Bandaged cubes theoretical underpinning - groupoids - visualized

[bandagedgroup]

Hi all,

it's well known that normal cube is mathematically modelled as a group. Bandaged cubes can't be represented this way, because you can't always compose moves (bandages block you). The move composition is therefore only a partial binary operation and we can observe this leads to the structure of a groupoid.

I don't want to lose readers, so I won't expand on the higher math terminology now. What I want to share is that these objects have a beautiful internal structure, that can be visualized. I wrote some code to help me analyse these cubes. Here's one output picture:

The bandage analysed is displayed in white, the almost invisible dots are drained-of-color shapes of the bandaged cube, and the colorful lines are face turns by the corresponding color face. The puzzle is a product of this colorful graph labyrinth and a small loops-generated group of block permutations.

To make this much more approachable, I made a detailed, slow paced explanatory video:

Note that I use the DIY bandaged 3x3 kit to test my theoretical analyses.

Let me know if you like the animations, want me to provide more theoretical explanations, or if I'm missing existing sources on bandaged cubes theory!

 

[xyzzy]

Hey, this is pretty cool!

Something that this video leaves me wondering about is: how can you be sure that every circuit starting at the central state can be decomposed into the "small" cycles and the "big" cycles? This seems like a non-obvious statement, and I think I can come up with a not-too-complicated proof for this specific bandaged cube configuration, but what about a differently-bandaged cube where the graph of the states looks messier?

(edit: Oh, it's something about fundamental groups and their generators, maybe?)

(I was going to say "you could post this on TwistyPuzzles to get more eyeballs on it" but it seems you posted this there first, haha. I should probably reset my TP account or something; I forgot my password there.)

 

[bandagedgroup]

Thanks! You're right, that's a very handwavy part.

Any circuit starting and ending at the central shape gives you a permutation of the movable blocks. I computationally explored the full puzzle, including colors, so I know there are 60. The three short loops generate all 60 of them, which translates into the statement about loops.

That's not very theoretical, of course. In general, I think you should first transform the graph a bit - get rid of the single color 4-cycles (4 consecutive single face turns). For example, create a new graph having a vertex for each such cycle in the original graph, and an edge for whenever the two cycles in the original graph had a common vertex. I have yet to see if this doesn't lose something. Then you can remove tree-like outgrowths (repeatedly cutting of degree-1 vertices), and find a cycle basis. Combining cycle basis cycles gives you all possible cycles, so combining all the permutations induced by them gives you all possible permutations.

Please keep in mind I just produced the above paragraph, I'll need to think it through properly. I haven't implemented something like that yet. So far I prove I have found all scrambles and solutions by either:

• full search colors included (good for heavily bandaged cubes with few total scrambles)
• if full search is infeasible (like normal Rubik's cube - just one bandage, but too many color scrambles), it's typically pretty easy to find enough loops to do everything, that's possible on a normal cube - like twist any two corners, etc. That shows you found everything similarly to normal cube - you showcase moves that generate lot of stuff, and prove remaining permutations impossible by parity arguments. If there are some things you couldn't do on your bandage by a not-completely-exhaustive loop collecting, but are possible on a normal cube (e.g. two edge flip), you have to resort to finding a parity argument specific to your bandage.

Yeah, I already had several posts about bandaged cubes on twistypuzzles over the years.

 

[abunickabhi]

Interesting concept, U' L S L' S2 L' S L U .

 

[qwr]

I took one class on abstract algebra and didn't understand anything past the isomorphism theorems. Sorry.
I am more interested in combinatorics but that's not my field of study either.

 

[bandagedgroup]

Well there are certainly interesting combinatorial questions to be asked. For example: how many distinct ways can you sticker a cube with only black and white stickers? We consider two stickerings distinct, if they can't be made to coincide using a sequence of face turns.

That's a kind of combinatorics that turns to algebra fast though. It would suffice to find the cycle index of Rubik's group and then use Pólya's enumeration theorem. The group is represented in terms of direct and semidirect products of groups for which cycle indices are known. I searched and there appear to be articles deriving cycle index of both direct and semidirect products from cycle indices of the factors, so the problem may be doable.

Even harder is to enumerate all possible bandagings of a 3x3 up to face turning. I don't know any theoretical approach here. Andreas Nortmann over at twistypuzzles has done it computationally.

What's your field of study though?

 

[qwr]

statistics lol.
but I have a passing interest in number theory and combinatorics. I did a bunch of project euler problems a while ago.