# Message #3712

From: Melinda Green <melinda@superliminal.com>

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

Date: Mon, 29 May 2017 18:57:41 -0700

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>

> ————————————

>

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> ————————————

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