   Date: Mon, 11 Nov 91 17:45:36 -0500 (EST)   From: Dan Hoey <hoey@aic.nrl.navy.mil >
~~~ ~~~ Subject: Rubik's Cube question

Jeff Baggett (baggett@mssun7.msi.cornell.edu) asks on the sci.math
newsgroup:

1. I am seeking a description of the group of symmetries associated
with Rubiks cube. I have some ideas but they aren't particularly
elegant. Can someone suggest a paper?

Jeff,

I have looked into this somewhat. As far as I know, the symmetries of
the 3^3 cube are just the symmetries of the cube, but in larger sizes
we can do better. The best way of looking at this is to imagine that
there is a (N-2)^3 cube sitting inside your N^3 cube, and smaller
cubes within, and you are trying to solve them all together.

Suppose we address each cubelet of the N^3 cube using cartesian
coordinates (x,y,z), where (0,0,0) is the center of the cube (for N
odd) and no cubelets have any coordinate zero if N is even. The
maximum absolute value of the coordinates is [N/2].

```Then for 1<=I<=[N/2], there is a symmetry
F[I]:(x,y,z)->(f(x),f(y),f(z)),
where f(I)=-I, f(-I)=I, and f(x)=x otherwise.

Then for 1<=I<J<=[N/2], there is a symmetry
E[I,J]:(x,y,z)->(e(x),e(y),e(z)),
where e(I)=J, e(J)=I, e(-I)=-J, e(-J)=-I, and e(x)=x otherwise.
```

These are symmetries of the cube group, and they map elementary moves
to elementary moves (provided we take an elementary move to be a
rotation of the slab of N^2 cubelets that have a particular nonzero
value of a particular coordinate). Symmetries of the cube group that
preserve elementary moves are useful in the study of local minima in
the cube group.

It turns out that if you only want to consider the outside of the cube
(ignoring the (N-2)^3 cube inside) all of these symmetries are still
present except F[[N/2]] and E[I,[N/2]].

I mentioned these symmetries in a note to the Cube-Lovers mailing list
in 1983. I called E[I,J] evisceration, F inflection, and F[[N/2]]
exflection in that note (where I was dealing explicitly with only the
4^3). The discussion of the relation to local minima took place in
1980. Let me know if you'd like a copy of these messages.

I ran into these symmetries earlier, though. They are symmetries of
the N^3 tic-tac-toe board! I would not be surprised if they arise in
some other connection in mathematics, but I have never run into them.
They generalize into larger dimensions, as well.

I've also taken the liberty of Cc'ing the Cube-Lovers list with this
note. If you'd like to be on that list, you may ask of
"Cube-Lovers-Request@AI.AI.MIT.Edu".

Dan Hoey
Hoey@AIC.NRL.Navy.Mil     