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Re: [MC4D] Lower dimensional cubes

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  • David Vanderschel
    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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