He Invented the Rubik’s Cube. He’s Still Learning From It | Hacker News Reader

He Invented the Rubik’s Cube. He’s Still Learning From It

nytimes.com

90 pointspseudolus47 comments

> Mathematicians later calculated that there are 43,252,003,274,489,856,000 ways to arrange the squares, but just one of those combinations is correct.

This is only true for cubes that have pictures printed on their faces. Color-only cubes have permutations that can be identical, and therefore correct. See http://www.alchemistmatt.com/cube/rubikcenter.html for algorithms solving these.

??? Are we talking about a physical cube? I don't think you can have multiple identical permutations; each of the 26 external pieces of a 3x3x3 cube is unique. You would have to somehow have more than one way to get, say, a Yellow-Blue-Red corner, or a White-Green edge for this to be true, no?

I believe it’s possible for the center faces to be rotated in the solution?

Now it’s up to you if you consider that a separate permutation or not. (I don’t)

But yeah, every other piece is two or three colors, and therefore has only one allowed orientation in a solved cube.

Yeah, I was talking about centers orientations, sorry for not being clear enough, I thought the link with pictures would have been explicit.

Yes, there are "mirror cubes" which have brushed-metal-like surface. They are not considered solved unless the brushing is aligned.

These cubes do not have any more state than a color cube with orientation markings on the center pieces.

If the brushed lines are uniform, they might have half the state, right? A center cube could be at 0˚ or at 180˚ and the pattern could still line up.

I've never seen one, so I don't know if that's the case or not. I also don't know if there are natural constraints in the topology of the cube that make it possible or not for center faces to be 180˚ out of alignment without others being off by a quarter turn. I do assume that any center face that's flipped would have to be paired with another one in the same orientation. But that's just my intuition.

A brushing pattern, even if perfectly straight, consists of scratches with random thicknesses. It does not have 180 degree symmetry; if the piece is reversed, the scratches do not match, and that is pretty obvious if held up to the light.

I believe all the center face re-orientations are paired, 180 degree rotations: there are no quarter turns.

> Now it’s up to you if you consider that a separate permutation or not. (I don’t)

This isn't really up to you; it's determined by the method you use to count how many permutations there are.

In this case, I strongly suspect that we count 43,252,003,274,489,856,000 and then may or may not say "but 4,096 of those are equivalent for any given coloration pattern, so divide by 4,096". I would bet we don't count 10,559,571,111,936,000. Unless you have a method that yields that result naturally, you should consider the rotated centers to be separate permutations.

> In this case, I strongly suspect that we count 43,252,003,274,489,856,000 and then may or may not say "but 4,096 of those are equivalent for any given coloration pattern, so divide by 4,096".

I don't think that's true. I went looking for the formula, and found this[1].

They include a term for the permutations of the corner cubes (8!∙3⁸), a term for the edge cubes (12!∙2¹²) and a term for the center cubes (1!∙1¹), then divide that by 12 (2∙3∙2) to eliminate impossible configurations. That term for the center cube permutations seems to agree with my view on it. There's no such thing as "rotation" for a plain colored center cube. It's the same no matter how you look at it.

I think a cube with designs on the center faces will have more permutations than the 43e18 we're discussing here.

[1] http://b.chrishunt.co/how-many-positions-on-a-rubiks-cube

> They include a term for the permutations of the corner cubes (8!∙3⁸), a term for the edge cubes (12!∙2¹²) and a term for the center cubes (1!∙1¹), then divide that by 12 (2∙3∙2) to eliminate impossible configurations.

But that matches exactly what I said was being done, modulo the particular number and an error in the formula you provide. (It should be 1·1⁶, not 1!·1¹.) You have 4 orientations for each center cube, and then you divide 4⁶ by 4,096 to get 1.

I agree that the way your link presents it, with a glaring error in the term for the centers, suggests that they didn't put any thought towards the centers beyond "they don't and can't change in any way".

The 43,252,003,274,489,856,000 number disregards the orientation of center tiles. You'd have to multiply it by 4^4 or something to do the faces. (I can't remember if there is one face or two that are determined by the others, but they're not independent.)

The number is derived here (with computer assistance), by considering the permutations of the non-center tiles:

https://www.gap-system.org/Doc/Examples/rubik.html

If we take a color-only cube, the only state information that is aliased away is the rotation of the center pieces of the faces. If we add orientation marks so that the solution has to have the center faces in a certain orientation, then we get what is sometimes called a "supercube":

https://en.wikipedia.org/wiki/Rubik%27s_Cube#Centre_faces

There are simple moves that will rotate pairs of centre faces.

Cubes from rubiks.com have a logo in the white centre, which gives it an orientation. Using that you can demonstrate moves which flip that centre 180 degrees while keeping the cube solved.

All other pieces have an orientation that is visible; they do not have multiple orientations that are indistinguishable by color.

My favorite class in college was a class on Motion Planning, and I remember using the Rubik's cube as an analogy for discrete motion planning problems. It was a fun way to talk about some of the less common graph search techniques, like iterative deepening. It's also interesting how the graph produced by the transitions of a Rubik's cube is so large, yet so uniform. There are 43 quintillion states, yet each state has the exact same number of transitions: 3 for each face, so 18 in total. And the fact that we actually can reliably search that space even using a relatively dumb graph search algorithm is astounding.

Here's how to analyze the Rubik's cube using GAP [1]. For example, they show that you can get any permutation of the eight vertices of the cube. It's a great way to get motivated to learn some group theory.

[1] https://www.gap-system.org/Doc/Examples/rubik.html

If you've never learnt the algorithms required to solve a Rubik's cube-- don't. Understanding and solving a cube by brute effort is a great little challenge to come back to over the years.

Because once you learn them, you will never unlearn. I took a cube in my hands after some 30+ years break and my hands just instantly knew the sequences...

I remember about 80% of them by muscle memory, but... I've forgotten some, so my party trick now is to get a cube 'nearly' solved in about 50 seconds, then give it to someone else, and say "I've done all the hard work already, you can finish it off".

Generally that's true for things learned as children, not as adults.

I managed to forget most of them after about 5 years of not doing it. I can intuitively get the first two layers now, but the top seems too difficult without knowing the sequences.

I like messing about with a Megaminx also, and it's the same for that. I can get the first couple of layers just thinking about it intuitively, but then I get stuck on the top layer.

At some point I'll relearn the sequences again, but I haven't yet.

Also picked up a 2x2 cube recently, and that one has me pretty stumped. You'd think it'd be easier to figure out but I haven't figured it out yet.

Apart from the fact that you get the happiness of having solved it without help. What else does it give?

Probably not much. A bit more intuition about non-commutative transforms, perhaps.

But you don't gain anything much from learning to solve them either. It's a fun party trick once or twice, but you'll likely feel uncomfortable not acknowledging that you read the answer, at which point it's comparatively unimpressive.

Is there any game or challenge that actually makes you smarter or better or helps you out in any practical way?

Of course. Games like go can help reduce the effects of Alzheimer[1]. I'm sure there are studies for chess too.

Sports are games with obvious health, mental and social benefits.

On a more intuitive level (aka I don't know if it's been researched), I would be surprised if most board games don't improve, at least a bit, the analytical thinking useful for many engineering fields (or even day to day life).

Challenges is a term too broad, you can imagine countless ones that have practical benefits. Learning-to-play-an-instrument-in-a-month challenge for example. 100 push-ups a day for a month. Read one book a month.

[1] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4548213/

Depends upon what one considers practical? I had some nice (here: both informative and entertaining) conversations with someone I'd consider a genius, not via the more evidently practical skill of juggling, but because we'd made each others' acquaintance due to a mutual interest in Rubik's Cube.

If you haven’t seen it, you may be interested in https://youtube.com/watch?v=q6AsllXpKBU, which combines solving Rubik’s cube with juggling (1 minute, 41 seconds per cube)

Speedcubing. It is very ignorant to claim that all you need is to know a bunch of algorithms. It is on par to claim that to win a basketball match you need to throw a ball into the basket. You need to think how to optimally solve the cross (or maybe to do extended cross), you need to be quick to spot the pieces for F2L and to choose optimal way to place them, you can choose different ways to solve the last layer to maybe gat an OLL or a PLL skip.

Millions of years of evolution and the presence of play in mammals suggests yes

> A bit more intuition about non-commutative transforms

Most (almost all?) state machines one encounters in software are non-commutative. Cosets and conjugation are cromulent CS concepts.

Edit: Grand Corps non Commutatif https://www.youtube.com/watch?v=SZXHoWwBcDc

Oh, for sure, I don't mean to suggest that non-commutativity isn't an important concept. But just as I think throwing a ball around is of limited value beyond a bit of intuition in studying aerodynamics, I don't think playing with a Rubik's cube is going to fundamentally advance your understanding of transforms.

If you can explain the various algorithms and how they work it may still seem impressive.

In my opinion, you don’t understand the Rubik’s cube if you don’t know how to solve larger ones. I’ve never held a 5×5×5 or larger in my hand, but I know I can solve it or any larger one easily.

Variants such as dodecagons can be a little trickier, as knowledge doesn’t transfer as easily, but if you understand the cube, you still should be able to solve them reasonably quickly.

> But you don't gain anything much from learning to solve them either.

It is very useful to be able to solve the cube yourself.

This allows you to start from a solved state easily, and then experiment with different patterns, move sequences, etc.

I was not one of those people who could completely track the state of the cube in my mind. But it was easy for me to see what I've done if I started from the solved state.

Telling people what kinds of intellectual effort are worthy of your respect is both obnoxious and interesting. Please stop.

Isn't that the whole point of puzzles?

That depends if you are doing the puzzle for the love of puzzles, or if you are doing the puzzle for your father's love of puzzles.

When my oldest was about six or seven, I showed her the Rubik's Cube. She was enthralled with it. But she had no drive to solve it, and showing her how (no algorithms) really didn't interest her. Now, at thirteen, she's picked it up again. And she absolutely loves solving it.

I am so glad that I've never pushed her to learn things, rather just having just _exposed_ her to interesting things without pushing her. Now she has the intrinsic drive to learn.

If you have learned them, it's not too late. Martin Gardner mentioned an algorithm that doesn't work layer by layer, but instead starts off with all manipulations available, and gradually restricts which ones are admissible. In principle, just at the point no moves are admissible, the cube is also solved.

In that oh so distant future when I have nothing else to do, I'd like to try to work out this approach for myself.

Sounds interesting, do you have any source on this? tried googling but couldn't find anything...

I messed up. Upon referring to http://martin-gardner.org/MGSAindex.html , I've learned it was a Hofstadter column: https://www.scientificamerican.com/article/metamagical-thema... (paywalled)

Does http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers... provide enough information? (At least it gives you some better search terms.)

> "indeed it is not visually obvious that any progress at all is being made until the cube is almost solved."

I do the edges first and then the corners. I changed it up for novelty and to force myself to come up with a different solution. So I've got permutations to manipulate edges that also mess with the corners, followed by ones that manipulate the corners.

I've wondered if there is a way to do it by restricting to smaller and smaller subgroups, but that'd probably be a lot of work to figure out.

I have a cube sitting on my desk; it helps me think. Over time I've learned more and more algorithms for it; I almost know all the PLLs. Honestly it's a lot of fun!

The competitions should be for minimal no of steps/moves. Not the time taken.

There are 'fewest moves' competition as well

It is almost the same thing. Less moves means less time to solve. I'm amazed by the performance of robots and people these days at this task.

It's not. There's a factor of over 100 between fastest solves and novice solves with a guide.

We're talking about competition.

Here is a previous world record solve from 2016, in about 31 moves over 4.73 seconds:

https://www.youtube.com/watch?v=SjOyaf2JKoE

You don't just pick that up from a book.

Optimal number of moves is 20, so ~30 is pretty good.

It’s already like that to be fair, people get faster by getting from A to B in fewer moves. The two go hand in hand, it’s like speedrunning.

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