Date: Tue, 10 Jan 95 18:14:15 -0500
From: der Mouse <mouse@collatz.mcrcim.mcgill.edu >
~~~ Subject: Difficulty of Skewb

If you're familiar with the SKEWB, I would like to know whether it's
harder or easier than the classic 3x3x3 Rubik's Cube (I suspect it's
simpler, but it has fewer symmetries).

I think I have a skewb. Each face is cut into a square turned 45
degrees, and four little 45-45-90 triangles, right? And there are four
cuts you can turn it about, all passing through the center of the cube?

In that case, here's my opinion. Mathematically, it verges on trivial
compared to the 3x3x3, but because it's deep-cut, there are no stable
portions to take the psychological place of the face centers on the
3x3x3, and each turn affects all six faces so, unlike the 2x2x2, there
are no unaffected faces to pick up the slack. These combine to make it
difficult for psychological reasons.

I called it verging on trivial, but that is relative to the 3x3x3.
There is some structure. If the 3x3x3 has 8!*3^8*12!*2^12 "imaginable"
configurations (of which only 1/12 are reachable from any given state),
the skewb has only 8!*3^8*6! "imaginable". Of these, far less than
1/12 are reachable; in my estimate, not having thought about it hard,
the reachable number is (4!*4!)*(3^8/3)*(6!/2), or only one in 420 of
the "imaginable" configurations, a reduction achieved largely due to
the constraints on corner mixing: they form two tetrahedral sets, and
it is not possible to mix the sets. Thus 8! turns into 4!*4!; this is
a factor of 70. The additional factor of 6 comes from the usual sort
of parity constraints, and again is only my guess.

The size of the thing is thus somewhere on the order of 500 million
configurations. This is why I called it trivial next to the 3x3x3. :-)
The group structure in terms of facicles, for what's-his-name to sic
GAP on should he care to, derived from the following facicle labeling

```           +----------+
|  6     7 |
|     8    |
|  9    10 |
+----------+----------+----------+----------+
|  1     2 | 11    12 | 21    22 | 31    32 |
|     3    |    13    |    23    |    33    |
|  4     5 | 14    15 | 24    25 | 34    35 |
+----------+----------+----------+----------+
| 16    17 |
|    18    |
| 19    20 |
+----------+

is:
```
```Cut 1: (15  7  4) (24 31 19) (17 22 35) (20 25 34) (18 23 33)
Cut 2: (12  1 20) (10 32 25) (21  6 34) (22  7 31) (23  8 33)
Cut 3: (11 19 22) ( 2 35  7) ( 9  4 31) ( 6  1 32) ( 8  3 33)
Cut 4: (14 25  6) (16 34  1) ( 5 20 32) ( 4 19 35) ( 3 18 33)
```

(This formulation holds facicle 13 fixed at all times.)

der Mouse

mouse@collatz.mcrcim.mcgill.edu