benbotto

Night.

Okay. Thanks again! Let me know if you figure it out, and I'll do the same.

Also, because the total change in orientation is always 6 for disorienting twists, which is divisible by 3, the corner orientation is always divisible by 3.

Yep yep.

Yes, I agree.

I'm over here hacking away on my calculator. It's just too convenient that 3^8-48 = 6513.

All right. Thanks again for the help.

Correct.

You mean that 4 of the sides change the orientations of corners with 90-degree twists, and 2 of the sides don't change the orientations?

I have various implementations of the RC, but I think this definition is most concise: github.com/benbotto/rubiks-cube-cracker/blob/master/Model/…

That's from Ryan Heise: ryanheise.com/cube/cube_laws.html

(And that definition is arbitrary.)

Any corner that has blue or green pointed up or down is disoriented by 2.

Any corner that has white or yellow pointed up or down is disoriented by 1.

If white is up top and red is front: any corner that has red pointed up or down is oriented.

So the total change in orientation is always 6 for 90-degree twists of faces that disorient corners.

From any state. Let's say that corners are oriented (0), rotated once (1), or rotated twice (2). From the solved state, moving the front face, for example, changes the orientations of the 4 corners cubies on that face from 0, 0, 0, 0, to 1, 1, 2, 2.

Does that make sense?

So 1 + 1 + 2 + 2 = 6, the change in orientation.

For the moves that disorient, two of the cubes are disoriented by 1, and two by 2.

Agreed.

I.e. none disoritented, or two disoriented by 1, and two disoriented by 2.

Yeah, and the number disoriented is always 0 or 6, right?

Right.

My program counts double moves as 1 turn, but no biggy.

Okay.

That's just the orientation I picked in my program.

Mind if we use the convention white up top, red up front?

Hah! Same here, if you can't tell from the repo I posted.

Hey SpaceDisgrace. Thank you for your help.

Also, the 6,513 corner position orientations I mentioned are reachable in 6 twists. For all orientations of the corner pieces, the 2187 states are reachable in 7 twists. All corner arrangements and permutations (8! * 3^7) can be reached in 11 or fewer moves.

Ah, yes, sorry I misunderstood the God's number bit.

In the search I'm using I'm not doing any pruning other than obvious trivial pruning of redundant and commutative moves, like {L L'}, {L L2}, {L R} (which is the same a {R, L}), etc. So, the branching factor is roughly 13, and it takes a long time to iterate to depth 9. In roughly 5 hours I will have iterated through depth 10. Put another way, my program just hasn't finished depth 10 yet.

I have the code available online, and it has been heavily peer reviewed (github.com/benbotto/rubiks-cube-cracker), but extracting the code that I used to come up with the 6513 number would be difficult due to the amount of code involved (the search code, the pattern database, the rubik's cube models, etc.). In that repository, I've implemented Korf's optimal solver (God's algorithm), and Thistlethwaite's algorithm, and both work well. I'm working on an improvement to the latter, though, and I'm struggling with this part of the math.

I added a few code snippets above, for what it's worth. As far as searching, I basically just iterate through all possible combinations of moves and count the number of unique permutations (github.com/benbotto/rubiks-cube-cracker/blob/master/Control‌​ler/…).

God's number is 20. I use Korf's algorithm for that.

I came up with 6513 using a program that I wrote. I used an iterative-deepening depth-first search up to depth 9 (all combinations of 9 twists).

I disagree with that statement in the comments, and further think it's worded in a rather confusing manner.

I have a bit of a debate going on in the comments section of an answer, and I'm wondering if someone with a bit more of a mathematics background can chime in. Here's the answer in question: security.stackexchange.com/a/161509/213456 The debate is surrounding a sentence in the "Interesting note" section at the bottom. It says, "a 1000 possibility combination lock, if guessed randomly, actually has a 1 in 630 chance of getting it correct."