if it were a face of
a 3^3 Rubik's cube. In a second model, you cannot move part of a
hyper-face, but can turn each hyper-face as if it were a solid cube in
space. A third model allows either kind of move.

These models are different from each other. The second model permits
the face centers of the hyper-faces to move around, whereas in the
first model only edges and corners move. In the first model, odd
permutations of corners are possible, which is not true in the second
model. Of course, the third model is the closure of the first two.

According to Hofstatder's column reprinted in _Metamagical_Themas_,
there is an unpublished 1982 manuscript by H J Kamack and T R Keane
entitled ``The Rubik Tesseract''. They calculated the size of the
group of the 3^4 puzzle, but I don't know which model was used.

Alan Christiansen indicates he has gone directly to the 4^4 puzzle. I
don't know which model he plans, or if the models become more similar
with the extra possibilities inherent in the larger cube. I don't
even know whether he plans to figure out how big the groups are or
whether they are identical.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil     