nxnxn Rubik's Cube Theory

cmowla · 2012-10-21 13:53:43

The first topic of this thread is about whether or not every nxnxn Rubik's cube scramble in the commutator subgroup can be generated/solved with a single commutator solution.

Last edited by cmowla (2014-05-09 04:40:48)

cmowla · 2014-05-09 05:07:37

(For those who have seen this thread before, I edited the first post to simply be a more intense problem on the same subject matter. Much has changed since then as well.)

I believe I have proved the answer to this question to be YES.

I believe that I have not only proved it, but I created a systematic human method to solve the nxnxn cube explicitly on paper with abstract solutions which then are translated into actual Rubik's cube moves.

If you don't know what the first post is mentioning or you would just like to see an example, post a 3x3x3 Rubik's cube scramble (with some known notation so that I can interpret it), and I will post a solution to it using my method.

If my proof is indeed correct, do any of you mathematicians (I have a BS in math myself) think it's worthy to be a PhD dissertation topic in some branch of pure mathematics? I didn't do any research in previous existing math libraries to gather any information (or used any previously existing results of proof) to write this proof, and thus I'm not sure which branch of math this falls under exactly. Certainly this topic has never been done before.

The proof expands on basic facts about permutations, perhaps some calculus (for the nxnxn cube in general), programming (in Mathematica), and perhaps some well known results from abstract algebra (which are "common talk" in the "cubing community"). My goal was to make this proof as simple as possible so that everyone could understand it if they devoted the time to just read it, and thus I'm beginning to wonder if it is "complex" enough to be a dissertation topic (perhaps it's easy for me to understand because I have been working on this proof for a year and a half).

The amount of new knowledge it brings is great (but I don't see any real world applications of it besides solving the Rubik's cube without reducing it into different subgroups...that is, my method-which is used to test my theory-can be used to solve all pieces of the nxnxn cube in one step), and thus that is why I thought I would kindly ask for some opinions from you guys (as well as other people).

Last edited by cmowla (2014-05-09 05:18:07)

Agnishom · 2014-05-09 05:44:52

Where is the proof?

cmowla · 2014-05-09 05:56:16

Agnishom wrote:

Where is the proof?

It's just on my PC (I didn't post it on the web). So, assuming that my proof is correct, would the result I proved be enough, or is the content of the proof the main issue with PhD dissertations (my proof is certainly long enough)?

Last edited by cmowla (2014-05-09 05:57:48)

Agnishom · 2014-05-09 06:10:00

If you are asking if it is worthy of a PhD topic, it most certainly is.

my proof is certainly long enough

Mathematicians like simplicity. They do not judge a proof by its length. In fact, the shorter the proof the more beautiful it is.

Anyhow, if you have truly done something like that, very good. Well Done!

cmowla · 2014-05-09 06:22:45

Agnishom wrote:

If you are asking if it is worthy of a PhD topic, it most certainly is.
my proof is certainly long enough
Mathematicians like simplicity. They do not judge a proof by its length. In fact, the shorter the proof the more beautiful it is.
Anyhow, if you have truly done something like that, very good. Well Done!

Wow, I thought it might be, but I wasn't positive. My proof is mainly proof by exhaustion, and thus, although lengthy, I kept it very easy to understand and as compact as I could without making it more complicated than it must be. The same goes for the code and its documentation. (I just programmed functions in Mathematica, nothing fancy).

Thanks for your opinion, and if you would like to post a any 3x3x3 scramble (I can do those quickly, but I could solve up to a 7x7x7 cube scramble if you like...that's as big as I will go, however) to see the results, feel free to post any scramble, and I will give a solution for it and link it to an online cube simulator so that you don't have to execute it yourself (the solutions are quite lengthy).

Last edited by cmowla (2014-05-09 06:28:52)

Agnishom · 2014-05-09 06:38:12

Nowadays, many mathematical works involve CAS tools and functions. It does not make it less fancy.

By the way, you have a BSc in Mathematics. I am only a hi-school guy. You certainly know better than me. However, there are experts here who probably have a more definite opinion

ShivamS · 2014-05-09 07:44:33

Obviously you have a thesis advisor who helped you develop this, so if he thinks this is okay then go on with it.

Last edited by ShivamS (2014-05-09 07:44:53)

cmowla · 2014-05-09 09:41:43

ShivamS wrote:

Obviously you have a thesis advisor who helped you develop this, so if he thinks this is okay then go on with it.

I actually don't have one, as I'm not in graduate school. I've been doing this research as a hobby, and thus I was interested in PhD by publication, possibly.

I actually got the topic from Per Kristen Fredlund in a thread he started on a Rubik's cube forum. (I have the same username there as I do here). I had a little help from members qqwref and Stefan, but not that much. Most of this was good old trial and error because I never had to do something quite like this for my undergraduate degree, and it seems like Rubik's cube theory so far has either been limited to pure group theory or pure computer brute force algorithms ("god's number is 20", for example), but I haven't heard too much about what can come from combining both (although I did not do a brute force search, as it wasn't necessary because the theory and brain power was a more natural substitute).

There are a few people in the world, at least, who are doing projects which involve both, but I don't believe any of these projects are as groundbreaking as this one (I'm not bragging, rather, I feel as if I am observing, not producing, this proof because the process of writing it and doing all calculations was a VERY humbling and out of this world experience), and thus I thought perhaps I should make it a dissertation so that it will be recognized by audiences who can possibly make derivative works from it which could lead to real world applications in the future.

Recently, however, I found out that at least 10 years ago, some people were wondering about this (for the 3x3x3 Rubik's cube, at least), and thus Per wasn't the first to pose the question online. Link

Last edited by cmowla (2014-05-09 09:50:00)

ShivamS · 2014-05-09 12:38:55

Hi cmowla;

You are doing a PhD by publication? That's nice! But I have a couple of concerns:
1. In your second post in this thread, you used "believe" a lot. For example, you believe that you proved it. This leads me to believe that obviously your dissertation has not been peer reviewed. This means it could potentially contain fallacies. I suggest you post some parts that you think may be incorrect here. I also highly suggest that you go to a local university and have your paper reviewed.

2. You say you have a BS already. You also say that your dissertation contains fairly elementary, undergraduate level mathematics. Are you sure you have the knowledge which is generally received while doing a MS or the first two years of doing coursework in a PhD?

3. Unfortunately, a PhD by publication is often considered to be of less value then a regular PhD (I have a couple of sources), including to your employer (in academia and the industry).

To counteract all of these easily, have you considered enrolling in a regular PhD program, completing the coursework in one year and giving your thesis defence and getting the PhD? I think that is the best method for you.

I am sure that your dissertation is interesting, but in the past 2 years I have never spoken to anyone submitting their dissertation which contains absolutely no graduate level mathematics.

Last edited by ShivamS (2014-05-09 12:39:06)

cmowla · 2014-05-09 15:37:00

ShivamS wrote:

I am sure that your dissertation is interesting, but in the past 2 years I have never spoken to anyone submitting their dissertation which contains absolutely no graduate level mathematics.

I'm glad you mentioned this because this IS my main concern. Now I say the following with all due respect:

Perhaps those who are fluent with graduate level topics related to the content in my proof can say the same thing I say in plain English with condensed symbols and technical language and interpret them in their relation to group theory, for example. In addition, perhaps there exists knowledge which may shorten my proof significantly.

However, my primary goal was to make it as simple as possible to understand without doing anything fancy (that is, I purposely did not study any academic work which is related to this, if there even is such work, so that I could explain the proof to almost anyone who was interested in simple language) or using what I call "encrypted" language to say something simple in an overcomplicated way in my eyes and other laymen. I also did this so that my proof would be completely independent of other previously published works.

ShivamS wrote:

1. In your second post in this thread, you used "believe" a lot. For example, you believe that you proved it. This leads me to believe that obviously your dissertation has not been peer reviewed. This means it could potentially contain fallacies. I suggest you post some parts that you think may be incorrect here.

I guess I shouldn't have used that term because I'm positive there are no fallacies, as I have tested my theory with actual Rubik's cube scrambles along the way, and I have at least triple checked everything. I am very hesitate to say the word "positive", but, based on how I proved it, I left no room for error, especially since I didn't do anything fancy which can certainly cause more room for error.

ShivamS wrote:

I also highly suggest that you go to a local university and have your paper reviewed.

Even though I doubt (with good reason) that I have a fallacy, it would certainly be good to have at least two separate reviews to gain the confidence of the public. This is certainly a necessity.

ShivamS wrote:

3. Unfortunately, a PhD by publication is often considered to be of less value then a regular PhD (I have a couple of sources), including to your employer (in academia and the industry).

I am aware of this, and it's completely understandable I suppose (I guess it really all depends on what traits your employer is looking for, what you have published to get a PhD, etc.). I have not planned on getting a PhD to actually use it, but if I have done something which can technically earn me one anyway, why not I suppose, as I might be able to use it to advance myself somehow in the future when my life goals change. In addition, as I mentioned already, if it is published as a dissertation, people who are familiar with graduate level mathematics will be interested to read it and possibly create unforeseen derivative works from it (who knows, I guess).

But please, if I have a misconception about any of this (which is very possible), please let me know.

ShivamS wrote:

To counteract all of these easily, have you considered enrolling in a regular PhD program, completing the coursework in one year and giving your thesis defence and getting the PhD? I think that is the best method for you.

That sounds nice, but is it only a year of course work? Even though I have potential dissertation after only doing less than 2 years of research and work, I can't say that I could finish advanced course work at an accelerated pace (although I'm not positive, I'm almost certain that I just have an average IQ...the process of writing this proof made me feel as though I have a below average IQ to be honest).

Thanks for the advice. I really appreciate your time, and I hope to hear from you again as well as others.

ShivamS · 2014-05-10 03:08:26

What concerns me is if your proof can be done by using such elementary mathematics, it may not be that significant. That is why it is especially important to discuss it with professors and students (not just to point of fallacies, but to get more guidance).

Now, here is the unfortunate part: You have a very big misconception. A PhD = MsC + research, not just research. In order to be awarded a PhD, you must have the coursework done or the knowledge at least. I thought you have self-studied it already and therefore I said you could complete it in one year. Note that in UK, most people do complete the BS in 3 years and the MS or the PhD coursework in one year.

I still suggest going to a graduate program for one year, completing the coursework and discussing your dissertation with professors, thesis advisers and other students.

Also, you said you want this PhD just because what you have done can "technically earn me [you] one anyway." If your backup plan is your BS in Math, be informed that a BS in Math is now barely employable - especially in the field of maths.

cmowla · 2014-05-12 07:48:37

Thanks for your reply, ShivamS.

The following is one concept which comes up in my proof which I am not sure is new in abstract algebra. It's obviously well-known that the 3x3x3 Rubik's cube (and the 2x2x2 Rubik's cube) can be studied using Group Theory, but...

Has anyone ever heard of a conditional identity element in a non-Abelian group? The reason I ask is that for a specific portion of my proof, there are three identity elements which can be used for a certain procedure, two of which can only be used on a condition. If we violate this condition, then we are no longer in the Rubik's cube group. (Note that I don't use such terminology in the proof to describe this phenomena, but this one of the concepts shown concretely and in simple English).

I guess what I'm really asking is if such a concept is considered undergraduate math (I'm not sure because I haven't heard of this before). This is one of the examples for which I had in mind when I said earlier

cmowla wrote:

Perhaps those who are fluent with graduate level topics related to the content in my proof can say the same thing I say in plain English with condensed symbols and technical language and interpret them in their relation to group theory, for example.

Last edited by cmowla (2014-05-12 07:52:13)

ShivamS · 2014-05-12 07:56:24

I have never heard of a conditional identity element in the first place. Neither have the people who are sitting beside me right now.

ShivamS · 2014-05-18 08:52:07

What did you decide to do?

cmowla · 2014-06-23 07:34:19

(ShivamS, you got my email about your question, right?)

Since the title of this thread is a general one, aside from my current project, I note that the following is an alternate formula which I found that calculates the number of positions of the nxnxn cube (I'm sure some of you all have seen it before using the modulo function):

And, it's derivative can be written as:

See my posts in this thread (and the thread other members from there linked to in which I also posted in) for further details, or, if anyone has questions, ask away.

Who would have thought prime numbers would be present in such formulas! (Perhaps those who are actually interested in prime numbers might be able to interpret this in a way I cannot).

Last edited by cmowla (2014-06-23 07:38:38)

ShivamS · 2014-06-23 09:26:26

I don't think I got the email, unless you sent it in the past 5 hours.

ShivamS · 2014-06-24 06:41:11

Yeah, I got the email. I just missed it.

Math Is Fun Forum

#1 2012-10-21 13:53:43

nxnxn Rubik's Cube Theory

#2 2014-05-09 05:07:37

Re: nxnxn Rubik's Cube Theory

#3 2014-05-09 05:44:52

Re: nxnxn Rubik's Cube Theory

#4 2014-05-09 05:56:16

Re: nxnxn Rubik's Cube Theory

#5 2014-05-09 06:10:00

Re: nxnxn Rubik's Cube Theory

#6 2014-05-09 06:22:45

Re: nxnxn Rubik's Cube Theory

#7 2014-05-09 06:38:12

Re: nxnxn Rubik's Cube Theory

#8 2014-05-09 07:44:33

Re: nxnxn Rubik's Cube Theory

#9 2014-05-09 09:41:43

Re: nxnxn Rubik's Cube Theory

#10 2014-05-09 12:38:55

Re: nxnxn Rubik's Cube Theory

#11 2014-05-09 15:37:00

Re: nxnxn Rubik's Cube Theory

#12 2014-05-10 03:08:26

Re: nxnxn Rubik's Cube Theory

#13 2014-05-12 07:48:37

Re: nxnxn Rubik's Cube Theory

#14 2014-05-12 07:56:24

Re: nxnxn Rubik's Cube Theory

#15 2014-05-18 08:52:07

Re: nxnxn Rubik's Cube Theory

#16 2014-06-23 07:34:19

Re: nxnxn Rubik's Cube Theory

#17 2014-06-23 09:26:26

Re: nxnxn Rubik's Cube Theory

#18 2014-06-24 06:41:11

Re: nxnxn Rubik's Cube Theory

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