# Expansion of Melinda's 2x2x2x2 Canonical Moves

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### Blobinati Cuber

Aug 17, 2022, 9:39:42 PM8/17/22
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Hi everyone. A lot of speedsolving the physical 2^4 has been happening recently, and has brought up old questions about what exactly the canonical moves are. I'd like to reopen this for discussion.

I value accuracy and analogy with the virtual puzzle. I think that moves on the physical puzzle that map to a single move on the virtual should be completely allowed. This means I would love to see the following twists be added to the canonical move set (which can be found here) ((even if I wouldn't use them in a speedsolve))

I made a video demonstrating the moves, which you can find here.

1. Iy2/IU2 and Iz2/IF2
Not much to say about this. Just rotations of the I cell that should be allowed in a solve.

2. Uz2/UF2 and Ux2/UR2 and Ux*/UR*
Melinda says "180 degree twists of one of the side faces" in the canonical moves video. If you've solved any virtual n^4, you know that there are actually 3 different 180 twists of a face. For example, Ux2, Uy2, and Uz2. The one that she shows is a Uy2/UO2. The other ones are completely valid moves on the virtual puzzle, and I know of fast ways of executing them on the physical puzzle (shown in my video). What I forgot to show in my video is a single Ux*/UR* move. This is the equivalent of Lz Rz' Ix Lz' Rz, for example. These moves I find incredibly useful during speedsolving of block building layers.

3. 1/2 gyro/ algorithm for Iz/IF
Hactar recently brought to my attention a fast algorithm that gives the I cell a z rotation but with affecting orientation in the correct way. Simply do R2 F2 U2 R2 z' on the I cell. Just straight up rotating the I cell like a z rotation permutes the pieces fine, it just orients them wrong due to the projection. But this 4 move algorithm can fix it, making it a longer way to do a single move, just like the full gyro (6+ snaps to do a 0 move reorientation)

Thanks for the consideration!
- Rowan

### Grant S

Aug 17, 2022, 9:49:17 PM8/17/22
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I am all for these moves, a bit back Luna suggested a simple rule that
"anything that maps to a single move on the real 2^4" should be allowed."
Which I think would be a nice general purpose way to say what is allowed or not.
I'm looking forward to seeing what everyone else thinks about all these moves Rowan presented.

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### Joel Karlsson

Aug 18, 2022, 3:40:29 AM8/18/22
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Completely agree! Any move that corresponds directly to a single move on the virtual puzzle should be allowed.

### Eduard Baumann

Aug 18, 2022, 4:19:03 AM8/18/22
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Wow. Very interesting and valuable considerations. I agree completely.
Best regards
Ed
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### Grant S

Aug 18, 2022, 11:01:38 AM8/18/22
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So far it seems like no one is opposed to these additions.
Is there anyone at all who is opposed to these changes?

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### Robert DuBeau

Aug 18, 2022, 12:25:49 PM8/18/22
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I am not opposed, all in favor, but just want to point out what Rowan has already alluded to: the “half gyro” will never save much or maybe any time in an actual solve! Having done a complete solve using half-gyros I can say it’s pretty satisfying in its own way, and for completeness should be allowed, but will “hardly ever” save anyone time.

### Grant S

Aug 18, 2022, 1:11:10 PM8/18/22
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I think that some of the methods used by speedsolvers for orienting the 2 cells at the start of the solve could be sped up by occasionally using half gyros. I am pretty sure that one of the quickest methods for this step (Used by both Rowan and I) could actually be sped up by using a half gyro move in a specific spot. I haven't actually tested this though, I will test when I get home later today and let you know.
But yeah, I also agree that just because a certain move may not be "useful" it should still be considered for inclusion just for the completeness of the canonical moveset.

### Grant S

Aug 18, 2022, 7:31:50 PM8/18/22
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Ok, just tried it and it is about the same time right now, but also I have 0 practice doing the half gyro moves quickly.... but definitely some room for improvement by using the half gyros in some specific scenarios.

### Robert DuBeau

Aug 19, 2022, 8:29:42 AM8/19/22
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That is amazing! I’ve never been a speed solver so just wasn’t sure. I have a solve i made a few months ago that uses half-gyros throughout. Slowest solve ever, but it made me get fairly quick at the half-gyro move I had found. If anyone watches the video I recommend it at 2x speed…

BTW this was my first solve ever of any 4D magic cube. I find the speed techniques you guys are using to be quite cool.

### Melinda Green

Aug 20, 2022, 5:21:51 PM8/20/22
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I suppose it's my job to drill in and act as devil's advocate to try to nail down exactly what is being proposed and express all the considerations I can think of. I hope everyone interested in the subject will help in that regard to make sure we don't overlook anything potentially important, rather than simply lobbying for inclusion of any personal favorite move.

This exercise was prompted I think by a suggestion by Luna that all moves resulting in a single twist of a single face in MC4D should be allowed. Let's call this the "maximally inclusive" approach for now. That's clearly a reasonable position. We just need to ask what it implies for the canonical move set and its uses for purposes of competitions, record keeping, and implications for newbies.

The proposition itself doesn't seem to be enough for these practical purposes. For example what should a competition official do regarding unfamiliar sequences in the middle of someone's solve? They can certainly ask, and the contestant can explain and demonstrate their moves, but that would require close examination of potentially any number of such sequences which seems impractical.

One solution to such custom moves might be to make people declare and demonstrate those moves at the outset. In videos that simply means demonstrating them before scrambling. One problem with that is that it's too easy to forget or conveniently not quite finish such an acceptable sequence when that creates a shortcut in a solution. It would be very difficult to catch these when anyone can use custom moves. It's hard enough as it is with custom gyros. Perhaps this suggests that we shouldn't allow custom moves at all? A good general solution is not at all clear to me.

My feeling is that the goal here should be a clear static list of allowable moves, and that it should be kept to a minimum. Maybe call this the "minimally inclusive" approach. So no custom moves, and no including moves simply because they exist. One strong argument in favor of the approach is that it makes it easier for newbies to learn. I fully understand that this will be unattractive to some or most speedsolvers who care about the subject, but remember that this puts all the burden on me or whoever is verifying solves. So if you want to argue for maximal inclusion, then I think you should be willing to take on that responsibility yourself, or at least to share it amongst those who feel the same.

But perhaps neither extreme works very well and some sort of compromise will do. I'm thinking of a 2-tiered system with a core set of moves that is the current canonical moves or something very close to that, plus a more malleable set of additional moves that we limit to only the most useful of the maximally inclusive moves. This set can be altered to suit the needs of the popular methods, a bit analogous to the US Constitution which has a minimal core set of governmental rules and a bunch of amendments that get added when current needs are strong enough.

Perhaps we need some specific proposals to get a sense of the best approach, and Rowan has provided exactly that, so let's take a look:

1. Iy2/IU2 and Iz2/IF2
We've always known that these make sense, but does anyone intend to actually use them if they're allowed?

2. Uz2/UF2 and Ux2/UR2 and Ux*/UR*
I'm unsure about the full set of moves suggested here plus what he says he didn't include in his video, but the first of these moves in his video (sliding a2x4x1 slab halfway down and moving the overhanging half to the other end) is one we've known from the start but which nobody has needed since it's easily done with a 180 degree twist of a side face plus a simple rotation. The other on in the video has similarly long been known (counter-rotating two 2x2x1 halves of a side face) which is equivalent to a 180 twist of a side face and a different simple rotation. Rowan calls (some of?) these moves "incredibly useful" though I don't see how they are any easier than the canonical ways involving a simple rotation. We probably just need a little more discussion about them.

3. 1/2 gyro/ algorithm for Iz/IF
The half gyros are an interesting case that I am sympathetic to, but what are the ramifications? Can they be reduced to a single sequence or is there a range of different preferences?

And the question of custom regular gyros still remains, so all-in-all there are quite a few overlapping issues, ideas, and options here. I don't have a particular proposal to make and only hope that I helped sketch out some of the opportunities, issues, and approaches we could take. Do you feel like you see a good way through, or a way to focus/simplify the topic? I would love to hear it.

Best,
-Melinda

### Grant S

Aug 20, 2022, 5:40:00 PM8/20/22
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Those are great things to consider and I hadn't really considered the perspective of ensuring only correct moves are used while watching a solve at all, so thank you for bringing that up.

But here is my take on some of the questions you asked.
1. Iy2/IU2 and Iz2/IF2
Personally I don't know when I would use them, they are slower to execute than the other 180 twists we have available.
2. Uz2/UF2 and Ux2/UR2 and Ux*/UR*
I have used Uz2 in solves when not timing them officially and for me it is just slightly easier & quicker to know where to line up the piece correctly with other turns in order to get the desired result from that turn. Also at times you might find yourself doing setup moves, 180 twist, undo setup moves, when this could instead be accomplished with just a single twist if the other 180 twist variations were allowed.
3. 1/2 gyro/ algorithm for Iz/IF
The only algorithm for the half gyros that I have seen is the R2 F2 U2 R2 z', but this is also likely just due to the fact that they aren't very widely used at all. I am not sure what other reasonable options (if any) exist for these moves.

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### Asa Kaplan

Aug 20, 2022, 6:01:38 PM8/20/22
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Throwing out a different direction altogether here:

I think the goals as people have outlined are

1. Identifiability of legal moves
2. Keeping the Identifiability to a limited set
3. Keeping a one move = pure rotation or a single turn on a 4D puzzle

I'm going to suggest something different altogether, since we're on the subject. Why should we really keep 3?

Spoiler: I really enjoy the ROIL moveset.

- Using it creates a clear set of moves that are unambiguous
- There is no need for gyros, custom or not. The discussion here would really not end up happening, because we'd already be complete.
- It's relatively close to pure moves. There aren't ridiculous things that are happening, just a few extra moves for each. It's not the equivalent of adding a sune or something to some pieces, or doing major orientation changes
- It increases the dimensionality of most stages of the solve. This is the biggest one for me. Instead of solving or orienting one 3d perspective, and then changing it with a gyro, then solving another... It all can happen at once

On the last point, there is a reason most of the methods for solving the puzzle as-is have taken similar paths. Being relatively constrained by gyros means that we have a limited number of choices for each stage in between a gyro.

I think with changing to ROIL, we'll end up having more interesting methods, and it won't stray too significantly from the 4d mapping

I can also say that from the BLD perspective, it makes things more clear. Tracking pieces through gyros is doable, but just... A lot, and a bit unnecessary. Half of the work I did with BLD was to make it easier to work in and out of a gyro (the success video I posted had 1 gyro [1/16 solves can be done this way], and most BLD solves in my method have 2 [nearly but not quite the remaining 15/16])

- Asa (she/her)

### Grant S

Aug 20, 2022, 6:11:43 PM8/20/22
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In terms of blind solving, yes, removing gyros entirely definitely will make things easier. Gyros being included are what scared me away from attempting bld a couple months ago. Adding the half gyro moves without removing the full gyros, that could make bld more feasible and still allow it for ease during non bld solves. If we move anything from the current moveset instead of only adding things, we are invalidating many previous solutions and records.

Also you said "On the last point, there is a reason most of the methods for solving the puzzle as-is have taken similar paths. Being relatively constrained by gyros means that we have a limited number of choices for each stage in between a gyro."
However, the quickest speedsolving techniques used for virtual 2^4 and physical 2^4 are actually very similar, first orienting the first 2 cells (there are a couple different paths here) and then permuting them either by orienting 4 sides and then doing P4L or by building 1 layer, and then another layer, matching those up and then an rkt 2x2x2 solve.
The extra flexibility in a virtual solve doesn't actually change much for the top speed solves, and the wr physical solve is less than 10 seconds slower than the wr virtual solve...

Great points though, always good to bring in a new perspective.

### Blobinati Cuber

Aug 21, 2022, 2:17:02 AM8/21/22
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Melinda said "Rowan calls (some of?) these moves "incredibly useful" though I don't see how they are any easier than the canonical ways involving a simple rotation"
I just want to point out that I'm obviously coming from a slightly biased speedsolving perspective. My main point is that it is 1 move on a virtual 2^4. You have to click twice. Whereas to execute that same exact twist on the physical 2^4, you must do 2 superfluous rotations in order to do it with I moves, which are less ergonomic. I just timed it and and the Ux2 takes me about 2 seconds to execute using RKT (rotating, Ix2, rotating), whereas the pure Ux2 takes me only 1 second. I honestly can't think of a reason why you should not allow them.

Imagine the 3D equivalent of this (3x3x3): U2 moves aren't canon, but everything else and rotations are.
In order to do a U2, you would have to rotate the whole puzzle (which takes valuable time), do an R2 or any other _2 move, then rotate that side back to U. That's honestly just ludicrous. I hope that helps everyone understand what I mean. The canonical way to do the Ux2 is the same. 4D rotation, Ix2, 4D rotation back. Taking 3 movements instead of just 1. Maybe I can do a video solely focusing on the Ux* moves, if that would help, just let me know :)

The 1/2 gyro I could definitely do without, but it just makes sense as we have a full gyro, which is a long physical sequence of moves to do a single 4D rotation. And the 1/2 gyro would also be a longer sequence of physical moves to do 1 virtual twist. But I strongly believe in the other types of moves to add to the current canon. I don't think it should span much beyond that, or to the point where we need a list of allowed gyro algorithms.

I understand your ideas about keeping the moves allowed to a minimum so that it is easier to understand for beginners. I would think that most solvers of the physical 2^4 would start in MC4D, and then if they like the puzzle, bought a physical one. Meaning that they would already have experience with it and how the virtual puzzle turns, and be able to understand and extrapolate those patterns to the physical puzzle.

I don't think I would use Iy2 or Iz2 in a solve - just because that's not how my method works - but someone might in the future. Should include them because they are perfectly valid.

- Rowan

### Melinda Green

Aug 21, 2022, 7:28:31 PM8/21/22
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There are a lot of different perspectives here including
• Speed solvers
• Blind solvers
• Math enthusiasts
• Puzzle enthusiasts
• Puzzle creators
• Students/educators/parents
• Collectors
This subject affects all of them and they each have different needs. You guess that "most solvers of the physical 2^4 would start in MC4D", but you are only talking about solvers, not owners. Only a small minority of owners are solvers, but the great majority of them still want to understand its operation and the meaning of its operations well enough to explain them to others. This is similar to the 3x3x3 where over 450 million have been sold and I bet an even smaller percentage can solve it.

The main sort of feedback I'm getting here--not just from you--is of the form "I just want my favorite moves included". You appear to be proposing a general solution which is a kind of "maximally inclusive" approach of accepting any sequence that results in a single move on the virtual puzzle. It's helpful that you are proposing something broader than just the moves you presently want, but you still haven't addressed the practical problem of administrating records and competitions. Are you volunteering to certify and adjudicate all HOF solutions and competitions with that rule? I'm not.

I'd really love to hear some concrete proposals of the form "These N moves because...", and I would love to keep N as small as possible.

Best,
-Melinda

### Nicole

Aug 21, 2022, 7:32:53 PM8/21/22
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I'd like to just further Melinda's point here, as I've solved the physical puzzle, but never the virtual one. To be completely honest, I find the physical puzzle more intuitive.

### Grant S

Aug 21, 2022, 7:36:32 PM8/21/22
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I agree that there is a lot to consider and that allowing absolutely everything that maps to a single move isn't feasible for verifying solutions.
I think I fall under multiple of those categories (speedsolver perspective but also puzzle creator and puzzle enthusiast and collector...) which is giving me a lot to think about and some mixed feelings on these moves myself.

Im going to think on this and get back to you on a set list that I would consider.

Its a tough decision.
-  Grant

### vasvaria

Sep 7, 2022, 3:22:16 AM9/7/22
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Hello HyperCubists,

I completely agree with Rowan regarding the Iy2/IU2 and Iz2/IF2 twists that these should be considered as canonical moves. These are useful moves and it is easy to execute and also easy to recognize during a solve.

The second Uz2/UF2 and Ux2/UR2 and Ux*/UR* twists I'm not sure because I can't see the benefits and it is harder to recognize during a solve if it was a legal move or not.

As for the third 1/2 gyro/ algorithm for Iz/IF moves I think it's definitely not a good idea to add them to the canonical moves because if we allow single twists on a single cell (even if it results in legal state of the cube) it can cause a lot of confusion later.

Happy Hypercubing!

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### miguelalo...@gmail.com

Feb 7, 2023, 11:12:20 AM2/7/23
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I completely agree with  the Iy2/IU2 and Iz2/IF2 twists   but not with the others

### Philip Strimpel

Dec 20, 2023, 10:29:39 AM12/20/23
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If I may add my input here, I would argue that any solution that doesn't allow twists of a hyperface on the W axis actually makes the puzzle far too trivial to solve. I actually try not to use any full gyro rotations because of such triviality, and it's a lot more fun in my opinion: just my two cents on the matter.

### Ben Prather

Jan 29, 2024, 1:23:29 AMJan 29
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My interest in this puzzle has been piqued.

In the 4D abstract puzzle there are 8*24=192 rotations that would not count as moves in a solve. Has anyone classified these moves for the 2x2x2x2?

24 of them are generated by separating the two cubes, performing one of the 24 possible 3D cube rotations on both and rejoining them. The gyro move appears to be just one of the many involving the 4th dimension.
A 3D rotation, gyro and a 2D rotation can generate all 4D rotations, so 3D x gyro x 3D rotations covers all cases.

Further, there are 8*24=192 rotations that correspond to a rotation of each face. The 24 for the right or left cube are obvious. Only 4 of the 24 possibilities for the top seem to be covered by the canonical move set. (rotation by 180, swapping the two squares, and composing these.) What are the other 20?

Just by combinatorics the rest are a gyro x twist or a gyro x twist x gyro combination.
Are any of there somehow short? The latter cases in particular seems likely.

Indeed, these are what opened this discussion.