Number of Cube States With No Same-Colored Stickers Adjacent?

[Dapianokid]

And did this test every case? and counting across face boundaries significantly lowers the count of states and is harder for a human to check. The reason I asked originally is because my friends generally go for states with no adjacencies. I have to tell them that it's about as hard as actually solving the cube. My mom often randomly gets scrambles that satisfy the rule about no same colored sticker adjacencies, so she is beating incredible odds without trying.

 

[cmhardw]

My mom often randomly gets scrambles that satisfy the rule about no same colored sticker adjacencies, so she is beating incredible odds without trying.

I really doubt this to be true. I don't think your mom is scrambling completely randomly without examining the cube at all. If she does that, and still gets the no-adjacencies scrambles, then this is beating incredible odds. I'll bet your mom is actively trying to break up blocks of the same color. This will drastically change the odds.

As a comparison, imagine you scramble a cube with your eyes closed, then open them up and see if you have a solved cube. Chances are 1 in 43252003274489856000 that you do have a solved cube.

Now, solve the first two layers of your cube. Do the last block/pair with your eyes closed, then open them. Chances are 1/62208 that you have a solved cube. Actively giving the cube a little bit of order will drastically improve the probability of having a solved cube by chance.

In a similar way your mom drastically improves her chances of a no-adjacencies case by getting 2 or 3 sides to have no adjacent stickers of the same color.

 

[Dapianokid]

She also tends to do very few moves on the cubes. So, she isn't really decreasing the order of the cube by an insane amount.
She told me it was all by chance. I think she's playing a joke. I wouldn't say she was incredibly lucky to get attention but I seriously don't know how she does it. I have a hard time even scrambling the cube to such positions and she does it differently every time in 10 moves or so! Is there a way of quickly identifying how to create no adjacencies?

 

[Kirjava]

F U' R2 F2 U2 R' F'

omg that sucks

 

[Dapianokid]

I'm guessing the number of positions that satisfy this rule are spread evenly accross the cube groups.

 

[Stefan]

I'm guessing the number of positions that satisfy this rule are spread evenly accross the cube groups.

What does that mean?

 

[Dapianokid]

That the distribution of these states is regular accross all size cubes and classified states. They appear as often with 2x2x2's as they do 3x3x3's as they do 4x4x4's.

 

[cmhardw]

That the distribution of these states is regular accross all size cubes and classified states. They appear as often with 2x2x2's as they do 3x3x3's as they do 4x4x4's.

Are you saying that the proportion of states on a 2x2x2 that don't have adjacent stickers of the same color to the total number of 2x2x2 states is the same as the proportion of states on a 3x3x3 that don't have adjacent stickers of the same color to the total number of 3x3x3 states (and the same proportion on a 4x4x4, 5x5x5, etc.)?

This does not seem at all intuitive to me that such a thing would be true. Do you have a proof or at least line of reasoning that would lead you to this conclusion?

 

[Dapianokid]

Are you saying that the proportion of states on a 2x2x2 that don't have adjacent stickers of the same color to the total number of 2x2x2 states is the same as the proportion of states on a 3x3x3 that don't have adjacent stickers of the same color to the total number of 3x3x3 states (and the same proportion on a 4x4x4, 5x5x5, etc.)?

This does not seem at all intuitive to me that such a thing would be true. Do you have a proof or at least line of reasoning that would lead you to this conclusion?

It's obvious I need to word my posts better in this forum.

I think it's proportional, but not in a linear sense. I think there is an eponential curve to the number of states that is constant. I think that there are more positions which satisfy this condition (and occur more often, as in they are easier to obtain) for a 4x4x4 than there are for a 3x3x3. I think the same for a 5x5x5 compared to a 4x4x4. I have no idea how to go about doing this math, though, but the interest is killing me.

 

[cmhardw]

It's obvious I need to word my posts better in this forum.

I think it's proportional, but not in a linear sense.

If two things vary proportionately, then by definition they vary linearly. I don't understand what you mean when you say that things are proportional, but not linearly. Can you explain your thoughts further?

I think there is an eponential curve to the number of states that is constant. I think that there are more positions which satisfy this condition (and occur more often, as in they are easier to obtain) for a 4x4x4 than there are for a 3x3x3. I think the same for a 5x5x5 compared to a 4x4x4. I have no idea how to go about doing this math, though, but the interest is killing me.

I think what you're saying can be phrased as:

The probability of getting a no-adjacent stickers of the same color case on a 2x2x2 from a random scramble is the same as the probability of getting a no-adjacent stickers of the same color case on a 3x3x3 from a random scramble (and the same on a 4x4x4, 5x5x5, etc.).

Does that sound right?

 

[Stefan]

I think there is an eponential curve to the number of states that is constant.

An eponential curve that is constant?

 

[Dapianokid]

An exponential curve that remains consistently exponential.
I've decided to simply browse the forums until I have more cubing experience and more understanding of this place

 

[Stefan]

An exponential curve that remains consistently exponential.

What is "consistently exponential"? And can you show an example of an exponential curve that isn't "consistently exponential"?

 

[TMOY]

I don't know what he has in mind, but I can propose the folliowing mathematical definition of "consistently exponential":
A function \( f \) is consistently exponential if it is of the form \( f(x)=\lambda e^{\mu xg(x)} \), where \( \lambda \) and \( \mu \) are constants and \( g \) is a function whose limit in \( +\infty \) is 1.

For example, with this definition \( x\mapsto e^{x+\sqrt x} \) is consistently exponential but not exponential.

 

[Christopher Mowla]

Hmm. Interesting math problem. Tell me when somebody decides to have a go at it.

This was from 4 months ago (so maybe you know about it). He had the constraint that no more than 2 colors are on each face though (which is more strict than the restriction stated in this thread).

(And noobs who think that it's against the rules to bump old threads... don't dirty this thread up with posts shaming me for doing so. It's perfectly acceptable if it's a good bump.)

 

[Herbert Kociemba]

Accidently I posted a quite similar problem here three months ago...

 

[qq280833822]

I did the calculation about four years ago (but didn't verify it) and there were 234,173,555,841,235 states that no same color stickers adjacent, which is about 1 in 180,000.
As a simple verification, I randomly generated 1B states and found 5350 states that met the conditions.