Message #3636

From: Christopher Locke <project.eutopia@gmail.com>
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
Date: Thu, 09 Feb 2017 16:09:36 -0800

Hello Melinda,

Congratulations! Looks very nice.

I noticed when looking at your puzzle, that the rough correspondence is
the following:

• Each left and right 2x2x2 block corresponds to the +/- w direction
  faces of the 4D cube, with the respective stickers on the 8 corners of
  each block. So the w-direction stickers are initially on the outer
  corners. For reference, let’s call the left - and right +

• By arbitrarily setting the central 8 stickers that bridge the gap
  between the two halves to be the -y face, then the +y face is split
  between the left and right open faces

• The stickers in the top layer that are facing upwards are the split +z
  face stickers, and those in the bottom are the split -z face stickers
  (can be brought together by rotation).

• Similarly, for the front (+x) and back (-x) faces

• The faces seem to live as follows: (-w leftmost 8 blocks, +w rightmost
  8 blocks, +z topmost 8 blocks, -z bottommost 8 blocks, +x frontmost 8
  blocks, -x backmost 8 blocks, +y middle 8 blocks, -y leftmost 4 and
  rightmost 4 blocks).

• Stickers are as follows: (-w are on outer corners of leftmost 8
  blocks, +w on outer corners of leftmost 8 blocks, +z in the middle of
  the top face of the left and right 2x2x2 blocks, and -z in the middle of
  the bottom face, +x in the middle of the front face of the left and
  right 2x2x2 blocks, and -x in the middle of the back facing part, +y the
  covered up middle 8 stickers, and -y the leftmost and rightmost middle
  stickers.

• From this point of view, comparing to the 4D model in MagicCube4D
  (where the -w face is in the center, and the +/- x,y,z faces are
  surrounding it, with +w hidden), then 90 degree twists of the -w face
  are achieved by rotating the left block arbitrarily, and 90 degree
  twists of the +w face by rotating the right block arbitrarily.

  • From this point of view, your rotations of both faces
    simultaneously are two twists of the inner and outermost halves,
    resulting in a 3D rotation of the MagicCube4D model that keeps the
    central -w face in the center (plain 3D rotations)

• Similarly, it seems that 180 degree rotations of, say, the topmost +z
  face, seems to be a 180 degree rotation of the topmost +z face in
  MagicCube4D around the y-axis (keeps the +/-y stickers on the +/-y face)

• 180 degree rotation of front block is a rotation of +x face around the
  y-axis

  • Note that although you only have 180 degree twists about the
    y-axis, a rotation of the puzzle can bring it into an orientation to
    probably do any 180 degree twist of the x, y, or z faces.

The problem here is that the outer +/-w stickers always stay on the 8
corners of the left and right 2x2x2 blocks. This means that you cannot
mix up w stickers with x, y, or z stickers yet, without doing some
inversion of the small 3D cubes.

In conclusion, without yet considering the "inversion" type moves, you
have all 90 degree twists of the w faces, and 180 degree twists of the
x, y, z faces. The trick is to find out how to legally mix the w
stickers up with x, y, z stickers. If you can do this, then the 90
degree flexibility of the w face twists should make it possible to reach
any permutation. I will think about this more on the weekend, using
some pen and paper instead of just thinking while walking to work (it is
hard to visualize the inversion moves in my head :D).

Best regards,
Chris

On 2017年02月09日 15:37, Melinda Green melinda@superliminal.com [4D_Cubing]
wrote:
>
> Hello Liam,
>
> Yes, a new perspective can sometimes change everything. In this case,
> I wouldn’t guess that this version will be easier to solve or
> understand than a virtual version, but there is undoubtedly something
> satisfying about puzzles you can hold in your hands. One way it may be
> helpful is in showing the difference between stickers and pieces. The
> physical pieces are now obvious, though the stickers are no longer
> cubes, so that is a little misleading. It may make for a gateway
> puzzle that gets some people to look more deeply into the virtual
> puzzles. I’ll be curious to see how people in the general puzzle
> community react when I show it around.
>
> And yes, the magnet restrictions definitely seems like a kind of
> bandaging. Extending the Mathologer’s arrangement allows two main
> implementations that will support the basic moves. (Mathologer video
> for reference. <https://www.youtube.com/watch?v=Xb8ENlS-5Go>) I chose
> the one that seems to allow for more unusual moves such as the
> restacking move, but the other version might allow for some other
> legal, breakthrough move. Maybe I’ll just need to prototype that one too.
>
> Thanks!
> -Melinda
>
> On 2/9/2017 8:38 AM, liamjwright@btinternet.com [4D_Cubing] wrote:
>> Congratulations!
>>
>> If there’s anything I’ve learned exploring different puzzles, it’s
>> that a different perspective can totally change the difficulty of a
>> puzzle. It would definitely be interesting to see if this version of
>> a 2^4 makes it any easier or harder to solve/understand it.
>>
>> When you split the cube down the middle, whether it produces a legal
>> permutation of the cube or not, I think it definitely has the
>> potential to create a slightly odd new puzzle. The way the magnets
>> limit rotation after the split could perhaps act in the same way that
>> a bandage does on a regular 3D puzzle?
>>
>> -Liam
>
>