Date: Sat, 07 Oct 95 23:55:00 -0500
From: Mark Longridge <mark.longridge@canrem.com >
~~~ ~~~ Subject: Picture Cubes

David Singmaster (ZINGMAST@VAX.SBU.AC.UK) writes on
Wed Sep 13 11:50:09 1995:
------------------------------------------------
I think the ranking is not quite fair because the puzzles are of
very different types. E.g. the 15 puzzle has nearly as many patterns
as the 3^3 but no one would claim it was anywhere near as difficult.
Indeed the Babylon Tower has 36! = 3.72 x 10^41 basic positions.
One can divide by some small value such as 2 or 6 or perhaps more,
depending on what one considers the same position. This puts it
between 3 and 4 in your list, but it is not a difficult puzzle, except
that it is hard to see the gradations of the colors! Indeed, the
commercial 7 x 7 'fifteen puzzles' have 49! =6.08 x 10^62 basic
patterns - again one has to divide by something, in this case 2.
This falls between 2 and 3 in your list, but again it is hardly a
difficult puzzle.

So I think you are comparing puzzles which are of such different
type that the number of patterns is not a fair comparison of their
difficulties.I would group them in three (or perhaps 2) types.
Rubik Cube, etc.
Fifteen Puzzles, etc. in the plane.
Cylindrical Puzzles - barrels, etc.
------------------------------------------------

I quite agree. One of my reasons for making that list was to simply
rank all the puzzles by number of combinations only, to show the
feasibility (or lack of) for a brute force search to find
God's Algorithm.

The major drawback, as you point out, is that difficulty in solving
is not only a function of the number of combinations.

```Dr. Singmaster continues:
------------------------
Re your Case E.  Almost all the picture cubes have all four
orientations distinct on the face centres - both those with nine little
pictures on each face and those with a big picture spread over all
nine facelets.  These are actually pretty common.
-------------------------------------------------
```

Case E, that is cases that have only a fraction of the total possible
number of combinations for a Rubik's picture cube, are unfortunately
well represented in my own cube collection.

The following cubes are all in Case E:

Rubik's Calendar Cube, Rubik's Cube 4th Dimension, Rubik's World,
Blind Man's Cube (from Germany), Royal Wedding Cube (with Charles
& Di).

Although I don't doubt that, over all, these cases are exceptional.
In the case of Rubik's World there are 3 blank centre pieces, and
in the Royal Wedding Cube only 2 opposite faces can show all 4
possible orientations.

```Name                  Combinations      Inventor
```
``` 8. Picture Cube (3x3x3) (E) 8.8*10^22      Erno Rubik, Dan Hoey
9. Calendar Cube (3x3x3)(F) 4.4*10^22      Marvin Silbermintz
10. Rubik's Cube 4th Dim.(D) 1.1*10^22      Erno Rubik
11  Rubik's World (G)        2.7*10^21      Erno Rubik
12. Royal Wedding Cube       6.9*10^20      Unknown
13. Rubik's Cube (3x3x3)     4.3*10^19      Erno Rubik

-> Mark <-
```