# Message #3695

From: Joel Karlsson <joelkarlsson97@gmail.com>

Subject: Re: [MC4D] Physical 4D puzzle V2

Date: Fri, 28 Apr 2017 08:14:48 +0200

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 melinda@superliminal.com

[4D_Cubing]" <4D_Cubing@yahoogroups.com>:

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 joelkarlsson97@gmail.com [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 melinda@superliminal.com

[4D_Cubing] <4D_Cubing@yahoogroups.com>:

>

>

> Dear Cubists,

>

> I’ve finished version 2 of my physical puzzle and uploaded a video of it

> here:

> https://www.youtube.com/watch?v=zqftZ8kJKLo

> 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: http://superliminal.com/cube/d

> essert_cube.jpg I don’t know how practical a solution this is but it sure

> looks delicious! Welcome Marc!

>

> -Melinda

>

>