We have a 6th druid in the party.  He can wildshape twice per short rest, and for each use, maintain his shape for half his level, thus for 3 hours.    Can a druid effectively stay wildshaped indefinitely by taking short rests in wildshape form, thus allowing them to regain their wildshape uses?  For example, the fore-mentioned druid wants to wildshape into a cat and sit on the back of the horse while the party is traveling.  Would the druid be able to simply continue his wildshape form every 3 hours while resting on in a cart or on a horse?  Thanks.
Do the druid's uses of the Wild Shape feature recharge while still in Wild Shape?  Say a 10th-level Circle of the Moon Druid spends two uses of Wild Shape to transform into an Air Elemental, whose form she can stay in for the next 5 hours. After 2 hours, she takes a short rest, but does not drop the Wild Shape. She has another couple of hours to keep the Air Elemental form.  My main concern: Does she get her two uses of Wild Shape back with that short rest? Or, does she have to drop the Air Elemental form first before taking the short rest to recharge Wild Shape?
I don't want to do this through trial and error, and the best way I have found so far was to start dividing from 1.     $n! = \text {a really big number}$   Ex. $n! = 9999999$   Is there a way to approximate n or solve for n through a formula of some sort?  Update (Here is my attempt):  Stirling's Approximation: $$n! \approx \sqrt{2 \pi n} \left( \dfrac{n}{e} \right ) ^ n$$  So taking the log:     $(2\cdot \pi\cdot n)^{1/2} \cdot n^n \cdot e^{-n}$       $(2\cdot \pi)^{1/2} \cdot n^{1/2} \cdot n^n \cdot e^{-n}$       $.5\log (2\pi) + .5\log n + n\log n \cdot -n \log e$       $.5\log (2\pi) + \log n(.5+n) - n$   Now to solve for n:     $.5\log (2\pi) + \log n(.5+n) - n = r$       $\log n(.5+n) - n = r - .5 \log (2\pi)$   Now I am a little caught up here. Kernels and reduced row echelon form - explanation The following text is written in my textbook and I don't really understand it:     If $A = (a_{ij}) \in$ Mat$(m x N, F)$ is a matrix in reduced row echelon form with $r$ nonzero rows and pivots in the columns numbered $j_1 &lt; ... &lt; j_r$, then the kernel ker$(A)$ is generated by the $n-r$ elements $ w_k = e_k - \sum\limits_{1 \le i \le r, j_i \le k} a_{ik}e_{j_i}$ for $k \in {1, \cdots , n} \ {j_1, \cdots, j_r}$, where $e_1, \cdots, e_n$ are the standard generators of $F^n$.    There are no computational examples and the notation is a bit overwhelming so I don't know how I can use it. Can somebody give an example of this in practice? I've tried almost any method to root my ASUS ZenFone Live, but no one seems to work. Also I only need root, no unlocked bootloader (don't want to wipe data) or custom recovery (not essential ATM). "rsync: failed to set permissions on ..." error with rsync -a or -p option Clearing NSUserDefaults I'm using +[NSUserDefaults standardUserDefaults] to store application settings. This consists of roughly a dozen string values. Is it possible to delete these values permanently instead of just setting them to a default value? What do repeating '^@' symbols in a logfile mean? A few times I have come across this in a log file:  Sep 17 09:35:57 00187d378926 ovpn-00187d378926[1701]: /sbin/ifconfig tun0 0.0.0.0 Sep 17 09:35:57 00187d378926 kernel: Kernel logging (proc) stopped. Sep 17 09:35:57 00187d378926 rsyslogd: [origin software="rsyslogd" swVersion="4.6.4" x-pid="1707" x-info="http://www.rsyslog.com"] exiting on signal 15. ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@ ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@ ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@   What does it mean?  Sometimes it is a single line, other times it is lots of lines.  I usually see it happen on Debian 6 machines. Is it possible to conditionally format a cell if the cell contains formula (to alert myself and other users when updating the cell)? GNOME has grey bar on top and bottom after upgrade from 17.04 to 17.10, how do I fix it to how it's supposed to look? This is how my desktop looks after the update, the top bar turning grey and the addition of the bottom bar happened after the update.    From my searching of screenshots this does not appear to be how it is supposed to be. Is there any way to fix it? Pressure inside a closed room Hiding axis text in matplotlib plots Pascal's triangle in tikz I would like to typeset the top part of Pascal's triangle. To get the triangle with the names of the binomial coefficients, i.e., {n \choose k}, I used the following code  \begin{tikzpicture} \foreach \n in {0,...,4} {   \foreach \k in {0,...,\n} {     \node at (\k-\n/2,-\n) {${\n \choose \k}$};   } } \end{tikzpicture}   The result is this   Now I want to be equally lazy and do something like this for the values of the binomial coefficients, i.e., replace {\n \choose \k} in the node label with \CalculateBinomialCoefficient{\n}{\k} where \CalculateBinomialCoefficient is a hypothetical macro that calculates the binomial coefficient. Has anyone done something like that?  The result should look like this: How to deal with perfect separation in logistic regression? If you have a variable which perfectly separates zeroes and ones in target variable, R will yield the following "perfect or quasi perfect separation" warning message:  Warning message: glm.fit: fitted probabilities numerically 0 or 1 occurred    We still get the model but the coefficient estimates are inflated.   How do you deal with this in practice? Shell script works when saved with nano but not when saved with Notepad++ When I copy my bash script across from Notepad++ into a new file with nano editor inside SSH and save. It runs fine. (sh ./install).   But if I save the file (Exact same contents), upload to my web server, download it using Wget on the same machine. I get syntax errors. I've checked the encoding and they seem to be the same. I have since used a plethora of character encoding to see if it will resolve the issue. I am also setting the file to executable once I download it using wget!  The file runs fine and I have zero errors when copy and pasting using nano. Any idea what this may be? As figured out in  localized SE sites must not be displayed in the HNQ's. But  actually is.        Has to be fixed I guess? Living potato clock? Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$ Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$.    Challenge: Prove  $$\sum_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}.$$   I have two proofs, one of which is partially combinatorial.    I'm posing this problem partly because I think some folks on this site would be interested in working on it and partly because I would like to see a purely combinatorial proof.  (But please post any proofs; I would be interested in noncombinatorial ones, too.  I've learned a lot on this site by reading alternative proofs of results I already know.)  I'll wait a few days to give others a chance to respond before posting my proofs.  EDIT: The two proofs in full are now given among the answers. Use XeLaTeX in Texmaker instead of pdfLaTeX Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $S_3=D_6$ Evaluating $\int P(\sin x, \cos x) \text{d}x$ Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$.   For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$.  Is there a general method which allows us to evaluate the indefinite integral     $$ \int P(\sin x, \cos x) \text{d} x$$   What about the case when $\displaystyle P(x,y)$ is a rational function (i.e. a ratio of two polynomials)?   Example of a rational function: $\displaystyle \frac{x^2y + y^3}{x+y}$.    This is being asked in an effort to cut down on duplicates, see here:   and here: Radical change of favourite tags doesn't reflect in questions list