Math Is Fun Forum

I know that I'm bumping an old thread, but I recently found a video where a guy explains (proves) why the integration method that knighthawk posted about works.

Here is the video where he shows the method knighthawk posted about.
Here is the video where he (apparently) proves the method.

I actually commented on the video (showing the method), but he was "kind enough" to delete my comment (wherein I just mentioned that I saw this method posted here 10 years ago and also asked him what he thought of my Pascal's Triangle method).  So he's nice on screen, but maybe a little insecure . . .

I guess the spam filter block time is up.

For those interested, I replied to almighty100 with this pdf in one the comments in the third video of my video series.  In the hyperlinked thread which contained my first online post about this topic, if you go both backward and forwards from that post, you can see other proofs of specific powers of i.  (For example, if you go ahead, you will find how I proved Sum i^1 by writing a linear equation from two "data points", and you can see a system of equations approach for calculating i^2 from someone else a few posts before that.)

I forgot to mention this, but for those who don't have Mathematica, see !6 versus !6 (with this formula) to compute subfactorial.

You can substitute any integer in place of the 6's and get the correct subfactorial value.

Yesterday I created and uploaded a three part video series on the content I have posted in this thread previously (with the exception of the Bernoulli number stuff).

I present the method in "Part 1", abstract it in "Part 2", and prove it in "Part 3".  Enjoy!

I realized that I made a few mistakes in the proof(s) in the messages/spoilers in this post, but my video proof is correct.  In addition, I have created my own proof of

Therefore my proof is now both complete and correct.

Let me know what you guys think!

May I suggest to overwrite Google Translate's "Integers and sets us." at the end of *every* post you make here and on the math.stackexchange forum with "Integers set by us".

Perhaps that's why you are not being understood and not appreciated.  Miscommunication.

I also would suggest doing this in your second post of this thread, as I didn't understand what was going on until I realized that is what you meant to say.

(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).

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.

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.

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.

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.

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.

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).

(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).