
He Invented the Rubik’s Cube. He’s Still Learning From It - pseudolus
https://www.nytimes.com/2020/09/16/books/erno-rubik-rubiks-cube-inventor-cubed.html
======
Aissen
> 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](http://www.alchemistmatt.com/cube/rubikcenter.html)
for algorithms solving these.

~~~
0_____0
??? 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?

~~~
function_seven
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.

~~~
thaumasiotes
> 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.

~~~
function_seven
> _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](http://b.chrishunt.co/how-many-positions-on-a-rubiks-cube)

~~~
thaumasiotes
> 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".

------
throw149102
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.

------
mathgenius
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](https://www.gap-
system.org/Doc/Examples/rubik.html)

------
Tomminn
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.

~~~
aivisol
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...

~~~
cableshaft
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.

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

~~~
ansible
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.

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

~~~
ansible
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](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.

