question about new Rubik's cube proof (all solutions require <= 20 moves)

[Arnold_Winkelried]

Some people have proved that “Every position of Rubik’s Cube™ can be solved in twenty moves or less.”.
http://www.cube20.org/

My question is: how do they count 20 moves?

They have two examples at the site:

1. Example 1. The site says:
   “Superflip” The first position proven to require 20 moves
   You get to superflip from a solved position using these moves
   R-U2B- FU-BDFUD-LD2F-RB-DF-U-B-UD

I say:
R U2 B F U B D F U D L D2 F R B D F U B U D = 1 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 23 moves; not 20

2. Example 2. The site says:
   The hardest position for our programs:
   you get to the “hardest position” from a solved position using these moves
   FU-F2D-BUR-F-LD-R-U-LUB-D2R-FU2D2

I say:
F U F2 D B U R F L D R U L U B D2 R F U2 D2 = 1 + 1 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 + 1 + 1 + 2 + 2 = 24 moves; not 20

Obviously I am making a mistake in my counting; but where?

[Chronos]

They’re counting a 180 degree turn of a face as a single move, not two.

[Arnold_Winkelried]

I thought of that, but then, their first example (which I copied wrong)
R-U2BL-FU-BDFUD-LD2F-RB-DF-U-B-UD
would be a 22 move solution; not 20

[Chronos]

Huh, I just tried that 22-move sequence, and it does indeed produce a superflip. My best guess is that there’s some shorter sequence that also produces a superflip, but that they just didn’t list that one for some reason.

[Enright3]

So can someone explain the math to me on how they say there are 43,252,003,274,489,856,000 positions of the Cube?

I realize it’s flawed, but my first inclination to determine the number of positions would be 8! * 12! = 479,001,600.
I reasoned 8! was number of possible positions for the 8 corner pieces; and 12! for the number of possible positions for the edge pieces.

[Arnold_Winkelried]

@Chronos

Huh, I just tried that 22-move sequence, and it does indeed produce a superflip. My best guess is that there’s some shorter sequence that also produces a superflip, but that they just didn’t list that one for some reason.

Either that, or their computer doesn’t know how to count, which would make their proof bogus!

I suppose it’s possible that they are just showing one way to create a superflip. But since their whole website is about “20 moves or less”, I say it would behoove them to only show 20 move solutions.

I’m sure is the definition everyone else must use when setting upper limits on Rubik’s cube solutions, but counting a 180° turn (two quarter turns) as one move seems like cheating to me.

[Arnold_Winkelried]

@Enright3

So can someone explain the math to me on how they say there are 43,252,003,274,489,856,000 positions of the Cube?

Wikipedia says:

The original (3×3×3) Rubik’s Cube has eight corners and twelve edges. There are 8! (40,320) ways to arrange the corner cubes. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 3[sup]7[/sup] (2,187) possibilities. There are 12!/2 (239,500,800) ways to arrange the edges, since an odd permutation of the corners implies an odd permutation of the edges as well. Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 2[sup]11[/sup] (2,048) possibilities.

8! × 3[sup]7[/sup] × 12! × 2[sup]10[/sup] ≈ 4.33 × 10[sup]19[/sup]

There are exactly 43,252,003,274,489,856,000 permutations, which is approximately forty-three quintillion. The puzzle is often advertised as having only “billions” of positions, as the larger numbers could be regarded as incomprehensible to many. To put this into perspective, if every permutation of a 57-millimeter Rubik’s Cube were lined up end to end, it would stretch out approximately 261 light years.

The preceding figure is limited to permutations that can be reached solely by turning the sides of the cube. If one considers permutations reached through disassembly of the cube, the number becomes twelve times as large:

8! × 3[sup]8[/sup] × 12! × 2[sup]12[/sup] ≈ 5.19 × 10[sup]20[/sup]

The full number is 519,024,039,293,878,272,000 or 519 quintillion possible arrangements of the pieces that make up the Cube, but only one in twelve of these are actually solvable. This is because there is no sequence of moves that will swap a single pair of pieces or rotate a single corner or edge cube. Thus there are twelve possible sets of reachable configurations, sometimes called “universes” or “orbits”, into which the Cube can be placed by dismantling and reassembling it.

[ChrisBooth12]

This is not new information it has been known for some time. Whenever I buy a Rubik’s cube for a present I always pop off a corner and replace it with another piece and mix it up. My gift for many children have been unsolvable…I wonder if this makes them feel worthless or stupid.

[suranyi]

@Enright3

So can someone explain the math to me on how they say there are 43,252,003,274,489,856,000 positions of the Cube?

I realize it’s flawed, but my first inclination to determine the number of positions would be 8! * 12! = 479,001,600.
I reasoned 8! was number of possible positions for the 8 corner pieces; and 12! for the number of possible positions for the edge pieces.

**Arnold Winkelried **explained it. The key is that you counted the different postions of the pieces, but not their different orientations.

[Chronos]

With a bit of Googling, I did find a 20-move Superflip:

F B U^2 R F^2 R^2 B^2 U’ D F U^2 R’ L’ U B^2 D R^2 U B^2 U

Translated into the same notation the OP’s link used, I think that would be

FBU2RF2R2B2U-DFU2R-L-UB2DR2UB2U

I haven’t yet tested that one, though. And I agree that it was sloppy of the folks in the OP’s link to give a 22-move sequence for it, when the point of the page is that everything can be done in 20.

[BigT]

@ChrisBooth12

This is not new information it has been known for some time. Whenever I buy a Rubik’s cube for a present I always pop off a corner and replace it with another piece and mix it up. My gift for many children have been unsolvable…I wonder if this makes them feel worthless or stupid.

I know I’d be rather pissed at you ruining a gift. Giving somebody a puzzle that cannot be solved is rather sadistic, don’t you think?

[Steve_MB]

@ChrisBooth12

This is not new information it has been known for some time. Whenever I buy a Rubik’s cube for a present I always pop off a corner and replace it with another piece and mix it up. My gift for many children have been unsolvable…I wonder if this makes them feel worthless or stupid.

Another piece? You can get the same result by popping off one piece and putting it back in a different position:

The full number is 519,024,039,293,878,272,000 or 519 quintillion possible arrangements of the pieces that make up the Cube, but only one in twelve of these are actually solvable. This is because there is no sequence of moves that will swap a single pair of pieces or rotate a single corner or edge cube. Thus there are twelve possible sets of reachable configurations, sometimes called “universes” or “orbits”, into which the Cube can be placed by dismantling and reassembling it.

[Arnold_Winkelried]

@Chronos

With a bit of Googling, I did find a 20-move Superflip:

F B U^2 R F^2 R^2 B^2 U’ D F U^2 R’ L’ U B^2 D R^2 U B^2 U

Translated into the same notation the OP’s link used, I think that would be

FBU2RF2R2B2U-DFU2R-L-UB2DR2UB2U

I haven’t yet tested that one, though.

Yes, that is a 20-move solution to create a Superflip.

Maybe they didn’t mention that one in the webpage in my OP because it’s under copyright! Can you copyright a Rubik’s cube solution? I think I’ll e-mail the people listed on the page and ask them why they have a 22-move algorithm for the superflip.

[Enright3]

@suranyi

@Enright3

So can someone explain the math to me on how they say there are 43,252,003,274,489,856,000 positions of the Cube?

I realize it’s flawed, but my first inclination to determine the number of positions would be 8! * 12! = 479,001,600.
I reasoned 8! was number of possible positions for the 8 corner pieces; and 12! for the number of possible positions for the edge pieces.

**Arnold Winkelried **explained it. The key is that you counted the different postions of the pieces, but not their different orientations.

ding The light bulb just came on.
I’m actually a little surprised that all orientations are possible in all positions.
I was under the impression that if you took one apart it would be possible to re-assemble where it couldn’t be solved. I guess I was wrong on that one.

ETA: I can’t believe I didn’t think to go to Wikipedia first. Thanks everyone.

[Enright3]

Oops, I see know my original thought was correct about the unsolvable cube.

[Cheshire_Human]

@Steve_MB

@ChrisBooth12

This is not new information it has been known for some time. Whenever I buy a Rubik’s cube for a present I always pop off a corner and replace it with another piece and mix it up. My gift for many children have been unsolvable…I wonder if this makes them feel worthless or stupid.

Another piece? You can get the same result by popping off one piece and putting it back in a different position:

The easiest way is just peel off 2 stickers of different color from any 2 positions, and resticking them in the other’s position. No need to risk breaking a piece snapping it out and in again.

[Chronos]

Maybe they didn’t mention that one in the webpage in my OP because it’s under copyright! Can you copyright a Rubik’s cube solution?

You can copyright the presentation of data, but not the raw data itself. If I copied their Rubik’s Cube applet from their page and used it on my own, I’d probably be guilty of copyright infringement, but not just for retyping the move sequence in some simple notation.

[pulykamell]

@BigT

I know I’d be rather pissed at you ruining a gift. Giving somebody a puzzle that cannot be solved is rather sadistic, don’t you think?

shrug I would never have noticed myself, as I’ve never solved one of those damned things. And to know all possible permutations can be solved in 20 or fewer moves makes me feel even more of an idiot. ETA: Unless I’ve only managed to get my hands on ChrisBooth’s modified cubes…

[Chronos]

And to know all possible permutations can be solved in 20 or fewer moves makes me feel even more of an idiot.

Nobody but God and Google can actually solve it in 20 moves. Methods used by mortals tend to require many more moves than that. I just applied the method I use to the superflipped cube I still had lying around from post #4, and it took me 108 moves (though admittedly the method I use isn’t all that efficient, even by human standards).

[pulykamell]

@Chronos

And to know all possible permutations can be solved in 20 or fewer moves makes me feel even more of an idiot.

Nobody but God and Google can actually solve it in 20 moves. Methods used by mortals tend to require many more moves than that. I just applied the method I use to the superflipped cube I still had lying around from post #4, and it took me 108 moves (though admittedly the method I use isn’t all that efficient, even by human standards).

Yeah, I never was able to figure out any of the methods for the Rubik’s cube. On the other hand, I could always solve a Rubik’s Magic Puzzle (“Link the Rings” variety).