Date: Wed, 29 May 96 10:56:15 +0100
From: Andy Southern <a.southern@ic.ac.uk >
~~~ ~~~ Subject: Ultimate Rubik's Cube. (how to make a 4x4x4)

I was thinking about something similar to this a while back.

I asked (as did many others) about 4x4x4s, but soon after I sent that E-mail
I realised that I was holding one with a few extra pieces.

If we can consider a 3x3x3 Rubik's cube as a 2x2x2 with mid-edges and
centres, in effect we can reverse this and ignore the centres and mid-edges
of a 3x3x3 thus making it into a 2x2x2.

Anyone who wanted a 2x2x2 could just pull the stickers from the central
columns and rows of a 3x3x3. As this makes those pieces indistinguishable,
they are no longer part of the puzzle.

the relationship between 4x4x4 and 3x3x3 is slightly harder, but the above
is true for 4x4x4 and 2x2x2 (but if ANYONE tries that with a 4x4x4, I'll hit
them!!!!!!!).

A more common cube than the 4x4x4 is the 5x5x5 (which is still in
production, c/o Uwe Meffret). This can be transformed from a 5x5x5 into a
4x4x4 by removing the central lines of stickers. It can also be transformed
into a 3x3x3 (why anyone would want to.................) by removing columns
& rows 2&4 from each side, and 2x2x2 (I won't bother saying it.........) by
removing columns and rows 2,3,&4 from each side.

Higher orders of cubes aren't in production, but apparently do exist in
cyberspace, These would display similar properties.

The pattern is simple: smaller cubes with odd number of pieces per side can
be incorporated with other pieces to form larger cubes with odd number of
pieces per side. etc. etc. etc.

```2x2x2 + (extras) = 3x3x3
2x2x2 + (extras) = 4x4x4
3x3x3 + 4x4x4 + (extras) = 5x5x5
```

I'm sure there is some mathematical proof to what I am trying to say, but
I'm no mathematician.

It might start off:

```                        Pieces          Jump(from last)         used before:
2x2x2                   8                       8
0
3x3x3                   26                      18                             8
4x4x4                   56                      30
8
5x5x5                   98                      42
74
6x6x6                   152                     54
???
```

I would say that the ultimate Rubik cube was in fact the 2x2x2 because it
features in all the solutions. However, based on this logic, the 8086 is the
ultimate P.C. as any P.C. can run 8086 software, but an 8086 can't run 80386
software................

I hope this E-mail hasn't been too scatter brained...................

Andy.

Fact: did you know that British Airways has more Super-Sonic flying time
than any air force in the world??