Why does Dumbledore trust Snape?
What iron-clad reason did Dumbledore have for believing Snape was on his side
What methods can I use to create balance and consistency between a group of differing logos? A regular challenge that I face is balancing two (or more) logos. Whether they need to be equal or one needs to be given priority. Some logos however might be squares, others rectangles, circles, etc...   As a quick sample I snagged a nice high resolution photo from  and pretend its going to be an advertisement that needs to show some footwear brand logos on it. In this case I used Adidas and Saucony.  This is certainly not balanced even though they're height is the same:    Likewise this is not balanced even though they're width is the same:    So then the question becomes when are they in balance, is this balance:    Maybe a little more adjustment, is this balance:    And in these examples I'm only working with two Black Logos. Factor in color and it can get even more difficult to balance. Another issue being when there's tiny text as part of the logo.  So the question is, other than estimating / eye-balling it, what tips, methods or even tools are there to balance different sized logos? Taking screenshot of a specific area from the command line Forward/Inverse Search Does Not work on certain tex files I have this weird issue with forward and inverse searches. I am using WinEdt + Sumatra combination. The forward and inverse search works perfectly on certain some .tex files, but fails on some others.  Currently, I am working on a .tex file and the inverse search always points to the end of a section. Where ever I double click in a particular section of the pdf, the inverse search points to the end of that section in the .tex file.  Moreover, when I try to perform forward search (Shift+F8) from the .tex file, the section in the pdf file doesn't get highlighted.  There are other .tex files where both the searches work perfectly. What could be the issue with this .tex file? Expectation of truncated normal If $r,s,t$ are prime numbers and $p,q$ positive integers s.t. $\text{lcm}(p,q)$ is $r^2s^4t^2$, then the number of ordered pairs $(p,q)$ is? If $r,s,t$ are prime numbers and $p,q$ are the positive integers such that LCM of $p,q$ is $r^2s^4t^2$, then the number of ordered pairs $(p,q)$ is?  My attempt to the solution:  Let $r^a s^b t^c$ be the LCM where $a=2,b=4,c=2$  Then the cases that arise are  I case   1) $p$ has $r^as^bt^c$ then there are a total of $(2+1)(4+1)(2+1)$ options  II case   1) $p$ has $r^as^b$ or $s^bt^c$,$r^at^c$ then there are total of $(2+1)(4+1)(2+1)(3)$ options.  The answers comes out to be $45*4$ but the answer is $45*5$. Which case am I missing? Press-and-hold or Long-Press Gestures: Unintuitive? Often, I find people struggle with the usage of press-and-hold gestures on mobile/web platforms. The mere thought of having a user press on a defined object and hold for an arbitrary amount of time before the product reacts seems overtly complicated and unobtrusive. My questions are:   For what use cases do press-and-hold gestures best fit? Have there been published studies with regards to press-and-hold gesture intuition? Are there more viable alternatives to this system? The meaning of Bit-wise NOT in Python Real world uses of hyperbolic trigonometric functions Operand Should Contain 1 Column - MySQL NOT IN After Upgrade to Ubuntu 20.04: "Oh no! Something went wrong" "Was" or "were" for "half a dozen" In Microsoft Word, the following sentence is flagged.  It tells me to use "was" instead of "were"     There were half a dozen books strewn about the floor.   I would think that you would use "were" since it's a quantity more than one.  You wouldn't say, "There was twelve books strewn about the floor."  Does the use of the "half" modify it somehow? Is it possible to transfer Minecraft PE worlds to Minecraft PC? I have a really cool Minecraft PE world, and I would like to also have it on my Computer. Is there a way to move a Minecraft PE world onto the PC Edition?  My tablet is Android and my Computer is Windows 7. Thanks! The IPA symbol for the labiodental flap (ⱱ) How to use "`" in the same way it is used on the SE websites? How to run Java code using Java code? Proving $f(x,y,z)=\big(\frac{x}{a+x+y+z}, \frac{y}{a+x+y+z}, \frac{z}{a+x+y+z} \big)$ is injective Let $a\in \mathbb{R}$, $a\neq 0$, and $E=\{(x,y,z)\in \mathbb{R}^3:a+x+y+z\neq 0\}$ and $f:E\rightarrow \mathbb{R}^3$ defined by $$ f(x,y,z)=\big(\frac{x}{a+x+y+z}, \frac{y}{a+x+y+z}, \frac{z}{a+x+y+z} \big) $$ Show that $f$ is injective.    I've been trying to prove it directly by showing that if $f(x_1,y_1,z_1)=f(x_2,y_2,z_2)\Rightarrow$ $x_1=x_2, y_1=y_2,z_1=z_2$. But I get a system of equations which I'm find difficult to deal with. Any help would be greatly appreciated. How do I install wine? Finding the n-th lexicographic permutation of a string I have an ordered set of symbols S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }. I want to find the 1,000,000-th permutation in lexicographic order of S. It is a programming puzzle, but I wanted to figure out a way without brute-forcing the task.  So my thinking was like this:  For 10 variable symbol positions, we have 10! permutations, obviously. Now we want to find the first symbol.  If we fix the first symbol, then the remaining 9 symbols can have 9! combinations.  That means that 0 or 1 cannot be the first symbol, because the highest possible position is 2*9! = 725,760, which is lower than 1,000,000.  The lowest position for a leading 3 is 3*9! + 1 = 1,088,641, so it can't be 3 or higher, either.  Therefore, the first number must be 2. 2*9! is the largest multiple of 9! no greater than 1,000,000, so I need the 2nd symbol (zero-based) from the current set.  So now the question becomes, of the remaining S := S \{2}, which permutation of these symbols is at lexicographic position (1,000,000 - 2*9!) = 274,240?  6*8! = 241,920 is the largest multiple of 8! which is smaller than 274,240, so I need the 6th-smallest symbol of the remaining set, which is 7. So the prefix should be 27 by now.  That way, I keep going and finally arrive at: 1,000,000 = 2*9! + 6*8! + 6*7! + 2*6! + 5*5! + 1*4! + 2*3! + 2*2! + 0*1! + 0*0!  which results in "2783905614" as my solution.  However, according to the , (free reg. req.) that is incorrect.  Where did I go wrong in thinking or applying? Server 2012 Core (no GUI): Remote Server manager not working I am trying cmd --> MMC --> Server Manager and then connecting to my Server 2012 no GUI  but i keep getting the following error The WinRM client cannot process the request  I am unable to upload images right now  This is not related to other post "How to Remote Manage Windows Server 2012 in Server Manager?" because title and description of that post is quite different from what i am asking Why Do Jedi Use One Lightsaber in Combat? While I've seen Ahsoko Tano, in The Clone Wars, fight with two lightsabers (one in each hand), this seems to be extremely rare for Jedi.  We've seen Sith do this frequently, like Asajj Ventress, with two lightsabers.  Darth Maul used a dual lightsaber against Qui-Gon Jinn and Obi-Wan Kenobi and then there's General Grievous, who uses multiple lightsabers with his multiple limbs.  But most Jedi use just one blade, and their lightsaber hilts tend to be simple, just a straight cylinder without much adornment, while Sith sometimes use curved grips or other more exotic stylings.  Is there a reason Jedi stick with the simplest lightsaber designs and why the vast majority (at least in the movies and in The Clone Wars series) use only a single lightsaber?