# Message #3713

From: Joel Karlsson <joelkarlsson97@gmail.com>

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

Date: Tue, 30 May 2017 09:06:01 +0200

Well, usually we want to calculate distinct states and for the 2x2x2x2

(virtual) that become 16!12^16/(6*192), 16! for permutation of pieces,

12^16 for orienting the pieces, 1/6 to only get legal states (the puzzle

has 6 orbits) and 1/192 to fix one corner since the puzzle doesn’t have

fixed pieces like the 3x3x3x3. However, when I checked how many

representations there are of every state I didn’t count distinct states but

all states, even illegal ones. The count for the virtual 2x2x2x2 thus

becomes 16!*12^16 and with this count, every distinct state has 192

representations and only a sixth of them are legal (you can think of it as:

how many legal and illegal states are there if I’m not allowed to rotate

the puzzle? or: in how many ways can I put the pieces together?). The same

type of count for the physical puzzle would be 3*16!*24*12^15. The ratio of

these are 1:6 which means that even if you only count legal states and fix

the rotation of the puzzle the count for the physical puzzle, which would

be 3*16!*24*12^15/(6*192), is still too high with a factor 6 and thus, if

we claim that the physical puzzle is a representation of a 2x2x2x2, there

must be six representations of every distinct state that are not separated

with only rotations. Luckily, this is the case and these come from the

different projections (I agree that projection might be a better word than

representation although it might not be completely accurate).

Best regards,

Joel Karlsson

2017-05-30 3:57 GMT+02:00 Melinda Green melinda@superliminal.com

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

>

>

> Thanks for the clarification, Joel. Just to be complete, are you sure that

> that the count of MC4D states are being counted in the same way?

>

> One other thought I had regarding terminology: You called the two forms of

> the physical puzzle "representations", but I wonder whether a somewhat

> better term might be "projection". This puzzle is definitely not any sort

> of geometric projection into 3-space, but it seems to share a number of

> analogous properties with them. I often think of it as viewing the 4D

> object through a 2x2x4 "viewport".

>

> The half rotations are sort of like translating that viewport along or

> around the surface of that object. Since you point out that it is a 90

> degree rotation, perhaps "half rotation" isn’t the best term for that move.

> Whatever we call these rotations, the two forms feel to me like the

> difference between cell-first and face-first projections.

>

> -Melinda

>

>

> On 5/29/2017 2:57 AM, Joel Karlsson joelkarlsson97@gmail.com [4D_Cubing]

> wrote:

> > Hello,

> >

> > Just a quick correction regarding a previous statement. From the

> > calculation of the states of the puzzle, we can see that if we choose

> > a state and rotation there are still two representations of that state

> > and rotation in the physical puzzle. Previously, we also said that

> > these states are separated by a half-rotation such as Rx+ from an RL

> > rep or AKx rep. This is not the case. When I calculated the states of

> > the puzzle I assumed that the longer side of the puzzle should be

> > parallel with the x-axis. If we don’t make that assumption we find

> > that the number of states (not distinct states, some of these are just

> > separated by a rotation) are 3*16!*24*12^15 since there are three

> > possible choices for which axis should be parallel with the longer

> > side. This means that, in fact, each distinct state has 6

> > representations in the physical puzzle. This is actually what we

> > should expect since it should be possible to represent every state in

> > an RL rep, an UD rep, an FB rep, an AKx rep, an AKy rep and an AKz

> > rep. In the solved state, the rotation of the puzzle is determined by

> > which colour belongs to each face and given such a rotation there are

> > indeed six representations: first, choose an axis that the longer side

> > should be parallel with (3 alternatives) and then choose either the AK

> > rep or the non-AK rep (2 alternatives). However, the so-called

> > "half-rotation" changes which colour belongs to which face so this is

> > not a move that gets you from one representation of a state to

> > another. From an UD rep with:

> > P3 = Sy+x Vz+ Oxz’ Vz+ Oxz’ Vz+ Oz UD’x ,

> > P3 Ox’z’ takes you to an FB rep of the same state

> > Oy P3 Oz’ takes you to an RL rep of the same state

> > UD’x P3 UD’z’ Sy- Oy takes you to an AKy rep of the same state

> > UD’x P3 Oyx’ Sz+ takes you to an AKz rep of the same state and

> > UD’x P3 UD’x’ Oz’ Sx- takes you to an AKx rep of the same state

> > >From this, we can see that the AK states are closely related and it’s

> > possible to change between them without illegal moves. For example

> > going from an AKy rep to an AKz rep of the same state could be done

> > with:

> > (UD’x P3 UD’z’ Sy- Oy)’ UD’x P3 Oyx’ Sz+ = Oy’ Sy+ UD’z

> > P3’ UD’x’ UD’x P3 Oyx’ Sz+ = Oy’ Sy+ UD’z Oyx’ Sz+

> >

> > Note that these sequences don’t change the state nor the rotation of

> > the puzzle but they do change the representation. Let’s call these

> > types of sequences J moves. Note that what I have previously called a

> > type three rotation is actually a J move and a rotation. Further note

> > that J = I mod(rep) (modulo representation). The J moves changes which

> > pair of faces are symmetry breaking and this is the type of moves that

> > needs to be added to the set of legal moves to make the physical

> > puzzle an actual 2x2x2x2. A J move is a move that is equal to I

> > mod(rep, rot) but that isn’t equal to I mod(rot) (which means that it

> > has to preserve the state, is allowed to change the rotation and has

> > to change the representation/what faces (not in the left/right sense

> > but rather in the sense of colours) are symmetry-breaking ).

> >

> > Best regards,

> > Joel Karlsson

> >

> > 2017-05-22 22:32 GMT+02:00 Joel Karlsson <joelkarlsson97@gmail.com>:

> >> Hi Melinda,

> >>

> >> Thank you for the feedback. Regarding the coordinate system, it’s just

> >> a matter of preference. I thought it would be nice to have a

> >> xy-symmetry but understand that it might be more practical to follow

> >> conventions. So, adapting your suggestion, let’s redefine the axes as

> >> x pointing right, y up and z towards you. Note that this means that

> >> the longer side of the puzzle is parallel with the x-axis in the

> >> standard rotation. From here on, I will use this new coordinate

> >> system.

> >>

> >> Regarding the name of the faces. Since which faces are the "outer

> >> ones" changes with how you rotate the puzzle and what state the puzzle

> >> is in, I think that the labelling of the faces should be independent

> >> of which faces are the outer faces (forming what will be referred to

> >> as inverted octahedra). Since the puzzle is a representation of a

> >> 4d-cube in 3d-space it will (with our coordinate system) always have

> >> two faces "belonging" to every axis and two faces that don’t belong to

> >> any axis. Therefore, it makes sense to label the faces in such a way

> >> that the face in for example the positive x direction is R, always.

> >> So, the K face (which is one of the faces not belonging to a

> >> particular axis) is always the face only belonging to the center 2x2x2

> >> block (and this can be either an octahedron or an inverted octahedron

> >> (the outer corners) depending on the representation of the puzzle).

> >> The distinction between left and right never disappears; the R face is

> >> always the face only belonging to the right half of the puzzle and can

> >> be an octahedron, an inverted octahedron, two half octahedra (<><>) or

> >> one half and two quarter octahedra (><><). Note that the

> >> half-rotations are 90-degree rotations of the puzzle; for example, Sx+

> >> (physical puzzle) = Cx’ (virtual puzzle) is the rotation that does L

> >> -> K -> R -> A -> L so the face that previously was the L face is now

> >> the K face. From the standard rotation (where R and L are the inverted

> >> octahedra before the rotation), this would mean that the K and A faces

> >> are inverted octahedra after a Sx+ rotation. What faces are the

> >> "mysterious" (or more precisely symmetry breaking, forming inverted

> >> octahedra instead of regular octahedra) depend on the rotation of the

> >> puzzle and can be any two opposite faces (R and L, U and D, F and B or

> >> A and K). It might be useful to be able to describe what faces are

> >> inverted octahedra since this determines what moves are legal so let’s

> >> say that the puzzle is in an RL representation if R and L are inverted

> >> octahedra and similarly for other states. Thus, Sx+ can take you from

> >> an RL representation to an AK representation (what a long word let’s

> >> use rep for short). Note that the AK rep doesn’t specify which axis

> >> the longer side should be parallel with so let’s just add a lowercase

> >> letter to indicate this (an AKx rep is thus a state where the A and K

> >> faces are inverted octahedra and the longer side is parallel with the

> >> x-axis). In conclusion, I would like to keep the names of the faces as

> >> I first defined them and hope that it’s clearer what I mean with them

> >> and that the names of the faces are not related to what faces form

> >> inverted octahedra.

> >>

> >> You also wrote:

> >> "There are several, distinct types of rotations, none of which change

> >> the state of the puzzle, and I think we need a way to be unambiguous

> >> about them. The types I see are

> >>

> >> 1. Simple reorientation of the physical puzzle in the hand, no magnets

> >> involved. IE your ‘O’ moves and maybe analogous to mouse-dragging in

> >> MC4D?

> >> 2. Rolling of one 2x2x2 half against the other. Maybe analogous to

> ctrl-click?

> >> 3. Half-rotations. Maybe analogous to a "face first" view? (ctrl-click

> >> on a 2-color piece of the 3^4 with the setting "Ctrl-Click Rotates: by

> >> Cubie")

> >> 4. Whole-puzzle reorientations that move an arbitrary axis into the

> >> "outer" 2 faces. No MC4D analog."

> >>

> >> The rotations (as far as I know) are the O moves (type one rotation),

> >> which are indeed analogous to mouse-dragging in MC4D, rolling the

> >> 2x2x2 halves (type two rotation) (these are easily described as for

> >> example RL’y in an RL rep) which (in a non-AK rep) is a ctrl-click on

> >> a non-inverted face (that is ctrl-click on a face that is currently

> >> represented with an octahedron), half-rotations (also type two

> >> rotations) which is a ctrl-click on an inverted face and sequences

> >> that for example from an RL rep can take the R face to K without

> >> turning the puzzle into an AK rep (type three rotation). Type one and

> >> two rotations are legal moves but type three contain illegal 2x2x2x2

> >> moves according to hypothesis * in my previous email (however, they

> >> are very important and needed if we wish to be able to reach all

> >> states).

> >>

> >> Regarding fold moves, from the standard rotation (RL rep) are you

> >> saying that Vy+ = Vy’+ or something like Vy+ = Vy’+ FB’x2 = Vy’+

> >> mod(rot)? The former seems not to be correct but I believe the latter

> >> is, please correct me if I’m wrong. I don’t assume that you fold the

> >> puzzle back to the same representation. This is what + and -

> >> indicates, + preserve the representation mod(rot) (i.e AK rep both

> >> before and after or neither before nor after) and - changes it (going

> >> from AK rep to non-AK rep or vice versa).

> >>

> >> "I think the only legal 90 degree twists of the K face are those about

> >> the long axis. I believe this is what Christopher Locke was saying in

> >> this message. To see why there is no straightforward way to perform

> >> other 90 degree twists, you only need to perform a 90 degree twist on

> >> an outer (L/R) face and then reorient the whole puzzle along a

> >> different outer axis. If the original twist was not about the new long

> >> axis, then there is clearly no straightforward way to undo that

> >> twist."

> >>

> >> Yes, as pointed out further down in my previous email. The notation

> >> allows all 90-degree rotations after the section "extensions of some

> >> definitions" although only 180-degree rotations are legal for

> >> octahedral faces around axes not parallel with the longer side of the

> >> puzzle (see the section “some important notes on legal/illegal

> >> moves”).

> >>

> >> "I noticed something like this the other day but realized that it only

> >> seems to work for rotations along the long dimension (z in your

> >> example). These are already easily accomplished by a simple rotation

> >> to put the face in question on the end caps, followed by a double

> >> end-cap twist."

> >>

> >> It works for the other rotations as well although these are not legal

> >> moves. They could possibly be used instead of the illegal S moves

> >> (along axes not parallel with the longer side) and the illegal V moves

> >> (V- (minus) moves which are closely related to the illegal S moves,

> >> example from RL rep: Vy- = Vy+ Sx Oz2) but as I mentioned in my

> >> previous email I do believe that it’s better to use the S and V moves

> >> since you have found a relatively short way to perform a type three

> >> rotation with those.

> >>

> >> Let:

> >> P1 = Sy+ Uyz2 Sy-

> >> P1’ = P1 (P1 is its own inverse)

> >> P2 = (P1 UD’z P1 UD’z’ P1 UD’z’ P1 UD’z)2 (P2=P_2 not P^2

> >> whereas the two at the end means: perform twice)

> >> P2’ = (UD’z’ P1 UD’z P1 UD’z P1 UD’z’ P1)2

> >> P3 = Sy+x Vz+ Oxz’ Vz+ Oxz’ Vz+ Oz UD’x = I mod(rot) (type three

> rotation)

> >> P3’ = Oy’ P3 Oy’

> >> P4 = P2 P3 P2 P3’ P2’ P3 P2’ P3’

> >>

> >> P4 from UD rep is a 164 move sequence rotating only one corner in its

> >> place. The sequence is inspired by Roice “second four-color series”

> >> but I have changed the “Top 9” moves to pure rotations since it’s all

> >> that’s necessary and a bit shorter to perform. P3 is your type three

> >> rotation (with a rotation added at the end) and P2 is Roice “third

> >> three-color series”. Written with my notation but for the virtual

> >> puzzle, the sequences (Roice original since it’s easier to perform K

> >> twists than O rotations in MC4D) are:

> >> Q1 = (Kz2y’ Lz2y’ Kz2y’ Rz2y’)2 (analogue to P2)

> >> Q1’ = (Rz2y’ Kz2y’ Lz2y’ Kz2y’)2

> >> Q2 = Q1 Kxy’ Q1 Kyx’ Q1’ Kxy’ Q1’ Kyx’ (analogue to P4 but

> >> only 36 moves)

> >>

> >> How to read faster (the example applies to a 3x3x3x3): moves like

> >> Kz2y’ are clicking on 3C-pieces in MC4D. If the move is written in

> >> this way, [uppercase letter] [lowercase letter]2 [lowercase letter

> >> possibly with ‘ (prime)], there’s a quite quick way to realize which

> >> piece this is. The uppercase letter specifies which face the piece to

> >> press is on and the first lowercase letter (followed by 2) specifies

> >> one of the sides of that face that the piece belong to. There are then

> >> 4 possible pieces. Sadly, the piece do not lie in the direction of the

> >> last letter from the center of the side of the face but you have to

> >> move one edge clockwise from this. So, Kz2y’ is a click on the edge on

> >> the front side of the K face one step clockwise from the negative

> >> y-axis (thus, the front left 3C-piece on the K face). It’s a bit

> >> unfortunate that this “rule” isn’t even simpler but it’s at least true

> >> for all of these moves (as far as I know). Moves like Kxy’ are

> >> left/right-clicking on a corner piece but currently I don’t know a

> >> fast way to determine which corner. Any ideas?

> >>

> >> Best regards,

> >> Joel Karlsson

> >>

> >> 2017-05-19 3:33 GMT+02:00 Melinda Green melinda@superliminal.com

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

> >>>

> >>> Hello Joel,

> >>>

> >>> Thanks for drilling into this puzzle. Finding good ways to discuss and

> think

> >>> about moves and representations will be key. I’ll comment on some

> details

> >>> in-line.

> >>>

> >>> On 5/14/2017 6:16 AM, Joel Karlsson joelkarlsson97@gmail.com

> [4D_Cubing]

> >>> wrote:

> >>>

> >>>

> >>> Yes, that is correct and in fact, you should divide not only with 24

> for the

> >>> orientation but also with 16 for the placement if you want to calculate

> >>> unique states (since the 2x2x2x2 doesn’t have fixed centerpieces). The

> >>> point, however, was that if you don’t take that into account you get a

> >>> factor of 24*16=384 (meaning that the puzzle has 384 representations of

> >>> every unique state) instead of the factor of 192 which you get when

> >>> calculating the states from the virtual puzzle and hence every state

> of the

> >>> virtual puzzle has two representations in the physical puzzle. Yes

> exactly,

> >>> they are indeed the same solved (or other) state and you are correct

> that

> >>> the half rotation (taking off a 2x2 layer and placing it at the other

> end of

> >>> the puzzle) takes you from one representation to the same state with

> the

> >>> other representation. This means that the restacking move (taking off

> the

> >>> front 2x4 layer and placing it behind the other 2x4 layer) can be

> expressed

> >>> with half-rotations and ordinary twists and rotations (which you might

> have

> >>> pointed out already).

> >>>

> >>>

> >>> Yes, I made that claim in the video but didn’t show it because I have

> yet to

> >>> record such a sequence. I’ve only stumbled through it a few times. I

> talked

> >>> about it at 5:53 though I mistakenly called it a twist, when I should

> have

> >>> called it a sequence.

> >>>

> >>>

> >>> I think I’ve found six moves including ordinary twists and a

> restacking move

> >>> that is identical to a half-rotation and thus it’s easy to compose a

> >>> restacking move with one half-rotation and five ordinary twists. There

> might

> >>> be an error since I’ve only played with the puzzle in my mind so it

> would be

> >>> great if you, Melinda, could confirm this (the sequence is described

> later

> >>> in this email).

> >>>

> >>>

> >>> You mean "RLx Ly2 Sy By2 Ly2 RLx’ = Sz-"? Yes, that works. There do

> seem to

> >>> be easier ways to do that beginning with an ordinary rolling rotation.

> I

> >>> don’t see those in your notation, but the equivalent using a pair of

> twists

> >>> would be Rx Lx’ Sx Vy if I got that right.

> >>>

> >>>

> >>> To be able to communicate move sequences properly we need notation for

> >>> representing twists, rotations, half-rotations, restacking moves and

> folds.

> >>> Feel free to come with other suggestion but you can find mine below.

> Please

> >>> read the following thoroughly (maybe twice) to make sure that you

> understand

> >>> everything since misinterpreted notation could potentially become a

> >>> nightmare and feel free to ask questions if there is something that

> needs

> >>> clarification.

> >>>

> >>> Coordinate system and labelling:

> >>>

> >>> Let’s introduce a global coordinate system. In whatever state the

> puzzle is

> >>> let the positive x-axis point upwards, the positive y-axis towards you

> and

> >>> the positive z-axis to the right (note that this is a right-hand

> system).

> >>>

> >>>

> >>> I see the utility of a global coordinate system, but this one seems

> rather

> >>> non-standard. I suggest that X be to the right, and Y up since these

> are

> >>> near-universal standards. Z can be in or out. I have no opinion. If

> there is

> >>> any convention in the twisty puzzle community, I’d go with that.

> >>>

> >>> Note also that the wiki may be a good place to document and iterate on

> >>> terminology, descriptions and diagrams. Ray added a "notation" section

> to

> >>> the 3^4 page here, and I know that one other member was thinking of

> >>> collecting a set of moves on another wiki page.

> >>>

> >>>

> >>> Now let’s name the 8 faces of the puzzle. The right face is denoted

> with R,

> >>> the left with L, the top U (up), the bottom D (down), the front F, the

> back

> >>> B, the center K (kata) and the last one A (ana). The R and L faces are

> >>> either the outer corners of the right and left halves respectively or

> the

> >>> inner corners of these halves (forming octahedra) depending on the

> >>> representation of the puzzle. The U, D, F and B faces are either two

> diamond

> >>> shapes (looking something like this: <><>) on the corresponding side

> of the

> >>> puzzle or one whole and two half diamond shapes (><><). The K face is

> either

> >>> an octahedron in the center of the puzzle or the outer corners of the

> center

> >>> 2x2x2 block. Lastly, the A face is either two diamond shapes, one on

> the

> >>> right and one on the left side of the puzzle, or another shape that’s a

> >>> little bit hard to describe with just a few words (the white stickers

> at

> >>> 5:10 in the latest video, after the half-rotation but before the

> restacking

> >>> move).

> >>>

> >>>

> >>> I think it’s more correct to say that the K face is either an

> octahedron at

> >>> the origin (A<>K<>A) or in the center of one of the main halves, with

> the A

> >>> face inside the other half (>A<>K<). This was what I was getting at in

> my

> >>> previous message. You do later talk about octahedral faces being in

> either

> >>> the center or the two main halves, so this is just terminology. But

> about

> >>> "the outer corners of the center 2x2x2 block", this cannot be the A or

> K

> >>> face as you’ve labeled them. You’ve been calling these the L/R faces,

> but

> >>> the left-right distinction disappears in the half-rotated state, so

> maybe

> >>> "left" and "right" aren’t the best names. To me, they are always the

> >>> "outside" faces, regardless. You can distinguish them as the left and

> right

> >>> outside faces in one representation, or as the center and end outside

> faces

> >>> in the other. (Or perhaps "end" versus "ind" if we want to be cute.)

> >>>

> >>> I’m also a little torn about naming the interior faces ana and kata,

> not

> >>> because of the names themselves which I like, but because the

> mysterious

> >>> faces to me are the outermost ones you’re calling R and L. It only

> requires

> >>> a simple rotation to move faces in and out of the interior (octahedral)

> >>> positions, but it’s much more difficult to move another axis into the

> >>> L/R/outer direction.

> >>>

> >>> So maybe the directions can be

> >>>

> >>> Up-Down

> >>> Front-Back

> >>> Ind-End

> >>> Ana-Kata

> >>>

> >>> I’m not in love with it and will be happy with anything that works.

> Thoughts

> >>> anyone?

> >>>

> >>>

> >>> We also need a name for normal 3D-rotations, restacking moves and folds

> >>> (note that a half-rotation is a kind of restacking move). Let O be the

> name

> >>> for a rOtation (note that the origin O doesn’t move during a rotation,

> by

> >>> the way, these are 3D rotations of the physical puzzle), let S

> represent a

> >>> reStacking move and V a folding/clamshell move (you can remember this

> by

> >>> thinking of V as a folded line).

> >>>

> >>>

> >>> I think it’s fine to call the clamshell move a fold or denote it as V.

> I

> >>> just wouldn’t consider it to be a basic move since it’s a simple

> composite

> >>> of 3 basic twists as shown here. In general, I think there are so many

> >>> useful composite moves that we need to be able to easily make them up

> ad hoc

> >>> with substitutions like Let ↓ = Rx Lx’. These are really macro moves

> which

> >>> can be nested. That said, it’s a particularly useful move so it’s

> probably

> >>> worth describing in some formal way like you do in detail below.

> >>>

> >>>

> >>>

> >>> Further, let I (capital i) be the identity, preserving the state and

> >>> rotation of the puzzle. We cannot use I to indicate what moves should

> be

> >>> performed but it’s still useful as we will see later. Since we also

> want to

> >>> be able to express if a sequence of moves is a rotation, preserving the

> >>> state of the puzzle but possibly representing it in a different way,

> we can

> >>> introduce mod(rot) (modulo rotation). So, if a move sequence P

> satisfies P

> >>> mod(rot) = I, that means that the state of the puzzle is the same

> before and

> >>> after P is performed although the rotation and representation of the

> puzzle

> >>> are allowed to change. I do also want to introduce mod(3rot) (modulo

> >>> 3D-rotation) and P mod(3rot) = I means that if the right 3D-rotation (a

> >>> combination of O moves as we will see later) is applied to the puzzle

> after

> >>> P you get the identity I. Moreover, let the standard rotation of the

> puzzle

> >>> be any rotation such that the longer side is parallel with the z-axis,

> that

> >>> is the puzzle forms a 2x2x4 (2 pieces thick in the x-direction, 2 in

> the

> >>> y-direction and 4 in the z-direction), and the K face is an octahedron.

> >>>

> >>> Rotations and twists:

> >>>

> >>> Now we can move on to name actual moves. The notation of a move is a

> >>> combination of a capital letter and a lowercase letter. O followed by

> x, y

> >>> or z is a rotation of the whole puzzle around the corresponding axis

> in the

> >>> mathematical positive direction (counterclockwise/the way your

> right-hand

> >>> fingers curl if you point in the direction of the axis with your

> thumb). For

> >>> example, Ox is a rotation around the x-axis that turns a 2x2x4 into a

> 2x4x2.

> >>> A name of a face (U, D, F, B, R, L, K or A) followed by x, y or z

> means:

> >>> detach the 8 pieces that have a sticker belonging to the face and then

> turn

> >>> those pieces around the global axis. For example, if the longer side

> of the

> >>> puzzle is parallel to the z-axis (the standard rotation), Rx means:

> take the

> >>> right 2x2x2 block and turn it around the global x-axis in the

> mathematical

> >>> positive direction. Note that what moves are physically possible and

> allowed

> >>> is determined by the rotation of the puzzle (I will come back to this

> >>> later). Further note that Ox mod(rot) = Oy mod(rot) = Oz mod(rot) = I,

> >>> meaning that the 3D-rotations of the physical puzzle corresponds to a

> >>> 4D-rotation of the represented 2x2x2x2.

> >>>

> >>>

> >>> There are several, distinct types of rotations, none of which change

> the

> >>> state of the puzzle, and I think we need a way to be unambiguous about

> them.

> >>> The types I see are

> >>>

> >>> Simple reorientation of the physical puzzle in the hand, no magnets

> >>> involved. IE your ‘O’ moves and maybe analogous to mouse-dragging in

> MC4D?

> >>> Rolling of one 2x2x2 half against the other. Maybe analogous to

> ctrl-click?

> >>> Half-rotations. Maybe analogous to a "face first" view? (ctrl-click on

> a

> >>> 2-color piece of the 3^4 with the setting "Ctrl-Click Rotates: by

> Cubie")

> >>> Whole-puzzle reorientations that move an arbitrary axis into the

> "outer" 2

> >>> faces. No MC4D analog.

> >>>

> >>>

> >>> Inverses and performing a move more than once

> >>>

> >>> To mark that a move should be performed n times let’s put ^n after it.

> For

> >>> convenience when writing and speaking let ‘ (prime) represent ^-1 (the

> >>> inverse) and n represent ^n. The inverse P’ of some permutation P is

> the

> >>> permutation that satisfies P P’ = P’ P = I (the identity). For example

> (Rx)’

> >>> means: do Rx backwards, which corresponds to rotating the right 2x2x2

> block

> >>> in the mathematical negative direction (clockwise) around the x-axis

> and

> >>> (Rx)2 means: perform Rx twice. However, we can also define powers of

> just

> >>> the lowercase letters, for example, Rx2 = Rx^2 = Rxx = Rx Rx. So x2=x^2

> >>> means: do whatever the capital letter specifies two times with respect

> to

> >>> the x-axis. We can see that the capital letter naturally is

> distributed over

> >>> the two lowercase letters. Rx’ = Rx^-1 means: do whatever the capital

> letter

> >>> specifies but in the other direction than you would have if the prime

> >>> wouldn’t have been there (note that thus, x’=x^-1 can be seen as the

> >>> negative x-axis). Note that (Rx)^2 = Rx^2 and (Rx)’ = Rx’ which is

> true for

> >>> all twists and rotations but that doesn’t have to be the case for other

> >>> types of moves (restacks and folds).

> >>>

> >>> Restacking moves

> >>>

> >>> A restacking move is an S followed by either x, y or z. Here the

> lowercase

> >>> letter specifies in what direction to restack. For example, Sy (from

> the

> >>> standard rotation) means: take the front 8 pieces and put them at the

> back,

> >>> whereas Sx means: take the top 8 pieces and put them at the bottom.

> Note

> >>> that Sx is equivalent to taking the bottom 8 pieces and putting them

> at the

> >>> top. However, if we want to be able to make half-rotations we

> sometimes need

> >>> to restack through a plane that doesn’t go through the origin. In the

> >>> standard rotation, let Sz be the normal restack (taking the 8 right

> pieces,

> >>> the right 2x2x2 block, and putting them on the left), Sz+ be the

> restack

> >>> where you split the puzzle in the plane further in the positive

> z-direction

> >>> (taking the right 2x2x1 cap of 4 pieces and putting it at the left end

> of

> >>> the puzzle) and Sz- the restack where you split the puzzle in the plane

> >>> further in the negative z-direction (taking the left 2x2x1 cap of 4

> pieces

> >>> and putting it at the right end of the puzzle). If the longer side of

> the

> >>> puzzle is parallel to the x-axis instead, Sx+ would take the top 1x2x2

> cap

> >>> and put it on the bottom. Note that in the standard rotation Sz

> mod(rot) =

> >>> I. For restacks we see that (Sx)’ = Sx’ = Sx (true for y and z too of

> >>> course), that (Sz+)’ = Sz- and that (Sz+)2 = (Sz-)2 = Sz. We can

> define Sz’+

> >>> to have meaning by thinking of z’ as the negative z-axis and with that

> in

> >>> mind it’s natural to define Sz’+=Sz-. Thus, (Sz)’ = Sz’ and (Sz+)’ =

> Sz’+.

> >>>

> >>> Fold moves

> >>>

> >>> A fold might be a little bit harder to describe in an intuitive way.

> First,

> >>> let’s think about what folds are interesting moves. The folds that

> cannot be

> >>> expressed as rotations and restacks are unfolding the puzzle to a 4x4

> and

> >>> then folding it back along another axis. If we start with the standard

> >>> rotation and unfold the puzzle into a 1x4x4 (making it look like a 4x4

> from

> >>> above) the only folds that will achieve something you can’t do with a

> >>> restack mod(rot) is folding it to a 2x4x2 so that the longer side is

> >>> parallel with the y-axis after the fold. Thus, there are 8 interesting

> fold

> >>> moves for any given rotation of the puzzle since there are 4 ways to

> unfold

> >>> it to a 4x4 and then 2 ways of folding it back that make the move

> different

> >>> from a restack move mod(rot).

> >>>

> >>>

> >>> Assuming you complete a folding move in the same representation (<><>

> or

> >>>> <><), then there are only two interesting choices. That’s because it

> >>> doesn’t matter which end of a chosen cutting plane you open it from,

> the end

> >>> result will be the same. That also means that any two consecutive

> clamshell

> >>> moves along the same cutting plane will undo each other. It further

> suggests

> >>> that any interesting sequence of clamshell moves must alternate

> between the

> >>> two possible long cut directions, meaning there is no choice involved.

> 12

> >>> clamshell moves will cycle back to the initial state.

> >>>

> >>> There is one other weird folding move where you open it in one

> direction and

> >>> then fold the two halves back-to-back in a different direction. If you

> >>> simply kept folding along the initial hinge, you’d simply have a

> restacking.

> >>> When completed the other way, it’s equivalent to a restacking plus a

> >>> clamshell, so I don’t think it’s useful though it is somewhat

> interesting.

> >>>

> >>>

> >>> Let’s call these 8 folds interesting fold moves. Note that an

> interesting

> >>> fold move always changes which axis the longer side of the puzzle is

> >>> parallel with. Further note that both during unfold and fold all

> pieces are

> >>> moved; it would be possible to have 8 of the pieces fixed during an

> unfold

> >>> and folding the other half 180 degrees but I think that it’s more

> intuitive

> >>> that these moves fold both halves 90 degrees and performing them with

> >>> 180-degree folds might therefore lead to errors since the puzzle might

> get

> >>> rotated differently. To illustrate a correct unfold without a puzzle:

> Put

> >>> your palms together such that your thumbs point upward and your fingers

> >>> forward. Now turn your right hand 90 degrees clockwise and your left

> hand 90

> >>> degrees counterclockwise such that the normal to your palms point up,

> your

> >>> fingers point forward, your right thumb to the right and your left

> thumb to

> >>> the left. That was what will later be called a Vx unfold and the folds

> are

> >>> simply reversed unfolds. (I might have used the word “fold” in two

> different

> >>> ways but will try to use the term “fold move” when referring to the

> move

> >>> composed by an unfold and a fold rather than simply calling these moves

> >>> “folds”.)

> >>>

> >>> To specify the unfold let’s use V followed by one of x, y, z, x’, y’

> and z’.

> >>> The lowercase letter describes in which direction to unfold. Vx means

> unfold

> >>> in the direction of the positive x-axis and Vx’ in the direction of the

> >>> negative x-axis, if that makes any sense. I will try to explain more

> >>> precisely what I mean with the example Vx from the standard rotation

> (it

> >>> might also help to read the last sentences in the previous paragraph

> again).

> >>> So, the puzzle is in the standard rotation and thus have the form

> 2x2x4 (x-,

> >>> y- and z-thickness respectively). The first part (the unfolding) of

> the move

> >>> specified with Vx is to unfold the puzzle in the x-direction, making

> it a

> >>> 1x4x4 (note that the thickness in the x-direction is 1 after the Vx

> unfold,

> >>> which is no coincidence). There are two ways to do that; either the

> sides of

> >>> the pieces that are initially touching another piece (inside of the

> puzzle

> >>> in the x,z-plane and your palms in the hand example) are facing up or

> down

> >>> after the unfold. Let Vx be the unfold where these sides point in the

> >>> direction of the positive x-axis (up) and Vx’ the other one where these

> >>> sides point in the direction of the negative x-axis (down) after the

> unfold.

> >>> Note that if the longer side of the puzzle is parallel to the z-axis

> only

> >>> Vx, Vx’, Vy and Vy’ are possible. Now we need to specify how to fold

> the

> >>> puzzle back to complete the folding move. Given an unfold, say Vx,

> there are

> >>> only two ways to fold that are interesting (not turning the fold move

> into a

> >>> restack mod(rot)) and you have to fold it perpendicular to the unfold

> to

> >>> create an interesting fold move. So, if you start with the standard

> rotation

> >>> and do Vx you have a 1x4x4 that you have to fold into a 2x4x2. To

> >>> distinguish the two possibilities, use + or - after the Vx. Let Vx+ be

> the

> >>> unfold Vx followed by the interesting fold that makes the sides that

> are

> >>> initially touching another piece (before the unfold) touch another

> piece

> >>> after the fold move is completed and let Vx- be the other interesting

> fold

> >>> move that starts with the unfold Vx. (Thus, continuing with the hand

> >>> example, if you want to do a Vx+ first do the Vx unfold described in

> the end

> >>> of the previous paragraph and then fold your hands such that your

> fingers

> >>> point up, the normal to your palms point forward, the right palm is

> touching

> >>> the right-hand fingers, the left palm is touching the left-hand

> fingers, the

> >>> right thumb is pointing to the right and the left thumb is pointing to

> the

> >>> left). Note that the two halves of the puzzle always should be folded

> 90

> >>> degrees each and you should never make a fold or unfold where you fold

> just

> >>> one half 180-degrees (if you want to use my notation, that is).

> Further note

> >>> that Vx+ Sx mod(3rot) = Vx- and that Vx+ Vx+ = I which is equivalent to

> >>> (Vx+)’ = Vx+ and this is true for all fold moves (note that after a Vx+

> >>> another Vx+ is always possible).

> >>>

> >>> The 2x2x2x2 in the MC4D software

> >>>

> >>> The notation above can also be applied to the 2x2x2x2 in the MC4D

> program.

> >>> There, you are not allowed to do S or V moves but instead, you are

> allowed

> >>> to do the [crtl]+[left-click] moves. This can easily be represented

> with

> >>> notation similar to the above. Let’s use C (as in Centering) and one

> of x, y

> >>> and z. For example, Cx would be to rotate the face in the positive

> >>> x-direction aka the U face to the center. Thus, Cz’ is simply

> >>> [ctrl]+[left-click] on the L face and similarly for the other C moves.

> The

> >>> O, U, D, F, B, R, L, K and A moves are performed in the same way as

> above

> >>> so, for example Rx would be a [left-click] on the top-side of the right

> >>> face. In this representation of the puzzle almost all moves are

> allowed; all

> >>> U, D, F, B, R, L, K, O and C moves are possible regardless of rotation

> and

> >>> only A moves (and of course the rightfully forbidden S and V moves) are

> >>> impossible regardless of rotation. Note that R and L moves in the

> software

> >>> correspond to the same moves of the physical puzzle but this is not

> >>> generally true (I will come back to this later).

> >>>

> >>> Possible moves (so far) in the standard rotation

> >>>

> >>> In the standard rotation, the possible/allowed moves with the

> definitions

> >>> above are:

> >>>

> >>> O moves, all of these are always possible in any state and rotation of

> the

> >>> puzzle since they are simply 3D-rotations.

> >>>

> >>> R, L moves, all of these as well since the puzzle has a right and left

> 2x2x2

> >>> block in the standard rotation.

> >>>

> >>> U. D moves, just Ux2 and Dx2 since the top and bottom are 1x2x4 blocks

> with

> >>> less symmetry than a 2x2x2 block.

> >>>

> >>> F, B moves, just Fy2 and By2 since the front and back are 2x1x4 blocks

> with

> >>> less symmetry than a 2x2x2 block.

> >>>

> >>> K moves, all possible since this is a rotation of the center 2x2x2

> block.

> >>>

> >>>

> >>> I think the only legal 90 degree twists of the K face are those about

> the

> >>> long axis. I believe this is what Christopher Locke was saying in this

> >>> message. To see why there is no straightforward way to perform other 90

> >>> degree twists, you only need to perform a 90 degree twist on an outer

> (L/R)

> >>> face and then reorient the whole puzzle along a different outer axis.

> If the

> >>> original twist was not about the new long axis, then there is clearly

> no

> >>> straightforward way to undo that twist.

> >>>

> >>>

> >>> A moves, only Az moves since this is two 2x2x1 blocks that have to be

> >>> rotated together.

> >>>

> >>> S moves, not Sx+, Sx-, Sy+ or Sy- since those are not defined in the

> >>> standard rotation.

> >>>

> >>> V moves, not Vz+ or Vz- since the definition doesn’t give these

> meaning when

> >>> the long side of the puzzle is parallel with the z-axis.

> >>>

> >>> Note that for example Fz2 is not allowed since this won’t take you to a

> >>> state of the puzzle. To allow more moves we need to extend the

> definitions

> >>> (after the extension in the next paragraph all rotations and twists

> (O, R,

> >>> L, U, D, F, B, K, A) are possible in any rotation and only which S and

> V

> >>> moves are possible depend on the state and rotation of the puzzle).

> >>>

> >>>

> >>> Extension of some definitions

> >>>

> >>> It’s possible to make an extension that allows all O, R, L, U, D, F,

> B, K

> >>> and A moves in any state. I will explain how this can be done in the

> >>> standard rotation but it applies analogously to any other rotation

> where the

> >>> K face is an octahedron. First let’s focus on U, D, F and B and

> because of

> >>> the symmetry of the puzzle all of these are analogous so I will only

> explain

> >>> one. The extension that makes all U moves possible (note that the U

> face is

> >>> in the positive x-direction) is as follows: when making an U move first

> >>> detach the 8 top pieces which gives you a 1x2x4 block, fold this block

> into

> >>> a 2x2x2 block in the positive x-direction (similar to the later part

> of a

> >>> Vx+ move from the standard rotation) such that the U face form an

> >>> octahedron, rotate this 2x2x2 block around the specified axis (for

> example

> >>> around the z-axis if you are doing an Uz move), reverse the fold you

> just

> >>> did creating a 1x2x4 block again and reattach the block.

> >>>

> >>>

> >>> I noticed something like this the other day but realized that it only

> seems

> >>> to work for rotations along the long dimension (z in your example).

> These

> >>> are already easily accomplished by a simple rotation to put the face in

> >>> question on the end caps, followed by a double end-cap twist.

> >>>

> >>> This is as far as I’m going to comment for the moment because the

> >>> information gets very dense and I’ve been mulling and picking over your

> >>> message for several days already. In short, I really like your attempt

> to

> >>> provide a complete system of notation for discussing this puzzle and

> will be

> >>> curious to hear your thoughts on my comments so far. I hope others will

> >>> chime in too.

> >>>

> >>> One final thought is that a real "acid test" of any notation system

> for this

> >>> puzzle will be attempt to translate some algorithms from MC4D. I would

> most

> >>> like to see a sequence that flips a single piece, like the second

> 4-color

> >>> series on this page of Roice’s solution, or his pair of twirled

> corners at

> >>> the end of this page. One trick will be to minimize the number of

> >>> whole-puzzle reorientations needed, but really any sequence that works

> will

> >>> be great evidence that the puzzles are equivalent. I suspect that this

> sort

> >>> of exercise will never be practical because it will require too many

> >>> reorientations, and that entirely new methods will be needed to

> actually

> >>> solve this puzzle.

> >>>

> >>> Best,

> >>> -Melinda

> >>>

> >>> The A moves can be done very similarly but after you have detached the

> two

> >>> 2x2x1 blocks you don’t fold them but instead you stack them similar to

> a Sz

> >>> move, creating a 2x2x2 block with the A face as an octahedron in the

> middle

> >>> and then reverse the process after you have rotated the block as

> specified

> >>> (for example around the negative y-axis if you are doing an Ay’ move).

> Note

> >>> that these extended moves are closely related to the normal moves and

> for

> >>> example Ux = Ry Ly’ Kx Ly Ry’ in the standard rotation and note that

> (Ry

> >>> Ly’) mod(rot) = (Ly Ry’) mod(rot) = I (this applies to the other

> extended

> >>> moves as well).

> >>>

> >>> If the cube is in the half-rotated state, where both the R and L faces

> are

> >>> octahedra, you can extend the definitions very similarly. The only

> thing you

> >>> have to change is how you fold the 2x4 blocks when performing a U, D,

> F or B

> >>> move. Instead of folding the two 2x2 halves of the 2x4 into a 2x2 you

> have

> >>> to fold the end 1x2 block 180 degrees such that the face forms an

> >>> octahedron.

> >>>

> >>> These moves might be a little bit harder to perform, to me especially

> the A

> >>> moves seems a bit awkward, so I don’t know if it’s good to use them or

> not.

> >>> However, the A moves are not necessary if you allow Sz in the standard

> >>> rotation (which you really should since Sz mod(rot) = I in the standard

> >>> rotation) and thus it might not be too bad to use this extended

> version. The

> >>> notation supports both variants so if you don’t want to use these

> extended

> >>> moves that shouldn’t be a problem. Note that, however, for example Ux

> >>> (physical puzzle) != Ux (virtual puzzle) where != means “not equal to”

> (more

> >>> about legal/illegal moves later).

> >>>

> >>> Generalisation of the notation

> >>>

> >>> Let’s generalise the notation to make it easier to use and to make it

> work

> >>> for any n^4 cube in the MC4D software. Previously, we saw that (Rx)2 =

> Rx Rx

> >>> and if we allow ourselves to rewrite this as (Rx)2 = Rx2 = Rxx = Rx Rx

> we

> >>> see that the capital letter naturally can be distributed over the

> lowercase

> >>> letters. We can make this more general and say that any capital letter

> >>> followed by several lowercase letters means the same thing as the

> capital

> >>> letter distributed over the lowercase letters. Like Rxyz = Rx Ry Rz

> and here

> >>> R can be exchanged with any capital letter and xyz can be exchanged

> with any

> >>> sequence of lowercase letters. We can also allow several capital

> letters and

> >>> one lowercase letter, for example RLx and let’s define this as RLx =

> Rx Lx

> >>> so that the lowercase letter can be distributed over the capital

> letters. We

> >>> can also define a capital letter followed by ‘ (prime) like R’x = Rx’

> and

> >>> R’xy = Rx’y’ so the ‘ (prime) is distributed over the lowercase

> letters.

> >>> Note that we don’t define a capital to any other power than -1 like

> this

> >>> since for example R2x = RRx might seem like a good idea at first but it

> >>> isn’t very useful since R2 and RR are the same lengths (and powers

> greater

> >>> than two are seldom used) and we will see that we can define R2 in

> another

> >>> way that generalises the notation to all n^4 cubes.

> >>>

> >>> Okay, let’s define R2 and similar moves now and have in mind what

> moves we

> >>> want to be possible for a n^4 cube. The moves that we cannot achieve

> with

> >>> the notation this far is twisting deeper slices. To match the notation

> with

> >>> the controls of the MC4D software let R2x be the move similar to Rx but

> >>> twisting the 2nd layer instead of the top one and similarly for other

> >>> capital letters, numbers (up to n) and lowercase letters. Thus, R2x is

> >>> performed as Rx but holding down the number 2 key. Just as in the

> program,

> >>> when no number is specified 1 is assumed and you can combine several

> numbers

> >>> like R12x to twist both the first and second layer. This notation does

> not

> >>> apply to rotations (O) folding moves (V) and restacking moves (S) (I

> suppose

> >>> you could redefine the S move using this deeper-slice-notation and use

> S1z

> >>> as Sz+, S2z as Sz and S3z as Sz- but since these moves are only

> allowed for

> >>> the physical 2x2x2x2 I think that the notation with + and – is better

> since

> >>> S followed by a lowercase letter without +/- always means splitting

> the cube

> >>> in a coordinate plane that way, not sure though so input would be

> great).

> >>> The direction of the twist R3x should be the same as Rx meaning that

> if Rx

> >>> takes stickers belonging to K and move them to F, so should R3x, in

> >>> accordance with the controls of the MC4D software. Note that for a

> 3x3x3x3

> >>> it’s true that R3z = Lz whereas R3x = Lx’ (note that R and L are the

> faces

> >>> in the z-directions so because of the symmetry of the cube it will

> also be

> >>> true that for example U3x = Bx whereas U3y=By’).

> >>>

> >>> What about the case with several capital letters and several lowercase

> >>> letters, for instance, RLxy? I see two natural definitions of this.

> Either,

> >>> we could have RLxy = Rxy Lxy or we could define it as RLxy = RLx RLy.

> These

> >>> are generally not the same (if you exchange R and L with any allowed

> capital

> >>> letter and similarly for x and y). I don’t know what is best, what do

> you

> >>> think? The situations I find this most useful in are RL’xy to do a

> rotation

> >>> and RLxy as a twist. However, since R and L are opposite faces their

> >>> operations commute which imply RL’xy (1st definition) = Rxy L’xy = Rx

> Ry

> >>> Lx’ Ly’ = Rx Lx’ Ry Ly’ = RL’x RL’y = RL’xy (2nd definition) and

> similarly

> >>> for the other case with RLxy. Hopefully, we can find another useful

> sequence

> >>> of moves where this notation can be used with only one of the

> definitions

> >>> and can thereby decide which definition to use. Personally, I feel

> like RLxy

> >>> = RLx RLy is the more intuitive definition but I don’t have any good

> >>> argument for this so I’ll leave the question open.

> >>>

> >>> For convenience, it might be good to be able to separate moves like

> Rxy and

> >>> RLx from the basic moves Rx, Oy etc when speaking and writing. Let’s

> call

> >>> the basic moves that only contain one capital letter and one lowercase

> >>> letter (possibly a + or –, a ‘ (prime) and/or a number) simple moves

> (like

> >>> Rx, L’y, Ux2 and D3y’) and the moves that contain more than one capital

> >>> letter or more than one lowercase letter composed moves.

> >>>

> >>> More about inverses

> >>>

> >>> This list can obviously be made longer but here are some identities

> that are

> >>> good to know and understand. Note that R, L, U, x, y and z below just

> are

> >>> examples, the following is true in general for non-folds (however, S

> moves

> >>> are fine).

> >>>

> >>> (P1 P2 … Pn)’ = Pn’ … P2’ P1’ (Pi is an arbitrary permutation for

> >>> i=1,2,…n)

> >>> (Rxy)’ = Ry’x’ = R’yx

> >>> (RLx)’ = LRx’ = L’R’x

> >>> (Sx+z)’ = Sz’ Sx’+ (just as an example with restacking moves, note that

> >>> the inverse doesn’t change the + or -)

> >>> RLUx’y’z’=R’L’U’xyz (true for both definitions)

> >>> (RLxy)’ = LRy’x’ = L’R’yx (true for both definitions)

> >>>

> >>> For V moves we have that: (Vx+)’ = Vx+ != Vx’+ (!= means “not equal

> to”)

> >>>

> >>> Some important notes on legal/illegal moves

> >>>

> >>> Although there are a lot of moves possible with this notation we might

> not

> >>> want to use them all. If we really want a 2x2x2x2 and not something

> else I

> >>> think that we should try to stick to moves that are legal 2x2x2x2

> moves as

> >>> far as possible (note that I said legal moves and not permutations (a

> legal

> >>> permutation can be made up of one or more legal moves)). Clarification:

> >>> cycling three of the edge-pieces of a Rubik’s cube is a legal

> permutation

> >>> but not a legal move, a legal move is a rotation of the cube or a

> twist of

> >>> one of the layers. In this section I will only address simple moves and

> >>> simply refer to them as moves (legal composed moves are moves composed

> by

> >>> legal simple moves).

> >>>

> >>> I do believe that all moves allowed by my notation are legal

> permutations

> >>> based on their periodicity (they have a period of 2 or 4 and are all

> even

> >>> permutations of the pieces). So, which of them correspond to legal

> 2x2x2x2

> >>> moves? The O moves are obviously legal moves since they are equal to

> the

> >>> identity mod(rot). The same goes for restacking (S) (with or without

> +/-) in

> >>> the direction of the longest side of the puzzle (Sz, Sz+ and Sz- in the

> >>> standard rotation) since these are rotations and half-rotations that

> don’t

> >>> change the state of the puzzle. Restacking in the other directions and

> fold

> >>> moves (V) are however not legal moves since they are made of 8

> 2-cycles and

> >>> change the state of the puzzle (note that they, however, are legal

> >>> permutations). The rest of the moves (R, L, F, B, U, D, K and A) can be

> >>> divided into two sets: (1) the moves where you rotate a 2x2x2 block

> with an

> >>> octahedron inside and (2) the moves where you rotate a 2x2x2 block

> without

> >>> an octahedron inside. A move belonging to (2) is always legal. We can

> see

> >>> this by observing what a Rx does with the pieces in the standard

> rotation

> >>> with just K forming an octahedron. The stickers move in 6 4-cycles and

> if

> >>> the puzzle is solved the U and D faces still looks solved after the

> move. A

> >>> move belonging to set (1) is legal either if it’s an 180-degree twist

> or if

> >>> it’s a rotation around the axis parallel with the longest side of the

> puzzle

> >>> (the z-axis in the standard rotation). Quite interestingly these are

> exactly

> >>> the moves that don’t mix up the R and L stickers with the rest in the

> >>> standard rotation. I think I know a way to prove that no legal 2x2x2x2

> move

> >>> can mix up these stickers with the rest and this has to do with the

> fact

> >>> that these stickers form an inverted octahedron (with the corners

> pointing

> >>> outward) instead of a normal octahedron (let’s call this hypothesis *

> for

> >>> now). Note that all legal twists (R, L, F, B, U, D, K and A) of the

> physical

> >>> puzzle correspond to the same twist in the MC4D software.

> >>>

> >>> So, what moves should we add to the set of legal moves be able to get

> to

> >>> every state of the 2x2x2x2? I think that we should add the restacking

> moves

> >>> and folding moves since Melinda has already found a pretty short

> sequence of

> >>> those moves to make a rotation that changes which colours are on the R

> and L

> >>> faces. That sequence, starting from the standard rotation, is: Oy Sx+z

> Vy+

> >>> Ozy’ Vy+ Ozy’ Vy+ = Oy Sx+z (Vy+ Ozy’)3 mod(3rot) = I mod(rot)

> (hopefully I

> >>> got that right). What I have found (which I mentioned previously) is

> that

> >>> (from the standard rotation): RLx Ly2 Sy By2 Ly2 RLx’ = Sz- and since

> this

> >>> is equivalent with Sy = Ly2 RLx’ Sz- RLx Ly2 By2 the restacking moves

> that

> >>> are not legal moves are not very complicated permutations and

> therefore I

> >>> think that we can accept them since they help us mix up the R and L

> stickers

> >>> with other faces. In a sense, the folding moves are “more illegal”

> since

> >>> they cannot be composed by the legal moves (according to hypothesis

> *). This

> >>> is also true for the illegal moves belonging to set (1) discussed

> above.

> >>> However, since the folding moves is probably easier to perform and is

> enough

> >>> to reach every state of the 2x2x2x2 I think that we should use them

> and not

> >>> the illegal moves belonging to (1). Note, once again, that all moves

> >>> described by the notation are legal permutations (even the ones that I

> just

> >>> a few words ago referred to as illegal moves) so if you wish you can

> use all

> >>> of them and still only reach legal 2x2x2x2 states. However (in a strict

> >>> sense) one could argue that you are not solving the 2x2x2x2 if you use

> >>> illegal moves. If you only use illegal moves to compose rotations

> (that is,

> >>> create a permutation including illegal moves that are equal to I

> mod(rot))

> >>> and not actually using the illegal moves as twists I would classify

> that as

> >>> solving a 2x2x2x2. What do you think about this?

> >>>

> >>> What moves to use?

> >>>

> >>> Here’s a short list of the simple moves that I think should be used

> for the

> >>> physical 2x2x2x2. Note that this is just my thoughts and you may use

> the

> >>> notation to describe any move that it can describe if you wish to. The

> >>> following list assumes that the puzzle is in the standard rotation but

> is

> >>> analogous for other representations where the K face is an octahedron.

> >>>

> >>> O, all since they are I mod(rot),

> >>> R, L, all since they are legal (note Rx (physical puzzle) = Rx (virtual

> >>> 2x2x2x2)),

> >>> U, D, only x2 since these are the only legal easy-to-perform moves,

> >>> F, B, only y2 since these are the only legal easy-to-perform moves,

> >>> K, A, only z, z’ and z2 since these are the only legal easy-to-perform

> >>> moves,

> >>> S, at least z, z+ and z- since these are equal to I mod(rot),

> >>> S, possibly x and y since these help us perform rotations and is easy

> to

> >>> compose (not necessary to reach all states and not legal though),

> >>> V, all 8 allowed by the rotation of the puzzle (at least one is

> necessary to

> >>> reach all states and if you allow one the others are easy to achieve

> >>> anyway).

> >>>

> >>> If you start with the standard rotation and then perform Sz+ the

> following

> >>> applies instead (this applies analogously to any other rotation where

> the R

> >>> and L faces are octahedra).

> >>>

> >>> O, all since they are I mod(rot),

> >>> R, L, only z, z’ and z2 since these are the only legal easy-to-perform

> >>> moves,

> >>> U, D, only x2 since these are the only legal easy-to-perform moves,

> >>> F, B, only y2 since these are the only legal easy-to-perform moves,

> >>> K, A, at least z, z’ and z2, possibly all (since they are legal)

> although

> >>> some might be hard to perform.

> >>> S, V, same as above.

> >>>

> >>> Note that (from the standard rotation): Ux2 Sz+ Ux2 Sz- != I mod(rot)

> (!=

> >>> for not equal) which implies that the Ux2 move when R and L are

> octahedra is

> >>> different from the Ux2 move when K is an octahedron. (Actually, the

> sequence

> >>> above is equal to Uy2).

> >>>

> >>> Regarding virtual n^4 cubes I think that all O, C, R, L, U, D, F, B, K

> and A

> >>> moves should be used since they are all legal and really the only

> thing you

> >>> need (left-clicking on an edge or corner piece in the computer program

> can

> >>> be described quite easily with the notation, for example, Kzy2 is

> >>> left-clicking on the top-front edge piece on the K face).

> >>>

> >>> I hope this was possible to follow and understand. Feel free to ask

> >>> questions about the notation if you find anything ambiguous.

> >>>

> >>> Best regards,

> >>> Joel Karlsson

> >>>

> >>> Den 4 maj 2017 12:01 fm skrev "Melinda Green melinda@superliminal.com

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

> >>>

> >>>

> >>>

> >>> Thanks for the correction. A couple of things: First, when assembling

> one

> >>> piece at a time, I’d say there is only 1 way to place the first piece,

> not

> >>> 24. Otherwise you’d have to say that the 1x1x1 puzzle has 24 states. I

> >>> understand that this may be conventional, but to me, that just sounds

> silly.

> >>>

> >>> Second, I have the feeling that the difference between the "two

> >>> representations" you describe is simply one of those half-rotations I

> showed

> >>> in the video. In the normal solved state there is only one complete

> >>> octahedron in the very center, and in the half-rotated state there is

> one in

> >>> the middle of each half of the "inverted" form. I consider them to be

> the

> >>> same solved state.

> >>>

> >>> -Melinda

> >>>

> >>>

> >>> On 5/3/2017 2:39 PM, Joel Karlsson joelkarlsson97@gmail.com

> [4D_Cubing]

> >>> wrote:

> >>>

> >>> Horrible typo… It seems like I made some typos in my email regarding

> the

> >>> state count. It should of course be 16!12^16/(6*192) and NOT

> >>> 12!16^12/(6*192). However, I did calculate the correct number when

> comparing

> >>> with previous results so the actual derivation was correct.

> >>>

> >>> Something of interest is that the physical pieces can be assembled in

> >>> 16!24*12^15 ways since there are 16 pieces, the first one can be

> oriented in

> >>> 24 ways and the remaining can be oriented in 12 ways (since a corner

> with 3

> >>> colours never touch a corner with just one colour). Dividing with 6 to

> get a

> >>> single orbit still gives a factor 2*192 higher than the actual count

> rather

> >>> than 192. This shows that every state in the MC4D representation has 2

> >>> representations in the physical puzzle. These two representations must

> be

> >>> the previously discussed, that the two halves either have the same

> color on

> >>> the outermost corners or the innermost (forming an octahedron) when the

> >>> puzzle is solved and thus both are complete representations of the

> 2x2x2x2.

> >>>

> >>> Best regards,

> >>> Joel Karlsson

> >>>

> >>> Den 30 apr. 2017 10:51 em skrev "Joel Karlsson" <

> joelkarlsson97@gmail.com>:

> >>>

> >>> 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,

> >>> Joel

> >>>

> >>> 2017-04-29 1:04 GMT+02:00 Melinda Green melinda@superliminal.com

> [4D_Cubing]

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

> >>>>

> >>>>

> >>>> 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 roice3@gmail.com [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

> joelkarlsson97@gmail.com

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

> 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/dessert_cube.jpg I don’t know how

> practical a

> >>>>>> solution this is but it sure looks delicious! Welcome Marc!

> >>>>>>

> >>>>>> -Melinda

> >>>>>>

> >>>>>

> >>>>>

> >>>>>

> >>>>>

> >>>>

> >>>

> >>>

> >>>

> >

> > ————————————

> > Posted by: Joel Karlsson <joelkarlsson97@gmail.com>

> > ————————————

> >

> >

> > ————————————

> >

> > Yahoo Groups Links

> >

> >

> >

> >

>

>

>