I have a task to collect data about how students are doing their homework. I have colours to determine the processes of every individual homework, to each individual student. I have one tab for each day at school.  Some of the cells have text in them, but I'm only interested in counting the colors.  I also don't know how to script or to code.
I am using Google Spreadsheets to create a matrix (time/activity) as a daily schedule for staff to sign into.  Is there a formula to calculate the number of cells in a column that are shaded (or not shaded) a certain color?  For example: 20 of the cells are gray and 31 are not gray (shaded any other color) and instead of manually counting them, it calculates this at the bottom of reach column.
What is the rule to convert the 'mm' notation to the 'optical zoom' notation? I searched a bit and found this one:  optical zoom = maximum focal length / minimum focal length   For example a 18-55mm lens would have a 3x optical zoom, and a 18-200mm lens would have a 11x optical zoom. Is that right? Do sets, whose power sets have the same cardinality, have the same cardinality? Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$ Today, we had a math class, where we had to show, that $a_{100} &gt; 14$ for  $$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$  Apart from this task, I asked myself: Is there a closed form for this sequence? Since I didn't find an answer by myself, can somebody tell me, whether such a closed form exists, and if yes what it is? Extract all numbers from a single string in R Create a Symbolic link to the merged contents of 3 folders I have 3 folders with large amount of data in each which has to be maintained seperately. Can I create a sym link or shortcut.in windows which should show me the merged contents of the 3 folders. I am not looking for library conf in Win7.   Any suggestion is highly appreciated. probability question ("birthday paradox") $n$ people attend the same meeting, what is the chance that two people share the same birthday? Given the first $b$ birthdays, the probability the next person doesn't share a birthday with any that went before is $(365-b)/365$. The probability that none share the same birthday is the following: $\Pi_{0}^{n-1}\frac{365-b}{365}$. How many people would have to attend a meeting so that there is at least a $50$% chance that two people share a birthday?   So I set $\Pi_{0}^{n-1}\frac{365-b}{365}=.5$ and from there I manipulated some algebra to get $\frac{364!}{(364-n)!365^{n}}=.5\iff (364-n)!365^{n}=364!/.5=.....$   There has to be an easier way of simplifying this. What's a good back-up strategy for 1 desktop PC? Drag-and-drop functionality no longer working after upgrade to 19.04 After upgrading from 18.04 to 19.04 (with a necessary upgrade to 18.10 between) on a Lenovo Yoga 11e, I can no longer drag and drop anything (ie, files, windows, etc.) with the touchpad.  I can, however, move things around using the touchscreen.  Whenever I attempt to do so with the mouse, though, sometimes it turns into a hand.  Whether or not it turns into the hand or remains a pointer, the icon will not move anymore.  IOW, as soon as I left-click, the pointer freezes in place until I release the touchpad. Aborting evaluation when the memory exceeds a certain limit ... namely that if $n &gt; 1$ and $(n − 1)!\equiv−1\pmod{n}$, then $n$ is prime.  This is for a number theory class I'm in at Penn State. My idea is to follow accordingly, but I can't get it concretely..  Reason by contradiction, suppose that $n$ is not prime.   Then $n = ab$, where $1 &lt; a, b &lt; n$  In particular $(n-1)! = ai$ for some $i\in\Bbb{Z}$. $\\$ If $(n-1)! = nj - 1$ for some $j\in\Bbb{Z}$ then it follows that a probably doesn't divide $-1$? "Could not identify document class." error when trying to count words I'm trying to count words in a cyrillic LaTeX document with .  I run the command perl texcount.pl index.tex and get following output:  D:\wd&gt;perl texcount.pl index.tex  !!! Could not identify document class. !!! File: index.tex Encoding: utf8 Words in text: 0 Words in headers: 5 Words outside text (captions, etc.): 0 Number of headers: 1 Number of floats/tables/figures: 0 Number of math inlines: 0 Number of math displayed: 0  (errors:1)   index.tex starts with following line:  \documentclass[a5paper,pagesize,10pt,bibtotoc,pointlessnumbers, normalheadings,DIV=9,twoside=false]{scrbook}   So documentclass tag is present.  How can I fix this error? How to Install Virtualbox on Windows 10 host? Can I ever play Undertale again normally? The ending I got is :      the genocide   When I reset my file, my past choices still take effect. I want to start a clean run.  I don't want to buy the game again, what do I do? Assume two black holes in the most common size range, spiraling into each other until they merge. The event releases significant amounts of energy via gravitational waves, which warp the space-time.  If the distortion is powerful enough, it would get noticed in daily life, or perhaps even detected by the proprioceptors in the human body.  How close, as an order of magnitude estimate, would you have to be to the merger to perceive it like that, without any instrument? Restrict the access to suggested edit reviews on deleted posts on an answer deleted as rude/abusive is visible, even without logging in (and for users with less than 10k reputation). These users can't see the answer , via the , or a direct link to the .    This is a potential avenue for spammers, just like the suggested edit spam used to be, but is ; according to the :     In addition, in order to reduce the incentive for spammers to make spam edits that get rejected, suggested edit spam is hidden from logged-out users, so they can't show their clients public-facing pages with spam (so they won't get paid).   For this to work, the spammer would need two accounts, one to post the spam and a second one to suggest an edit. Even if the spam post is quickly nuked, the review link will be active forever. One could argue that logged-in users without 10k reputation shouldn't be able to see the suggested edits either; I'm partial to that. Can I list the resources in a given package? Why does the CRT formula yield a solution of a congruence system? How can I create Light Streaks in Cycles? In Section 9 of Chapter 6, Lang constructrs for the first time a non-abelian extension as a splitting field $K$ of $X^n - a$ when the ground field $k$ does not contain primitive $n$-th roots of unity $\zeta_n$. He constructs an injective homomorphism of the Galois group $G(K/k)$ into the matrices of type   $$ M=\begin{pmatrix} 1 &amp; 0 \\ b &amp; d \end{pmatrix} \quad \hbox{with}\ b\in \mathbb{Z}/n\mathbb{Z} \quad \hbox{and} \quad d\in (\mathbb{Z}/n\mathbb{Z})^*,$$  where $(\mathbb{Z}/n\mathbb{Z})^*$ the multiplicative group of integers modulo n.  Afterwards, he states in Theorem 9.4 that if $n$ is an odd positive integer prime to the characterstic and $[k(\zeta_n):k]=\varphi(n)$ (i.e, the primitive roots of unity are not in the ground field), then the above homomorphism is, in fact, an isomorphism. Besides, the commutator group is $G(K/k(\zeta_n))$, so $k(\zeta_n)$ is the maximal abelian subextension of $K$.  Lang proves the theorem inductively over the number of prime factors in the decomposition of $n$ (i.e. if $n=9= 3\cdot 3$, then the proof "would take two steps" to complete). The induction start with $n$ prime is written clearly and I agree with it.   Now come the questions:   I do not understand Lang's argument to prove that the commutator subgroup is the subgroup $\begin{pmatrix}1 &amp; 0 \\ b &amp; 1\end{pmatrix}$, $b\in \mathbb{Z}/n\mathbb{Z}$. What is this projection on the diagonal and how does one see that the factor group of the commutator group is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^*$?   By the way, direct computation of $x\, y\, x^{-1}\, y^{-1}$ for two arbitrary matrices $x=\begin{pmatrix} 1 &amp; 0 \\ b &amp; d \end{pmatrix}$, $y=\begin{pmatrix} 1 &amp; 0 \\ c &amp; e \end{pmatrix}$ yields a matrix $\begin{pmatrix} 1 &amp; 0 \\ b(1-e)+c(d-1) &amp; 1 \end{pmatrix}$. As $n$ is odd, $2 \in (\mathbb{Z}/n\mathbb{Z})^*$, so taking $e=1$ and $d=2$, we indeed reconstruct the above-mentioned group.   The second (inductive) part of the proof is a complete mystery to me with virtually every sentence being a Theorem which needs an extra proof. Can someone give me some hints?   2.a. Lang writes $n=p\,m$ and takes $\alpha$ to be the root of $X^n-a$. It is clear that $\beta=\alpha^p$ is a root of $X^m -a$. Now comes the argument "By induction we can apply the theorem to $X^m-a$". I agree that we can apply it here but the result is that $K=k(\beta,\zeta_m)$ is the splitting field of $X^m-a$. In the diagram below, there appears $k(\beta,\zeta_n)$ out of nowhere.   2.b. Now Lang proceeds to apply the theorem on $X^p-\beta$ over $k(\beta)$. Bergman's "Companion" reminds to check that $[k(\beta,\zeta_p):k(\beta)]=\varphi(p)$ before applying the theorem and gives a hint "$[k(\zeta_p):k]$ and $[k(\beta):k]$ are relatively prime". I disagree with this hint. Take $n=21= m p$ where $m=3$ and $p=7$. Then $[k(\zeta_p):k]=\varphi(p)=p-1=6$ but $[k(\beta):k]=m=3$ and both are not co-prime. Can we apply the theorem here at all? Or if so what is the argument? If Bergman's argument is valid, where is the fault in my logic?  2.c. Even assuming that we can apply the theorem in 2.b., then we can say something about $k(\alpha, \zeta_p)$ and not $k(\alpha,\zeta_n)$. How do we proceed to checking everything Lang states in the last section of the proof (basically, everything after the diagram)?  EDIT: I have found the essentially same question How to prove ${n \choose p} \equiv\left\lfloor \frac{n}{p} \right\rfloor\pmod p$ Prove ${n \choose p} \equiv \left\lfloor \frac{n}{p} \right\rfloor \pmod p$   ($n$ is natural and $p$ is a prime number)  How can i prove this statement?