Member Introduction / 2D cubes

[Blobinati Cuber]

Hello everyone! My name is Rowan and I have a YouTube channel called Blobinati Cuber. I became interested in Hypercubing when I discovered magic cube 4d in January. I made a video on my channel of the learning process of solving the 3^4.

Recently I was bored and was looking through the magic cube superliminal website, and I discovered this page about 2-Dimensional Rubik's cubes (Rubik's Squares??), and became super intrigued! I hope whoever made that doesn't mind me borrowing the idea, because I started to make my own iOS app for 2D twisty puzzles.

My question is how can I calculate the number of scrambles, or the number of legal unique positions (without caring for rotations or reflections) for the 3^2? Because if I remember correctly, you just take the number of pieces that can move around (the 4 edges) and ! factorial that, then divide the whole equation by 2 to filter out parity/impossible positions. So the whole formula would be 4!/2 which equals 24/2 which equals 12, but on the 2d cube webpage, is says there are only 8 scrambles for the 3^2. And also, how would I calculate that for the 2^2 or 4^2?

Thanks for your help :) glad to be part of the hyper cubing group!

[John Bailey]

Thanks for reminding me. Attached find my own adaptation of the cube, reducing its dimensionality by one.

John Bailey

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[Melinda Green]

Dear Rowan,

I love your 3x3x3x3 solution video! I only realized just now that there's a real need for MC4D tutorials aimed at speedsolvers. There's no need to bore them with descriptions of the Rubik's cube, and they'll want instruction in the terminology they already understand. On top of all that, your video is super funny, encouraging, relatable, and entertaining. Are you on reddit? If so, I encourage you to post it to the r/cubers subreddit where it may encourage cubers to follow your example. If not, I'll be happy to post it there for you. It's just better for you to do it so you can respond to any questions about it.

Regarding the 2D cube, that's mine. I just now added my name. It's a real shame that modern browsers no longer support Java Applets, but the state diagram on that page shows pretty much the whole story. I support your borrowing from it for your iOS app. I don't know how you're going to make it interesting, as I expect that all such puzzles will be relatively trivial, but I hope that's not true. Can anyone here think of puzzles of this type which could be challenging?

Regarding the size of the state space for this puzzle, first I wouldn't call them "scrambles" since that sounds like the number of random twists needed to fully scramble a puzzle. Instead just think of them as states. There are multiple ways of defining the possible states for a given puzzle. The most common method doesn't account for the "color symmetry" that an actual user imagines. For example, if a Rubik's cube is one twist away from being solved, you probably wouldn't care which face that is. A cuber probably wouldn't say "There are 6 states that are 1 clockwise twist from solved", but that's how mathematicians like to think of it. The state space diagram on the MC2D page compresses all such color 'patterns' into one, resulting in 8 unique states, rather than the 24 when counted like a mathematician. I think the cuber's perspective is the better one, but that's just my opinion. See n-dimensional sequential move puzzle for a more complete description.

Best,
-Melinda

[Tomas Rokicki]

There are plenty of two dimension "twisty puzzles" that are quite interesting and challenging.

I'm not sure they are strongly related to the Rubik's Cube, but they have in general made a

resurgence of late (although they tend to be more expensive than normal twisty puzzles).

Here's one of a set of popular ones:

   https://www.thecubicle.com/collections/geranium-puzzles/products/verypuzzle-butterfly-geranium

Here are a few more, older ones:

   https://www.jaapsch.net/puzzles/turnstil.htm

For many of these puzzles (but not all) the standard approach to calculating the size of the

state space is to represent the puzzle as a group and use the Schreier-Sims algorithm (which

is implemented in GAP and other tools).  It's moderately non-trivial.  For puzzles that are

sufficiently simple, a computer program can literally enumerate the state space, but that's

rather brute force.  You can also calculate it by hand if you can verify that certain operations

exist (two-swaps and three-swaps) and others can't, but this can be prone to errors.

-tom

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[Rty Euzch]

Melinda, I think I have an idea. A  2D megaminx could be created; maybe even a 2D gigaminx. Puzzles that are face-turning can use reflections across faces, while vertex-turning puzzles can reflect across the vertex. 

Breaking from the subject, do you have any idea how Andrey Astrelin solved the 3^4? I have read some archives, and he said that he had reduced it to a 3^3, but I don’t know how he went about doing so. It’s a shame he can’t just tell me, since he had sadly passed away.

Vào lúc 21:15:13 UTC-5 ngày Thứ Ba, 13 tháng 4, 2021, rokicki đã viết:

[Rty Euzch]

I apologize for the double-post, but if you are looking for a speedsolver-oriented tutorial for solving MC4D, I have recently discovered a tutorial by Charles Doan!
He posted his tutorials on a youtube channel called 4D Puzzles, and many other MC4D-related things.

Vào lúc 23:22:24 UTC-5 ngày Thứ Ba, 13 tháng 4, 2021, Rty Euzch đã viết:

[Melinda Green]

Dear Tom,

I should have been more clear. I don't consider the puzzles you cite to be 2D puzzles in the same sense as MC2D. That's because the stickers of your puzzles have 2D areas whereas MC2D has D-1 = 1D stickers. So the Rubik's cube has 3-1 = 2D stickers, and MC4D has 4-1 = 3D stickers. That makes your flat puzzles to have the same order as the Rubik's cube, at least when examined locally. Does that make them 3D puzzles? No, but they are the same as the Rubik's cube in that the elements of each puzzle live on 2D manifolds, the same as how many of the famous physical twisty puzzles are realized in MagicTile. So when I talk about other possible puzzles in the same vane as MC2D, I'm talking about puzzles with 1D stickers. So when looking for candidate puzzles, we should be looking at various kinds of polygons, not polyhedrons, and I have the feeling that there are just not enough interesting things that can be done in that regard. Still, that may be a good thing in the same way that MC2D gives us some insights into state spaces of larger and more complex twisty puzzles.

Best,
-Melinda

[Roice Nelson]

Hi Rowan,

I much enjoyed your video too. Thanks Melinda for highlighting it!

On the topic of the number of permutations, I thought I'd make a few comments toward the original questions that I hope are helpful.

• I prefer myself not to collapse states by making an equivalence of states "mod colors", for the simple reason that the classic "43 quintillion" calculation for Rubik's cube doesn't do this.
• Note (as Melinda mentioned) that the number of states taking the classic approach is 24 (not 12). This is 4! and the size of the Symmetric Group S_4. Why not divide by 2 like in the 3^3? The difference here is because twists are reflections rather than rotations. If we didn't make that extension, the size of the state space would be 1. Why isn't there a contribution to the count from orientations? You can convince yourself that corner orientations can't be changed by twisting, at least on a square.
• What about the 2^2? It is also governed by S_4, except in this case there are no immovable centers so some states collapse from an overall rotation of the puzzle (not because of colors!). We'll need to divide by the order of rotations of a square to get 6 total states. Should we divide by reflections too, giving only 3 states? Maybe. Probably. Debate!
• It might be easier to think of the 5^2 next, since that has immovable centers again. This has two copies of S_4 now, and I believe the size of the state space will be 24*24.
• The 4^2 will be like the 5^2, but again reduced by a factor of the symmetries of a square.
• There was quite a discussion about the size of the MC2D state space a decade ago. Unfortunately, our old yahoo group doesn't have a good archive, but it can be read in the backup. If you want to go down that rabbit hole, search for "State graph of MC2D" in the 10714925.mbox.00002.mbox file.  
  

Someone please correct me if they see mistakes in any of this reasoning. That's likely :)

Best of luck with the iOS app.

Cheers,

Roice

P.S. Because of this thread, I spent a few hours last night trying to see if I could import old messages into our group (you can't, we don't have access to the needed API with this public group). I also search for a way to convert our MBOX files in the backup into nice archived web pages. I haven't found a good solution yet, and am not motivated enough to write code to do this. If anyone is interested to help with this, let me know. It would be nice to be able to search and point to old messages, since many topics come up again.

To view this discussion on the web visit https://groups.google.com/d/msgid/hypercubing/eead56e5-81f3-bf1b-637e-e1ede257f262%40superliminal.com.