A $3\times3$ cube made up of $1\times1$ pieces is painted red from all faces and broken in $27$ smaller pieces ($1\times1$). A Blind man comes and randomly arranges pieces to make a $3\times3$ cube. What is the probability that a cube similar to the original is formed (red faces from all sides)?
Suppose 27 cubes are stacked together, suspended in the air, to form a larger cube.  The cube is then painted on all the exposed surfaces and dried.  The smaller cubes are then randomly permuted in both spatial position and spatial orientation to form another large cube.  What is the probability that new larger cube is identical to the original?  This is not homework, it was passed on by a friend who found it somewhere on the internet, and they do not remember where.
Prove that $\lim_{n\to\infty}a_n\le \lim_{n\to\infty}b_n$ How do I install a different Python version using apt-get? Can we prevent titles with an unnecessary tag in them? I just downloaded a DVD which is for a movie. After uncompressing the DVD file I see .VOB, .IFO and .BUP files. If I open the first .VOB file it shows me the DVD menu but I can't interact with it. So, my question is... is Totem able to play those kind of files? If so, how can I achieve that? What other app could I use in order to reproduce those files?  Edit  Using VLC didn't work neither. I forgot to mention I have already tried that. Let me rephrase: if I open the first video file it shows the DVD menu, BUT I can't interact with it. I have a third party php library which I would like to use in a module. How do I do that in Drupal 8? In drupal 7, I have done things like the following code, but libraries_get_path() doesn't exist in Drupal 8, and I'm not exactly sure how things change, not that we use namespaces (of which I am a beginner).  require(libraries_get_path('twitter-api-php') . '/TwitterAPIExchange.php');   And, where should I place my library? Should I create a new libraries directory and put it in the root, just like modules and themes are now in the root? This should be a very easy question but it might just be that I'm confusing myself. So we have the definition of a function $f$ on $S$ being continuous at $x_0$:     For any $\epsilon$>0, there exists $\delta&gt;0$ such that: whenever $|x-x_0|&lt;\delta$, we have $|f(x)-f(x_0)|&lt;\epsilon$   And I assume the negation is     There exists $\epsilon$>0 such that for all $\delta&gt;0$, $|x-x_0|&lt;\delta$ yet $|f(x)-f(x_0)|\ge \epsilon$.   Now I want to show that the function $f(x)=\sin(\frac{1}{x})$ together with $f(0)=0$ cannot be made into a continuous function at $x=0$. So I need to show that there exists $\epsilon&gt;0$ such that for all $\delta&gt;0$, $|x|&lt;\delta$ yet $|f(x)|\ge\epsilon$.  Let $\epsilon = \frac{1}{2}$. Then no matter what $\delta$ we choose, let $|x|&lt;\frac{1}{2}$. It is certainly possible that $|f(x)|\ge \frac{1}{2}$, because, well, $\frac{1}{x}$ can really take on arbitrarily large value as $x$ is small.  Now, what confuses me is that, as $x$ gets small, $f(x)$ can certainly be greater than $\frac{1}{2}$ for infinitely many times, but it will be less than that infinitely many times, too. But I suppose it doesn't really matter. So I think there's something wrong with my negation but I couldn't figure out where.  Update: The correct version can be found . Watch for Lemma 4.6 Misplaced/Overlapping Navigation Item Text on iOS 10 The bug displayed in the attached screenshot occurs consistently for me while running the developer beta of iOS 10 using v1.5.2 of the Stack Exchange app on an iPhone 6.  I was able to recreate this by simply tapping on any question or answer in my feed with a long title. This doesn't appear to be a bug with iOS 10 itself since I haven't seen it in other apps, but I could be wrong. The definition of "Lie group" typically restricts to a smooth manifold.  If we instead define a "Lie group" to be a topological manifold such that multiplication and inversion are continuous, is the manifold necessarily smooth?  Is the smooth structure unique if we want a smooth Lie group?  I believe the answer is yes, since a connected Lie group structure is determined by the Lie algebra, but my search attempts failed. Is it ok to say "good morning" at anytime? How to find a program which is preventing sleeping? I am noticing that recently my laptop does not go to sleep/lockout after its configured X minutes of inactivity like it did in the past. I suspect some program (like media player does) is preventing it from going to sleep. I do not have any media player running, so how can I find which program is preventing it from sleeping?  It does go to sleep if I close the lid. But I would also like for it to go to sleep after X minutes of inactivity.  Update: Upgraded to Ubuntu 18.04 with default Gnome and Wayland enabled. The usual example of unitary ring without the  property is the ring of column finite matrices, and in this case we have $R\simeq R^2$ as (left) $R$-modules. (See also .) In particular, we have $R^2\simeq R^3$.      I wonder if there is an example of unitary ring (without the IBN property) such that $R^2\simeq R^3$, but $R\not\simeq R^2$ (as $R$-modules). In Java, how do I convert a byte array to a string of hex digits while keeping leading zeros? How to attach CGPDFDocument/CGPDFPage to MFMailComposeViewController GridView sorting: SortDirection always Ascending Underline my name in the bibliography For several documents I'm preparing, I need to highlight (or underline) occurences of my name within the bibliography.   Is there an easy way of doing this? I think this was more of a fantasy series, though it was more subtle than The Lord of the Rings IIRC. Set in modern times - though there was some alternate realm thing that was less modern I think.   I read it about ten years ago, I think, and there were a number of books at that time. My brother and I stopped reading it shortly after the main character summoned a small demon somehow who was a bad influence... I think.  It involved a boy who somehow was at some kind of training school/academy. Unsure where it was located, but it needed a classic portal to get to it I think. I clearly remember some scene involving a car driving at specific speeds to get past some barriers. There may have been something about how the school was (slowly) moving or the wards were shrinking... unsure. I don't think it was only magic based, but that was some element of it. The boy had some kind of strange power that made him an incredible fighter, but he wasn't sure how (at least from what I remember). The recordings of him training just showed a white flash I think. I remember some scene in an elevator going down to the training floor where he went to a level below some adult with him.  There was something else about him joining a group that had twelve members, one for each month, whose powers grew and lessened based on how close the date was to "their" month.  Oh, and there was something about him looking for his mother I think? Perhaps in another "realm"?   Any help is appreciated! A specified Include path is not valid. The EntityType does not declare a navigation property with the name * What is the most accurate way to retrieve a user's correct IP address in PHP? Take a sponge ball and compress it. The net force acting on the body is zero and the body isn't displaced.  So can we conclude that there is no work done on the ball? vertical alignment of a turned cell I know using turn command and a sample code like this:  \usepackage{rotating}  \begin{turn}{90} Supervised Learning \end{turn}}   we can rotate a text inside a cell, however, I couldn't find a way to vertically center this rotated text. Any suggestions is appreciated.  Here is the sample MWE:  \begin{table}[]     \centering     \begin{tabular}{|&gt;{\centering\arraybackslash}m{0.5cm}|c|&gt;{\centering\arraybackslash}m{8cm}|}         \hline         \multicolumn{2}{|c|}{Classification} &amp; References \\ \hline         %\multirow{2}{*}{{\begin{turn}{90}Supervised Learning\end{turn}}}            \multirow{2}{*}{\rotatebox[origin=c]{90}{Supervised Learning}}  \end{tabular}     \caption{A Classification}     \label{tab:Machine_Learning_models} \end{table}