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Why exactly did the Sorting Hat originally want to put Harry in Slytherin? The Impossible puzzle ("Now I know your product") I am trying to figure out this impossible problem by Martin Gardener and still havent found a suitable solution or link to it     Two mathematicians S and P are discussing two   unknown integers, both greater than 1. S knows only the sum of the   numbers, whereas P knows only their product.:      S: "I see no way you can determine my sum."    P (after a suitable delay): "That didn't help me. I still don't know your sum." S (after   another delay): "Now I know your product." What are the two numbers?   I have seen the answers on the xkcd wiki link for this at  say 13 and 16 and it checks out mathematically but then there are more ambiguous responses and most of the attempts seem brute force rather than logic.  My question is two fold   How to solve this problem logically? Are there more answers then 13,16 or this answer is the only one even as the upper bound k(upper bound on answer) shifts higher Viewer-independent copyable spaces at the beginning of a line? I need to include a couple of Python code listings, where the indentation of the lines (using some number of spaces) is significant. I would like for the code listings to be copyable, so the spaces at the beginning of the line need to be be copied along with the text.  This question has been asked in various ways before (e.g. , or ). Those questions focus on making line numbers uncopyable, though.  Making the spaces at the beginning of a line copyable seems to be harder: "I am not sure it is possible to specify in the PDF (at least in a viewer-independent way) that the indentation should be copied too" (). At , Philippe Goutet suggests a solution (turning the spaces into visible spaces, and coloring them in the background color so that they appear invisible) that works using Acrobat Reader, but not all readers. He says "It works under Acrobat Reader and it's extremely pleasant to be able to quickly copy/paste code without problem (perhaps the problem can be circumvented by writing direct PDF code to tell that it's a space, I've never had the time to try)".  Is it possible to produce a PDF with a code listing with copyable real spaces at the beginning of a line?  Minimal example: The line return x should start with four spaces.  \documentclass{article}  \begin{document} \begin{verbatim} def myfunction(x):     return x \end{verbatim} \end{document}   I know that I could attach the code to the PDF as a file, but that's not what I want. How to make horizontal lists? This might seem like a very simple question, but what is the easiest way to turn a normal vertical list into a horizontal one? Which are the spec bits in this cache? I remember being taught that the correct order of adjectives in English was something along the lines of "Opinion-Size-Age-Color-Material-Purpose."  However, it's been a long time and I'm pretty sure I've forgotten a few categories (I think there were eight or nine). Can anyone fill them in? What's the full-form & significance of USS & ISS prefixes in name of Star Trek starships? In the Star Trek normal universe, almost all of Starfleet's starships were named with the prefix USS: Count from . Some popular ones: USS Enterprise - series, USS Stargazer, USS Exeter, USS Defiant, USS Excelsior, USS Voyager, USS Titan, USS Reliant, USS Equinox, USS Grissom, USS Relativity, etc.  In the mirror universe, Starfleet's starships were named with the prefix ISS. We don't have a big list, but we do see these: ISS Enterprise NX-01, ISS Enterprise NCC-1701, ISS Avenger.  What's the full form of USS &amp; ISS in Star Trek canon?  In the real world, the US Navy uses USS in ship names, which is reasonable as "United States Ship", but why is it used in Star Trek canon (in which there's no US to control starships)?  In the mirror universe, USS is replaced with ISS. Why? Does ISS have a meaning opposite from that of USS? I came across a map and when I was playing I found some coloured and bolded item names. I thought it isn't possible as at most you can only use the custom name tag to rename things! Is there a way to colour and bold item names in vanilla Minecraft? External programs are OK. Oh, and the wiki page of formatting codes doesn't really do much. Can anyone tell me how to interpret the 'residuals vs fitted', 'normal q-q', 'scale-location', and  'residuals vs leverage' plots?  I am fitting a binomial GLM, saving it and then plotting it. Why does an open interval NOT have measure zero? Universal Chord Theorem Let $f \in C[0,1]$ and $f(0)=f(1)$.   How do we prove $\exists a \in [0,1/2]$ such that $f(a)=f(a+1/2)$?  In fact, for every positive integer $n$, there is some $a$, such that $f(a) = f(a+\frac{1}{n})$.  For any other non-zero real $r$ (i.e not of the form $\frac{1}{n}$), there is a continuous function $f \in C[0,1]$, such that $f(0) = f(1)$ and $f(a) \neq f(a+r)$ for any $a$.  This is called the Universal Chord Theorem and is due to Paul Levy.  Note: the accepted answer answers only the first question, so please read the other answers too, and also this answer by Arturo to a different question:     This is being repurposed in an effort to cut down on duplicates, see here: .  and here: . Extension of degree 2 with characteristic not 2 I am told that $K \leq L$ is a field extension of degree 2 and that $K$ has characteristic not 2. I am then asked to show that $\exists \: \alpha \in L$ such that $L = K(\alpha), \; \alpha^2\in K $  So far I have that:  $\exists \; \alpha_1, \alpha_2 \in L$ such that $L = K( \alpha_1, \alpha_2)$  $\Rightarrow |L:K| = |L:K(\alpha_1)| |K(\alpha):K| $ so $|K(\alpha):K| = 1$ or $2$  $|K(\alpha_1):K| = 2 \Rightarrow L = K(\alpha_1)$ and $deg(f_{\alpha_1}(t)) = 2  $ where $f_{\alpha_1}(t)$ is the minimal polynomial of $\alpha_1$ over $K$   So $f_{\alpha_1}(t) = t^2 + at + b$ for some $a,b \in K$  Also, $|K(\alpha_1):K|=1 \Rightarrow K(\alpha_1) = K$ and $L = K(\alpha_2)$ which means $degf_{\alpha_2}(t) = 2$ where $f_{\alpha_2}(t)$ is the minimal polynomial of $\alpha_2$ over $K$.   So $f_{\alpha_2}(t) = t^2 + at + b$ for some $a,b \in K$  Given that we consider the same polynomial each time, relabel to consider:   $f_{\alpha}(t) = t^2 + at+ b $ for some $a,b \in K$  We have already demonstrated that $L=K(\alpha)$ for some $\alpha \in L$ so now we much show that $\alpha^2 \in K$  $\alpha^2 \in K \Leftrightarrow a = 0_K$  I now have no idea how to show that this coefficient of $f_{\alpha}(t)$ is necessarily $0$.   I know that $f_{\alpha}(t)$ has two roots in $L$ and that because $K$ and thus $L$ has characteristic not 2, they cannot be the same root, but I don't know how to then conclude that they must sum to $0$  Any help would be greatly appreciated. The full prophecy by Trelawney, the part my question is about in bold:     "The one with the power to vanquish the Dark Lord approaches ... born to those who  have thrice defied him, born as the seventh month dies ... and the Dark Lord will mark him as his equal, but he will have power the Dark Lord knows not ... and either must die at the hand of the other for neither can live while the other survives ... the one with the power to vanquish the Dark Lord will be born as the seventh month dies ..."    – Sybill Trelawney (to Dumbledore)    I am interpreting      for neither can live while the other survives   as: Harry can't live while Voldemort survives and vice versa. But why? It clearly worked until the end of book 7. One could argue with Dumbledore's thery of the self-fulfilling prophecy. But clearly Voldemort was smart enough to see that there was no mechanism that would kill him just because Harry stayed alive.    and either must die at the hand of the other    Again: Why? If somebody else destroyed all Horcruxes and then killed Harry, Voldemort would be just as vulnerable and could be killed by anyone. On the other hand, Harry could have been killed by anyone - his mum's protection transferred to Voldemort by Harry's blood was only helping against Voldemort.  I guess that I am interpreting this part of the prophecy in the wrong way or maybe there is an alternate, less obvious and more tricky one. So my question is: What exactly does the highlighted part of Trelawney's prophecy state? How can magnets be used to pick up pieces of metal when the force from a magnetic field does no work? I learned that the force from a magnetic field does no work. However I was wondering how magnets can be used to pick up pieces of metal like small paperclips and stuff. I also was wondering how magnets can stick to pieces of metal like a refrigerator? TikZ \foreach loop with macro-defined list running a bash file states:date: command not found I've installed a version of Ubuntu Server and although I'm fairly still fresh with Ubuntu, I'm trying to create a bash file to run a program in the background. More importantly, I'm trying to get a file nameto display correct and create it before I start running the program (I like my debug stuff).  However, here is my bash script:    #!/bin/bash  PATH="/var/log/servers/Game/"  FILENAME="startup"  EXT=".log"  DATE=$(date +%d-%m-%Y)  NEWFILE=${PATH}${FILENAME}_${DATE}${EXT}   I have verified that the file does exist in the PATH directory. However, the issue is with the date command. When I run the script bash -x game.sh this is what is displays.   + PATH=/var/log/servers/Game/  + FILENAME=startup  + EXT=.log  ++ date +%d-%m-%Y  game.sh: line 5: date: command not found  + DATE=  + NEWFILE=/var/log/servers/Game/startup_.log  + echo /var/log/servers/Game/startup_.log   I don't understand why it would say the command doesn't work, when I can go to the main console and run date and get a valid response.  Any help is greatly appreciated! This question is a possibly harder version of: .  Question.  Let $f,g :\mathbb R\to\mathbb R$, such that \begin{align} f(x-y)=f(x)\, g(y)-f(y)\, g(x), \tag{1}\\ g(x-y)=g(x)\, g(y)+f(x)\, f(y), \tag{2} \end{align} for all $x,y \in \mathbb{R} $. If $g$ is continuous at $x=0$ and not identically zero, then there exists an $\alpha\in\mathbb R$, such  $$ f(x)=\sin \alpha x\quad\text{and}\quad g(x)=\cos \alpha x. $$ Is there a pair of discontinuous $f$ and $g$ satisfying $(1)$ and $(2)$?  Update. If $\ell :\mathbb R\to\mathbb R$ is a linear functional over $\mathbb Q$ (i.e., $\ell(qx+ry)=q\ell(x)+r\ell(y)$, for all $x,y\in\mathbb R$ and $q,r\in\mathbb Q$), then $$ \sin\big(\ell(x)\big), \quad \cos\big(\ell(x)\big), $$ satisfy $(1)$ and $(2)$. Discontinuous such functionals do exist, and they are obtainable using Zorn's Lemma (equivalently the Axiom of Choice.) This takes care of the second question. How to disable notifications for a single mailbox in Mail? When managing multiple mailboxes, is it possible to disable the notifications for certain mailboxes? Changing color and fonts of itemize wp_customize_image_control default value