Message #3701

From: Joel Karlsson <>
Subject: Re: [MC4D] Physical 4D puzzle V2
Date: Sun, 30 Apr 2017 22:51:41 +0200

I am no expert on group theory, so to better understand what twists are
legal I read through the part of Kamack and Keane’s *The Rubik Tesseract *about
orienting the corners. Since all even permutations are allowed the easiest
way to check if a twist is legal might be to:

  1. Check that the twist is an even permutation, that is: the same twist can
    be done by performing an even number of piece swaps (2-cycles).
  2. Check the periodicity of the twist. If A^k=I (A^k meaning performing the
    twist k times and I (the identity) representing the permutation of doing
    nothing) and k is not divisible by 3 the twist A definitely doesn’t violate
    the restriction of the orientations since kx mod 3 = 0 and k mod 3 != 0
    implies x mod 3 = 0 meaning that the change of the total orientation x for
    the twist A mod 3 is 0 (which precisely is the restriction of legal twists;
    that they must preserve the orientation mod 3).

For instance, this implies that the restacking moves are legal 2x2x2x2
moves since both are composed of 8 2-cycles and both can be performed twice
(note that 2 is not divisible by 3) to obtain the identity.

Note that 1 and 2 are sufficient to check if a twist is legal but only 1 is
necessary; there can indeed exist a twist violating 2 that still is legal
and in that case, I believe that we might have to study the orientation
changes for that specific twist in more detail. However, if a twist can be
composed by other legal twists it is, of course, legal as well.

Best regards,

2017-04-29 1:04 GMT+02:00 Melinda Green
[4D_Cubing] <>:

> First off, thanks everyone for the helpful and encouraging feedback!
> Thanks Joel for showing us that there are 6 orbits in the 2^4 and for your
> rederivation of the state count. And thanks Matt and Roice for pointing out
> the importance of the inverted views. It looks so strange in that
> configuration that I always want to get back to a normal view as quickly as
> possible, but it does seem equally valid, and as you’ve shown, it can be
> helpful for more than just finding short sequences.
> I don’t understand Matt’s "pinwheel" configuration, but I will point out
> that all that is needed to create your twin interior octahedra is a single
> half-rotation like I showed in the video at 5:29
> <>. The two main
> halves do end up being mirror images of each other on the visible outside
> like he described. Whether it’s the pinwheel or the half-rotated version
> that’s correct, I’m not sure that it’s a bummer that the solved state is
> not at all obvious, so long as we can operate it in my original
> configuration and ignore the fact that the outer faces touch. That would
> just mean that the "correct" view is evidence that that the more
> understandable view is legitimate.
> I’m going to try to make a snapable V3 which should allow the pieces to be
> more easily taken apart and reassembled into other forms. Shapeways does
> offer a single, clear translucent plastic that they call "Frosted Detail",
> and another called "Transparent Acrylic", but I don’t think that any sort
> of transparent stickers will help us, especially since this thing is chock
> full of magnets. The easiest way to let you see into the two hemispheres
> would be to simply truncate the pointy tips of the stickers. That already
> happens a little bit due to the way I’ve rounded the edges. Here is a
> close-up <> of a half-rotation
> in which you can see that the inner yellow and white faces are solved. Your
> suggestion of little mapping dots on the corners also works, but just
> opening the existing window further would work more directly.
> -Melinda
> On 4/28/2017 2:15 PM, Roice Nelson [4D_Cubing] wrote:
> I agree with Don’s arguments about adjacent sticker colors needing to be
> next to each other. I think this can be turned into an accurate 2^4 with
> coloring changes, so I agree with Joel too :)
> To help me think about it, I started adding a new projection option for
> spherical puzzles to MagicTile, which takes the two hemispheres of a puzzle
> and maps them to two disks with identified boundaries connected at a point,
> just like a physical "global chess
> <>" game I have.
> Melinda’s puzzle is a lot like this up a dimension, so think about two
> disjoint balls, each representing a hemisphere of the 2^4, each a "subcube"
> of Melinda’s puzzle. The two boundaries of the balls are identified with
> each other and as you roll one around, the other half rolls around so that
> identified points connect up. We need to have the same restriction on
> Melinda’s puzzle.
> In the pristine state then, I think it’d be nice to have an internal
> (hidden), solid colored octahedron on each half. The other 6 faces should
> all have equal colors split between each hemisphere, 4 stickers on each
> half. You should be able to reorient the two subcubes to make a half
> octahedron of any color on each subcube. I just saw Matt’s email and
> picture, and it looks like we were going down the same thought path. I
> think with recoloring (mirroring some of the current piece colorings)
> though, the windmill’s can be avoided (?)
> […] After staring/thinking a bit more, the coloring Matt came up with is
> right-on if you want to put a solid color at the center of each
> hemisphere. His comment about the "mirrored" pieces on each side helped me
> understand better. 3 of the stickers are mirrored and the 4th is the
> hidden color (different on each side for a given pair of "mirrored"
> pieces). All faces behave identically as well, as they should. It’s a
> little bit of a bummer that it doesn’t look very pristine in the pristine
> state, but it does look like it should work as a 2^4.
> I wonder if there might be some adjustments to be made when shapeways
> allows printing translucent as a color :)
> […] Sorry for all the streaming, but I wanted to share one more
> thought. I now completely agree with Joel/Matt about it behaving as a 2^4,
> even with the original coloring. You just need to consider the corner
> colors of the two subcubes (pink/purple near the end of the video) as being
> a window into the interior of the piece. The other colors match up as
> desired. (Sorry if folks already understood this after their emails and
> I’m just catching up!)
> In fact, you could alter the coloring of the pieces slightly so that the
> behavior was similar with the inverted coloring. At the corners where 3
> colors meet on each piece, you could put a little circle of color of the
> opposite 4th color. In Matt’s windmill coloring then, you’d be able to see
> all four colors of a piece, like you can with some of the pieces on
> Melinda’s original coloring. And again you’d consider the color circles a
> window to the interior that did not require the same matching constraints
> between the subcubes.
> I’m looking forward to having one of these :)
> Happy Friday everyone,
> Roice
> On Fri, Apr 28, 2017 at 1:14 AM, Joel Karlsson
> [4D_Cubing] <> wrote:
>> Seems like there was a slight misunderstanding. I meant that you need to
>> be able to twist one of the faces and in MC4D the most natural choice is
>> the center face. In your physical puzzle you can achieve this type of twist
>> by twisting the two subcubes although this is indeed a twist of the
>> subcubes themselves and not the center face, however, this is still the
>> same type of twist just around another face.
>> If the magnets are that allowing the 2x2x2x2 is obviously a subgroup of
>> this puzzle. Hopefully the restrictions will be quite natural and only some
>> "strange" moves would be illegal. Regarding the "families of states" (aka
>> orbits), the 2x2x2x2 has 6 orbits. As I mentioned earlier all allowed
>> twists preserves the parity of the pieces, meaning that only half of the
>> permutations you can achieve by disassembling and reassembling can be
>> reached through legal moves. Because of some geometrical properties of the
>> 2x2x2x2 and its twists, which would take some time to discuss in detail
>> here, the orientation of the stickers mod 3 are preserved, meaning that the
>> last corner only can be oriented in one third of the number of orientations
>> for the other corners. This gives a total number of orbits of 2x3=6. To
>> check this result let’s use this information to calculate all the possible
>> states of the 2x2x2x2; if there were no restrictions we would have 16! for
>> permuting the pieces (16 pieces) and 12^16 for orienting them (12
>> orientations for each corner). If we now take into account that there are 6
>> equally sized orbits this gets us to 12!16^12/6. However, we should also
>> note that the orientation of the puzzle as a hole is not set by some kind
>> of centerpieces and thus we need to devide with the number of orientations
>> of a 4D cube if we want all our states to be separated with twists and not
>> only rotations of the hole thing. The number of ways to orient a 4D cube in
>> space (only allowing rotations and not mirroring) is 8x6x4=192 giving a
>> total of 12!16^12/(6*192) states which is indeed the same number that
>> for example David Smith arrived at during his calculations. Therefore,
>> when determining whether or not a twist on your puzzle is legal or not it
>> is sufficient and necessary to confirm that the twist is an even
>> permutation of the pieces and preserves the orientation of stickers mod 3.
>> Best regards,
>> Joel
>> Den 28 apr. 2017 3:02 fm skrev "Melinda Green
>> [4D_Cubing]" <>:
>> The new arrangement of magnets allows every valid orientation of pieces.
>> The only invalid ones are those where the diagonal lines cutting each
>> cube’s face cross each other rather than coincide. In other words, you can
>> assemble the puzzle in all ways that preserve the overall diamond/harlequin
>> pattern. Just about every move you can think of on the whole puzzle is
>> valid though there are definitely invalid moves that the magnets allow. The
>> most obvious invalid move is twisting of a single end cap.
>> I think your description of the center face is not correct though. Twists
>> of the outer faces cause twists "through" the center face, not "of" that
>> face. Twists of the outer faces are twists of those faces themselves
>> because they are the ones not changing, just like the center and outer
>> faces of MC4D when you twist the center face. The only direct twist of the
>> center face that this puzzle allows is a 90 degree twist about the outer
>> axis. That happens when you simultaneously twist both end caps in the same
>> direction.
>> Yes, it’s quite straightforward reorienting the whole puzzle to put any
>> of the four axes on the outside. This is a very nice improvement over the
>> first version and should make it much easier to solve. You may be right
>> that we just need to find the right way to think about the outside faces.
>> I’ll leave it to the math geniuses on the list to figure that out.
>> -Melinda
>> On 4/27/2017 10:31 AM, Joel Karlsson
>> [4D_Cubing] wrote:
>> Hi Melinda,
>> I do not agree with the criticism regarding the white and yellow stickers
>> touching each other, this could simply be an effect of the different
>> representations of the puzzle. To really figure out if this indeed is a
>> representation of a 2x2x2x2 we need to look at the possible moves (twists
>> and rotations) and figure out the equivalent moves in the MC4D software.
>> From the MC4D software, it’s easy to understand that the only moves
>> required are free twists of one of the faces (that is, only twisting the
>> center face in the standard perspective projection in MC4D) and 4D
>> rotations swapping which face is in the center (ctrl-clicking in MC4D). The
>> first is possible in your physical puzzle by rotating the white and yellow
>> subcubes (from here on I use subcube to refer to the two halves of the
>> puzzle and the colours of the subcubes to refer to the "outer colours").
>> The second is possible if it’s possible to reach a solved state with any
>> two colours on the subcubes that still allow you to perform the previously
>> mentioned twists. This seems to be the case from your demonstration and is
>> indeed true if the magnets allow the simple twists regardless of the
>> colours of the subcubes. Thus, it is possible to let your puzzle be a
>> representation of a 2x2x2x2, however, it might require that some moves that
>> the magnets allow aren’t used.
>> Best regards,
>> Joel
>> 2017-04-27 3:09 GMT+02:00 Melinda Green
>> [4D_Cubing] <>:
>>> Dear Cubists,
>>> I’ve finished version 2 of my physical puzzle and uploaded a video of it
>>> here:
>>> Again, please don’t share these videos outside this group as their
>>> purpose is just to get your feedback. I’ll eventually replace them with a
>>> public video.
>>> Here is an extra math puzzle that I bet you folks can answer: How many
>>> families of states does this puzzle have? In other words, if disassembled
>>> and reassembled in any random configuration the magnets allow, what are the
>>> odds that it can be solved? This has practical implications if all such
>>> configurations are solvable because it would provide a very easy way to
>>> fully scramble the puzzle.
>>> And finally, a bit of fun: A relatively new friend of mine and new list
>>> member, Marc Ringuette, got excited enough to make his own version. He
>>> built it from EPP foam and colored tape, and used honey instead of magnets
>>> to hold it together. Check it out here:
>>> essert_cube.jpg I don’t know how practical a solution this is but it
>>> sure looks delicious! Welcome Marc!
>>> -Melinda