This question has been asked before on other sites, but none of the answers were satisfying to me.  I have heard people say that one can use group theory to solve rubik’s cubes. I understand why the set of rubik’s cube transformaions are a group.  But it’s unclear to me how using the conceptual apparatus of group theory helps you solve them.   E.g. the fact that many helpful operations are of commutator or conjugate form is claimed to entail that group theory is helpful here, but I discovered operations of this form in rubik’s cubes before I knew these more general terms, and it’s not clear to me how learning these terms adds to your understanding of Rubik’s cubes.  So what are concrete ways in which understanding group theory actually helps you in a non-trivial way to understand Rubik’s cubes and how to come up with solutions?
Can RUBIK's cube be solved using group theory? If yes, how can we use it to solve a $2\times2$ Rubiks Cube?
Anyone know a way to immediately show the seconds of a file's date modified property in the GUI? So if you create a file, any file in any directory, right-click and choose Properties, the date modified (if it's recent) will say something like "dd/mm/yyy hh:mm, one minute ago" - reminder this is in Windows 7. Windows XP did it normally. Then they changed something.  If you wait a while, eventually you'll see the seconds, I'm not sure how long a while is, but this is incredibly annoying if you want to troubleshoot something that relies on the seconds of timestamps... is there a setting? registry key I can change perhaps?  I'm literally using Chrome, pasting in the path of the directory to be able to see the seconds quickly (as a workaround) but would be nice to be able to use Win7. En Dashes & Hyphens in Same Phrase Without recasting and in lieu of unsightly suspensive hyphenation, could these work? The en dash (which means "to") is used between the figures, and the hyphens are used in the compound modifier. To me, the examples below are concise and clear.  Do you agree that the examples below are 100% punctuated correctly--yes or no?  45–50-year-old men  10–20-mile radius  $45–$55-million-per-year industry  $100,000–$150,000-a-year savings  10–15-degree temperature difference  $2–$5-per-day service fees  45–75-cent-a-week raises  55–65-mph speed zone  15–20-ounce steaks  20%–40%-a-year tax increases Pretty sure that my edit was correct but it was rejected? The question is if you have a situation where one of the singular values is equal to 0 in a singular value decomposition of a matrix, how to do you procede to find the column vector of U corresponding to this singular value?   Usually I found the columns of U with this relation, so let's say that we refer to $\vec{u_2}$ as the U column vector corresponding to singular value equal to 0:  $\vec{u_2} = (1/s_2)Av_2$  Now the as we can see the 0 is in an illegal place, since you can't divide by zero. But if you move it to the left side, I still can't quite see how to find the column of U corresponding to it. Any suggestions?  Here is an example of a matrix that provided this specific challenge  $ A = \left(\begin{array}{rrr} -3 &amp; 1 \\ 6 &amp; -2 \\ 6 &amp; -2 \end{array}\right). $  $ \lambda{_1} = 90$,  $\lambda{_2} = 0$,  $s_1 = 3 \sqrt{10}$,  $s_2 = 0$,  How can one find $\vec{v_2}$ when $\lambda{_2} = 0$? In this case, you know that the eigenvectors of $A^{t} A$ are orthogonal, solve for $\lambda{_2}$ and find the eigenvector if it's orthogonal to the eigenvector corresponding to the first eigenvalue you have your solution.  So we end up with the following normalized column vectors for V, which we can see are orthogonal:  $ \vec{v_1} = \left(\begin{array}{rrr} -3/\sqrt{10} \\ 1/\sqrt{10} \\ \end{array}\right). $  $ \vec{v_2} = \left(\begin{array}{rrr} 1/\sqrt{10} \\ 3/\sqrt{10} \\ \end{array}\right). $ Convert Month Number to Month Name Function in SQL I have months stored in SQL Server as 1,2,3,4,...12. I would like to display them as January,February etc. Is there a function in SQL Server like MonthName(1) = January? I am trying to avoid a CASE statement, if possible. Story featuring smart people sent on a space mission to nowhere The story in question featured, as the vague title suggests, a "space mission" that is really a cover for forcing a bunch of really smart people to be in close proximity to each other for long enough that they can truly evolve their collective intelligence. I recall them developing new maths and, e..g, "farseeing" and eventually figuring out that while there was no planet at the end of their journey as they were told, they were able to construct one for themselves. Change date format [YYYYMMDD] to [MM/DD/YYYY] I generated a report that formats date as 20160509 and my Google Sheets locale is already set to United States. I've played around with various options within Format > Number > Date Format but I can't change the dates to MM/DD/YYYY format without doing a simple find and replace. How do I accomplish this? Which Harry Potter works are considered canon? There are many different Harry Potter works including:   books short stories movies card games video games toys amusement parks web content   Which ones are considered to be canon? Prove that, for every natural n, $\sum_{d|n} \frac{\mu(d)}{d}=\prod_{p|n}(1-\frac{1}{p})$ Integrate $\int \frac{dx}{\sqrt{-ax^2 + bx +c}} $ How can a photon have no mass and still travel at the speed of light? How to nest code within a list using Markdown Oftentimes, I want to include code samples in a list, for example:   Item One Item Two, for example: private bool ItemTwo() {     return this.IFeelLucky; } Item Three   The problem is that there isn't a good and intuitive way to do this using . It's doable, but you have to do some formatting black-magic to get it that way, which is just not "nice" to deal with.  Source that produces this:  1. Item One 2. Item Two, for example: &lt;br/>&lt;br/>&lt;pre>&lt;code>private bool ItemTwo()     {         return this.IFeelLucky;     }&lt;/code>&lt;/pre> 3. Item Three    I would love to just be able to do it this way:  1. Item One 2. Item Two, for example:      private bool Test()     {         return this.IFeelLucky;     }  3. Item Three  Which results in, well, this:   Item One Item Two, for example:  private bool Test() {     return this.IFeelLucky; } Item Three Why are newer generations of processors faster at the same clock speed? Why, for example, would a 2.66 GHz dual-core Core i5 be faster than a 2.66 GHz Core 2 Duo, which is also dual-core?  Is this because of newer instructions that can process information in fewer clock cycles? What other architectural changes are involved?     This question comes up often and the answers are usually the same. This post is meant to provide a definitive, canonical answer for this question. Feel free to edit the answers to add additional details. Avoiding 'instanceof' in Java In the following code, I need to use +/- where I have used it, and plus/minus where I have used it.  \documentclass{amsart} \usepackage{lipsum} \usepackage{calc} \parindent=0 pt \parskip=12 pt plus 0 pt minus 0 pt \def\headsep{25pt} \newcommand{\LAYOUT}[5]{%   \setlength{\hoffset}{0pt}   \setlength{\voffset}{0pt}   \setlength{\topmargin}{#1 - 1in - \voffset - \headheight - \headsep}    \setlength{\oddsidemargin}{#2 - 1in - \hoffset}   \setlength{\evensidemargin}{\oddsidemargin}   \setlength{\textheight}{#3}   \setlength{\textwidth}{#4}   \setlength{\marginparwidth}{#5} } \newcommand{\FULLPAGE}{\LAYOUT{1in}{1in}{9in}{6.5in}{0.75in}}  \FULLPAGE \begin{document} \lipsum[1-10] \end{document}   If I replace the plus and minus in line 5 with + and - I get an enigmatic  "missing begin{document}" error.    If I replace the minus's in lines 10-11 with "-" I get an equally enigmatic error:  "Missing number, treated as zero. Illegal unit of measure (pt inserted)."   Can somebody please explain when one can use +/- and when one can use plus/minus?  Presumably there's a good reason why latex's error messages are so completely mystifying, and why inconsistencies like these keep appearing? It's really hard to explain to my coauthors, who regularly complain that latex is the most infuriating program ever devised by mankind.  Thanks very much for any advice!   Leo AMD GPU Material in "material preview mode" and "rendered mode", is entirely gray as background. Only outline is displayed I don't know why but now, when i go in material preview or rendered (in solid mode it works fine), my object is entirely gray, or invisible idk, because it has the same color of the background. I've been working for days on this project without having this issue, it suddenly showed up. I tried opening other old projects and suddenly they all do the same so i guess it is some blender program setting that applies for all the projects, but idk how it happened cuz i didn't change anything :/, anyone got any tips? Suppose $f:X\rightarrow Y$ is a surjection and you want to show that there exists $g:Y\rightarrow X$ s.t. $f\circ g=\mathrm{id}_Y$. You need the AC to show this.  However, suppose $f$ is a injection and you want show there is $g$ s.t. $g\circ f=\mathrm{id}_X$. Then, according to my textbook, you don't need the AC to show this.  This is counterintuitive to me, because it's like you need a special axiom to claim that an infinite product of big sets is nonempty, while you don't need one to claim that an infinite product of singleton sets is nonempty, which seems smaller than the former.  So why don't you need the AC to show the latter?  EDIT: $X$ should be nonempty.  EDIT 2: I realized (after asking this) that my question mostly concerns whether the AC is needed to say that an infinite product of finite sets is nonempty, and why. Question about preposition -of- Sentence:      A set of deer antlers lying on a flat stone just inside [of?] one cave reveals more recent visitors.   Should we use preposition -of- in that case?  I have checked Corpus of Contemporary American English and have found examples of both usages:        ...to plant another 20 acres of vines, finish building the inside of   an on-site cave, and work on building a retail tasting room (Fiction)   However, opened and remained blue as the inside of an ice cave, paler   than the shell of a robin's egg (Fiction)   It looked like the wall on the inside of a cave: flat and kind of   scale (Fiction)   ...-To protect the inside of the cave. " // He stepped inside, leaving   the gate open.  (Fiction)     Views outside and inside of Wind Cave in South Dakota (Magazine)   Above, Black Hills grasslands and, right, boxwork inside of the cave   (Magazine)   You can see Bill Elliott's great photos of what you can see inside a   cave. (Spoken)   So, when I am asked, what was your feeling inside the cave? (Spoken)   At some point, we found a small river running through the inside the cave (Spoken)   We were gathered around a small fire inside the cave (Fiction)   About 700 feet inside El Sidrn cave, a research team including   Lalueza-Fox excavated 1,700 bones (Academy)      I still cannot come up with a clear pattern why we are using -of- in some cases but are not using it in other. Is it more a question of style rather then a grammar? Or it has something with how speaker/writer sees things around and perceives a reality? Does absence of -of- make things sound weird/wrong for native speaker? Thanks. Functions Which are non differentiable on a Given Set. I have recently read the famous Weierstrass Non differentiable function which is continuous everywhere but nowhere differentiable.     But now my question is:    Given a countable set $S (\subset \mathbb{R})$ , can we construct a function which is continuous everywhere but non differentiable only at points of S..?    My Attempt: If $S$ finite say, $S=\{c_1,c_2,...,c_n\}$ Then the following function is the desired one:$$f(x)=\sum_{i=1}^{n}|x-c_i|$$   But I cannot construct the function for a arbitrary Enumerable set $S.$     Can we generalize the question for an arbitrary set $S$? Please Help.  Thank you...!! Why should I "watch" a nearby Pokémon?