   Date: Tue, 14 Nov 95 09:13:41 -0400   From: Jerry Bryan <jbryan@pstcc.cc.tn.us > ~~~ Subject: God's Algorithm for the 1x1x1 Rubik's Cube

Solving the 1x1x1 Rubik's cube is probably a bit silly and whimsical, but
let's look at it anyway.

I was led in this direction by rereading some articles in the archives
from Dan Hoey and others concerning NxNxN Rubik's cubes. For example,
consider Dan's discussion "Cutism, Slabism, and Eccentric Slabism" from
1 June 83 19:39:00. Sometimes degenerate cases are slightly interesting.
I guess the 1x1x1 case is the most degenerate we have, unless you want to
consider the 0x0x0.

It seems to me that either cutism or slabism, as Dan calls them, reduce to
whole cube rotations for the 1x1x1 case. For example, a quarter turn face
turn or a quarter turn slice would be interpreted as a whole cube quarter
turn for the 1x1x1. Hence, the cube group for the 1x1x1 is simply C, the
group of 24 rotations of the cube.

By analogy with some of our previous work, I can think of essentially
three different ways to model the 1x1x1.

1) With the 2x2x2, we normally wish to consider the puzzle solved if
each face is all of one color. That is, whole cube rotations are
to be considered equivalent. With the Singmaster fixed face
center view of the 3x3x3, the issue of whole cube rotations does
not arise. But with the 2x2x2 we would normally consider
(for example) RL' equivalent to I. The most common way to
accomplish this type of equivalence is to fix one of the corners.

If we fix one of the corners of the 1x1x1, then we have a most
remarkable puzzle. There is only one state, nothing can ever
move, and the puzzle is always solved.

2) A second way to model the 2x2x2 such that whole cube rotations are
considered to be equivalent is to consider the set of states to be
the set of cosets of C, that is, the set of all xC.

If we take this approach with the 1x1x1, then there is only one
coset, namely iC (or just C, if you prefer). The cube can rotate,
but all 24 states are considered to be equivalent and the puzzle
is always solved.

3) Finally, if you model the 2x2x2 in such a way that whole cube
rotations are considered to be distinct, then you are really
modelling the corners of the 3x3x3. Indeed, a naive program
that simply modelled the permutations of the 2x2x2 facelets would
in fact unwittingly be modelling the corners of the 3x3x3.

If you take the same approach of modelling the permutations of the
1x1x1 facelets, then you in effect are considering whole
cube rotations to be distinct. You have a very easy problem,
but the problem is not totally trivial as it is with approach #1
or approach #2. The rest of this note will therefore consider the
problem of the 1x1x1 cube where whole cube rotations are considered
to be distinct.

Since we need to deal with whole cube rotations, I will use lower case
letters as our standard E-mail simulation of Frey and Singmaster's script
notation for whole cube quarter turns -- t for Top, r for Right, etc. We
need only three of the six letters because, for example, we have l=r',
d=t', b=f', etc. I will use t, r, and f.

We know before we start that there are 24 states. We also know before we
start that these 24 states form 5 M-conjugacy classes, where M is the set
of 48 rotations and reflections of the cube. (There are 10 M-conjugacy
classes of M, of which 5 are rotations and 5 are reflections.) Hence, any
discussion of God's algorithm will involve 5 conjugacy classes and 24
states.

The obvious searches to look at are for qturns only, and for qturns plus
hturns. We may generate the qturn case as C=<t,r,f>. We may generate
the qturn plus hturn case as C=<t,r,f,t2,r2,f2>.

Qturns Only

Distance Conjugacy Positions
from Classes
Start

```   0          1        1  {i}
1          1        6  {t,t',r,r',f,f'}
2          2       11  {tt,rr,ff},{tr,tr',tf,tf',t'r,t'r',t'f,t'f'}
3          1        6  {ttf,ttf',ffr,ffr',rrt,rrt'}
---        ----     ----
Total         5       24
```

Qturns Plus Hturns

Distance Conjugacy Positions
from Classes
Start

```   0        1        1  {i}
1        2        9  {t,t',r,r',f,f'},{t2,r2,f2}
2        2       14  {tr,tr',tf,tf',t'r,t'r',t'f,t'f'},
{t2f,t2f',f2r,f2r',r2t,r2t'}
---      ----     ----
Total       5       24
```

There are some additional problems we can look at. For an example, an
interesting problem on the 3x3x3 is variously called the stuck axle
problem or the five generator problem. In the case of the 1x1x1, we have
the "two generator problem" because we certainly can generate C as C=<t,f>
(Proof: r=tft'). But can we generate C with only one generator? The
answer is no. (Proof: Order(i)=1, Order(t)=4, Order(tt)=2, Order(tf)=3,
and Order(ttf)=2. All the orders are less than 24. Note that it suffices
to calculate the order for one representative of each conjugacy class.) I
will leave it as an exercise for the reader to determine the lengths of
each of the 24 positions if we generate C as <t,f>, and to determine the
appropriate conjugacy classes to take into account the symmetry of C
generated as <t,f>.

By the way, do we know the minimum number of generators required to
generate the 3x3x3? Here I do not mean the minimum number of quarter
turns. I am asking the question if we are permitted to use as generators
any elements of G.

Here is one final item about the 1x1x1. We do not know how many subgroups
of G there are for the 3x3x3. But we do know how many subgroups of C
there are. There has been much discussion of the 98 subgroups of M which
can be arranged in 33 conjugacy classes. The subgroups of C are simply
those subgroups of M which consist entirely of rotations. There are 30
such subgroups, and they may be arranged in 11 conjugacy classes.

``` = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)                jbryan@pstcc.cc.tn.us
```     