How many solved positions for 3x3x3?

[cuBerBruce]

would you count the centers flipping (like in picto cubes) as a different position?

I guess that's what I mean. When the flip of the centres affects whether or not the cube is 'solved', the number of positions basically quadruples. However, in a solid-coloured or -patterned cube, where the center rotation makes little practical difference, those extra positions are only theoretical. So, the answer is that it depends on the cube.
Yerachmiel

Well, it quadruples if only one center's orientation matters. If all 6 matter, the number of position increases 2048 times, as I've said more than once I think.

 

[Jun Kim]

you have to add 2 seconds

 

[yboy403]

would you count the centers flipping (like in picto cubes) as a different position?

I guess that's what I mean. When the flip of the centres affects whether or not the cube is 'solved', the number of positions basically quadruples. However, in a solid-coloured or -patterned cube, where the center rotation makes little practical difference, those extra positions are only theoretical. So, the answer is that it depends on the cube.
Yerachmiel

Well, it quadruples if only one center's orientation matters. If all 6 matter, the number of position increases 2048 times, as I've said more than once I think.

Very true. I was thinking of my Rubik's.com cube, which only has one center design, but on a pictocube you're right, all the centres would matter,
Yerachmiel.

 

[yboy403]

you have to add 2 seconds

Why? The timer? If that's what you're thinking of, read the rest of the thread first.
Yerachmiel

 

[shelley]

There are 43 quintillion possible positions. That doesn't mean there are 43 quintillion positions in which the cube is completely scrambled. A cube that is one or two moves away from solved is NOT SOLVED. It counts in one of the 43 quintillion positions that is not solved. Why is there this much discussion?

If you wanted to know how many possible positions are one or two moves away from solved, you should ask that, don't try to redefine the solved state.

And to the person who said there are infinite positions... you're an idiot.

 

[yboy403]

There are 43 quintillion possible positions. That doesn't mean there are 43 quintillion positions in which the cube is completely scrambled. A cube that is one or two moves away from solved is NOT SOLVED. It counts in one of the 43 quintillion positions that is not solved. Why is there this much discussion?

If you wanted to know how many possible positions are one or two moves away from solved, you should ask that, don't try to redefine the solved state.

And to the person who said there are infinite positions... you're an idiot.

Wow, I guess you're just smarter than all of us (insert heavy sarcasm here).
If you'd actually read the thread, you'd realize that although the thread title is misleading, because I wrote it in a hurry, there are lines like:
Is there any way to include those in the tally of "solved" positions? (Note the quotes around "solved")

Also (the killer):

Interesting. anyway, this has nothing to do with speedcubing or the WCA, just cubing in general. What I'm saying is that when Rubik's prints on their packages, or Dan Harris writes on the back of his book (good book BTW), that the ratio of total positions to solved positions is 43,252,003,274,489,856,000:1, they're leaving out the fact that there are an unspecified number (ben1996123 says 2048, Chris says around 300) of positions that a person with no knowledge of Rubik's Cube could solve easily, and which most people (WCA aside) would not consider a scrambled state. And even the WCA acknowledges that one turn from solved is something, because they only give a penalty for it, not a DNF. All I want to know is, including up to three turns from solved, which is the maximum my 5-year-old sister has solved with no help, how many unstated "solved" positions are there besides the standard one with all sides the same colour. If you do come up with a number, I'd like to see how you got it, purely for educational purposes (my own)
Yerachmiel

It's clearly stated there that I'm looking for

[P]ositions that a person with no knowledge of Rubik's Cube could solve easily, and which most people (WCA aside) would not consider a scrambled state.

That's basically the definition of what this thread is about.
For everybody reading this: This thread is not talking about solved positions. Read the rest of this post before you reply. If you'd bothered to read it, you'd know that. This isn't about speed, it's about difficulty, in terms of which one or two turns from solved is barely more difficult than being solved, besided the physical effort of turning. That's also why this thread is in the theory section and not the speedcubing section.
anyway, back to the original topic. While I agree that there are clearly not an infinite number of positions, posting on a thread you haven't read completely is a surer sign of idiocy than a statement that happens to contain a mistake (which we all make). The reason there's so much discussion is because it's an interesting (at least to me) discussion about the difficulty and mathematics of Rubik's Cube.
Yerachmiel

Spoiler

Shelley: As with all my posts, if you find any of this offensive to you, tell me and I will edit it out

Sorry this post is so long, it's just that there's been a lot of confusion over it and I just wanted to clear it up.

 

[shelley]

I did read the thread. Your question was answered by Chris on the first page. The rest of it seemed like an unnecessarily long argument over your usage of the term "solved state" when that could have been easily avoided by saying what you meant in the first place.

 

[yboy403]

I did read the thread. Your question was answered by Chris on the first page. The rest of it seemed like an unnecessarily long argument over your usage of the term "solved state" when that could have been easily avoided by saying what you meant in the first place.

That's true. But then I asked a new question, "What about 3 turns?" That one seems to have been lost in the confusion over the my bad wording.
Yerachmiel

 

[cmhardw]

That's true. But then I asked a new question, "What about 3 turns?" That one seems to have been lost in the confusion over the my bad wording.
Yerachmiel

Unless I misinterpreted your question, Bruce already answered that.

If you allow a third turn from the solved position, you can reach 3240 additional positions. That added to the 262 positions of reachable in two moves or less makes a total of 3502 positions. A computer algorithm called a breadth-first search can be used to calculate these numbers. Such numbers have been computed out to 13 moves (face turns). See: God's Algorithm out to 13f*.

Chris worked this out by hand for two moves, and it is not too difficult to extend that type of argument to three moves to come up with the answer 3502. It perhaps just gets a little trickier to figure out all the duplicates. Going out further than 3 moves you can start reaching duplicate positions in "unexpected" ways, making it even trickier to avoid overcounting.

Chris

 

[yboy403]

Must have missed that one.
I think this thread is getting too confusing. I have my answer already. Anybody else want to kill it?
Yerachmiel

 

[cuBerBruce]

That's true. But then I asked a new question, "What about 3 turns?" That one seems to have been lost in the confusion over the my bad wording.
Yerachmiel

Unless I misinterpreted your question, Bruce already answered that.

If you allow a third turn from the solved position, you can reach 3240 additional positions. That added to the 262 positions of reachable in two moves or less makes a total of 3502 positions. A computer algorithm called a breadth-first search can be used to calculate these numbers. Such numbers have been computed out to 13 moves (face turns). See: God's Algorithm out to 13f*.

Chris worked this out by hand for two moves, and it is not too difficult to extend that type of argument to three moves to come up with the answer 3502. It perhaps just gets a little trickier to figure out all the duplicates. Going out further than 3 moves you can start reaching duplicate positions in "unexpected" ways, making it even trickier to avoid overcounting.

Chris

I gave the number. I didn't go through how you derive it manually.

Basically you go through a procedure like Chris did for 2 moves. To avoid overcounting I will simply avoid having a U-layer turn following a D-layer turn, avoid an L-layer turn following an R-layer turn, and avoid an F-layer turn following a B-layer turn. The table below then gives the possibilities when the first move is a U- or D-layer turn. Each letter represents three possible turns of the indicated layer.

1st  2nd  3rd     count (for 3rd move)

U     D   LRFB    4*3
U     L   UDRFB   5*3
U     R   UDFB    4*3
U     F   UDLRB   5*3
U     B   UDLR    4*3
D     L   UDRFB   5*3
D     R   UDFB    4*3
D     F   UDLRB   5*3
D     B   UDLR    4*3

The first and 2 moves have 3 choices for each case (3 ways to rotate the layer) so the total count is 3*3*(4*3+5*3+4*3+5*3+4*3+5*3+4*3+5*3+4*3) = 1080.

We can do the same thing for the first turn being on L/R, and for the first turn being on F/B. This triples the number to 1080*3=3240.

(Be careful applying this procedure to more than 3 moves, because you will start reaching duplicate positions in non-obvious ways.)

 

[yboy403]

Thanks. I wasn't planning to do more than 3 moves anyway, because most non-cubers would probably make at least one wrong move if they didn't inspect it first for a long time. Now this thread's purpose has been served. You guys can continue it if you want, but I'm gone.
Thanks everybody,
Yerachmiel

 

[FatBoyXPC]

There are 43 quintillion possible positions. That doesn't mean there are 43 quintillion positions in which the cube is completely scrambled. A cube that is one or two moves away from solved is NOT SOLVED. It counts in one of the 43 quintillion positions that is not solved. Why is there this much discussion?

If you wanted to know how many possible positions are one or two moves away from solved, you should ask that, don't try to redefine the solved state.

And to the person who said there are infinite positions... you're an idiot.

I was going to say nearly this exact thing, including the 43 quintillion possibilities. No offense to anybody, but I don't think a person of average intelligence could solve a cube that is 3 turns away. I realize this seems rather easy, but I've done this exact thing with many many people who said something to the effect of "Wow I couldn't solve it unless it was like less than 5 moves." I'll usually give them 4, when they can't do it I give them 3, they still usually can't do it. Don't get me wrong, I've seen people do 3 and even 4 move scrambles, but the overall average is they can't do it with 3 turns.

 

[yboy403]

I'll usually give them 4, when they can't do it I give them 3, they still usually can't do it. Don't get me wrong, I've seen people do 3 and even 4 move scrambles, but the overall average is they can't do it with 3 turns.

I don't know about the people you hang around with, but I've seen my 5-year-old sister do three moves several times in a row with no help-three random moves, too.

 

[iasimp1997]

One.

 

[yboy403]

One.

??

 

[TeddyKGB]

This may be a few days late, but I thought I would give my opinion, I dont think one move away from solved would be considered solve, it wouldn't nessisarily be scrambled but its not solved! maybe "slightly askew" so the real question is "how many slightly askew positions are there?"

 

[megaminxwin]

Well if you consider a +2 a solved state then count the ways:
U U2 U' R R2 R' L L2 L' F F2 F' B B2 B' D D2 D'
Thats 18 plus the solve state so 19 in 43,252,003,274,489,856,000 would be considered solved.

Don't forget the x24

 

[yboy403]

This may be a few days late, but I thought I would give my opinion, I dont think one move away from solved would be considered solve, it wouldn't nessisarily be scrambled but its not solved! maybe "slightly askew" so the real question is "how many slightly askew positions are there?"

Right, which is why I've said a number of times that all the confusion is over my bad wording of the thread title.
Yerachmiel