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• ## Re: [MC4D] Lower dimensional cubes

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• On Monday, August 21, Melinda Green ... The 2-puzzle has 4 corner 2-cubies - all distinct. A twist swaps a pair of them. Three such swaps will also swap
Message 1 of 3 , Aug 21, 2006
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On Monday, August 21, "Melinda Green"
<melinda@...> 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.
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