Date: Fri, 14 Jan 94 21:59:00 -0500
From: Mark Longridge <mark.longridge@canrem.com >
~~~ Subject: Higher Order Cubes

Anton Dovydaitis writes:
>I've been getting a lot of requests for 4x4x4 cubes, and we're
>looking into getting them. However, I have a couple questions.
>
>1) Why are 4x4x4 cubes so interesting? Do the additional symmetries
>make for interesting questions, are they more fun, or easier to
>solve?

A well lubed 4x4x4 cube is still relatively easy to physically
manipulate. As der Mouse suggests, it is arguably the largest
interesting cube from a solver's point of view. Once one starts
actually twisting with a 5x5x5 cube, the physical problems
become more severe, e.g. the stickers come off easier,
turning the slice you want to is more of a challenge, etc.

In the virtual realm of computer cubing the patterns you can
create are more elaborate, although I find in practice that
finding pretty patterns on the 5x5x5 can become wearisome
due to fact there are 9 centre pieces per side!

2) It appears to me that if you know how to solve the 3x3x3 Rubik's
cube, then you can easily solve the 5x5x5 rubiks (i.e., the
solution is derivative). For example, you can treat the inner 3x3
faces of the 5x5x5 as a single 3x3x3 cube.

Using the 4x4x4 cube we can produce a single exchange of centres
and an exchange of edge pairs, and we can invert a single edge pair.
Thus we can construct all the impossible 3x3x3 patterns except those
involving a twist of a single corner! That is why I think the 4x4x4
cube is a good cube to have. The individual centre cubies can
naturally wander all over the cube, and on the 3x3x3 cube they are
fixed.

In the case of the 5x5x5 cube, lots of the 3x3x3 knowledge does
help. When dealing with the 5x5x5's middlemost slice (let's call
one such slice "fm" for middlemost front slice) some of the
3x3x3 move sequences will move the appropriate edges, but now
these sequences will also move centre pieces, specifically the
ones which have no counterpart on the 3^3 and 4^3.

To solve cubes 4x4x4 and greater requires new sequences to
efficiently move centre cubies at will, and in the case of
the 5^3 there really is no standard language to interchange
move sequences and label individual cubies.

I find having a letter as a mnemonic helps, so I'll suggest the
following as an extension of Singmaster's 4x4x4 notation
for the 5x5x5 cube:

L left face l inner left slice lm left middlemost slice
R right face r inner right slice rm right middlemost slice
F front face f inner front slice fm front middlemost slice
B back face b inner back slice bm back middlemost slice
U up face u inner up slice um up middlemost slice
D down face d inner down slice dm down middlemost slice

```Again, we follow the alphabetic component by a number to
signify the rotation (1 = 90, 2 = 180, 3 = 270 or -90)
```

This is overkill, and we can dispense with rm, bm and dm.

Thus we could flip 2 middlemost edges at FD and BD with:

```(fm1 D1) ^3 + fm1 D2 + (fm1 D1) ^3 + fm1   (disturbs some centres)
```

followed by:

```(fm2 + u2) ^2 + (fm2 + lm2) ^2    (corrects the centres)
```

I believe this is correct, and I will double-check on my physical
5x5x5 at home. Definitely 5^3 cubing is a sport for the
specialist ;->

-> Mark
Email: mark.longridge@canrem.com