# Message #328

From: David Vanderschel <DvdS@Austin.RR.com>

Subject: Re: [MC4D] Lower dimensional cubes

Date: Mon, 21 Aug 2006 03:19:09 -0500

On Monday, August 21, "Melinda Green"

<melinda@superliminal.com> wrote about MC2D:

>I’m not sure how many unique positions it has but I bet that it’s less

>than 10. Considering color symmetry, it may be quite a bit less than

>that. It’s so small that I had to make sure that the number of

>scrambling twists was odd otherwise scrambling would sometimes randomly

>leave it in the solved state!! I’d love to see the complete state graph

>laid out graphically. God’s algorithm length seems to be 2 though it

>sometimes takes 3 in the applet since I didn’t implement a middle slice

>twist.

The 2-puzzle has 4 corner 2-cubies - all distinct. A

twist swaps a pair of them. Three such swaps will

also swap (just) a diagonally opposite pair. Thus all

permutations are possible. There are 24 such

permutations.

Without center slice twists, it takes four twists to

fix the arrangement in which both diagonally opposite

corner pairs are swapped. However, that arrangement

is an even permutation, so it cannot be generated with

the odd-only algorithm.

Note that Melinda’s MC2D is actually more nearly

analogous to my MC3D with its reflecting twist

extension than it is to a physical Rubik’s Cube, as

doing a twist with MC2D causes the swapped corners to

assume mirror image states. A corner 2-cubie has the

correct handedness only when it is in its home

position or the diagonally opposite one. Without

admitting mirror imaging of the corner 2-cubies in

MC2D, no moves would be possible at all.

In a private communication to me, Roice has pointed out

that MC3D with its possibility of reflecting ‘twists’

could be regarded as another sort of 4D extension of

the 3-puzzle. With a 3-puzzle embedded in 4-space, it

would be ‘physically’ possible to create the 3D slice

reflections by twisting a slice of the 3-puzzle in

4-space. Similarly, MC2D’s (reflecting) moves would

require a 3rd dimension to implement them physically.

I have not fully worked out all the group theoretic

implications of the 3-puzzle-with-mirroring, but I

have reached a few interesting conclusions: Any even

numbered subset of the 8 corner 3-cubies can be in a

reflected state. I think that introduces a multiplier

of 170 on the number of possible states for the

3-puzzle. Furthermore, it appears that the 12-fold

‘parity’, which leads to 12 orbits for the regular

3-puzzle, is reduced to just a single binary parity -

introducing another factor of 6 on the states which

can be achieved from a given starting position. It is

not yet clear to me whether the 3-puzzle with

mirroring twists should be regarded as easier or more

difficult. Could someone point me to a forum in which

such theoretical issues for the 3-puzzle would be of

interest? My searches for Rubik with mirroring (or

reflection) have only turned up references to

mirroring as it applies to the whole puzzle, not

single slices; so this may be a totally unexplored

area.

Recall that my MC3D program also allows a solver to

approach the 3-puzzle from the point of view of an

inhabitant of 2-space. I am disappointed that no one

has reported success (or lack thereof) at solving the

3-puzzle based solely on a 1D projection as can be

generated by MC3D. Surely among folks reading this

forum, there must be some takers for this challenge

which pushes downward from n=3 to n=2 the analogy of

an n-dimensional solver solving the (n+1)-puzzle based

on an (n-1)D projection. It may well be that solving

the 3-puzzle is more difficult for a Flatlander than

is solving the 4-puzzle for us solid 3D folks.

Regards,

David V.