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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