How to derive this formula?   $$n^2=\sum_{k=1}^n(2k-1).$$
I stumbled across this relationship while I was messing around. What's the proof, and how do I understand it intuitively? It doesn't really make sense to me that the sum of odd numbers up to $2x + 1$ should equal $x^2$.
What are the “spam” and “rude or abusive” (offensive) flags, and how do they work? When flagging, there are two special case options: spam and rude or abusive.   What is spam, and when should I flag content as such? What is considered offensive, rude or abusive content? How does the spam flag differ from the rude or abusive flag? What is the effect of these special flags? When are these flags removed? I'm bugged by the following that's summarized on p. 109 of .     False theorem: All horses are the same color.      Proof by induction:      $\fbox{$P(n)$ is the statement: In every set of horses of size $n$, all $n$ horses are the same color.}$   $\fbox{Base Case or $P(1)$:}$ One horse is the same color as itself. This is true by inspection.   $\fbox{Induction Step:}$ Assume $P(k)$ for some $k \geq 1$.   $\fbox{Proof of $P(k + 1) :$}$   Since $\{H_1, H_2, ... , H_n\}$ is a set of $n$ horses, the induction hypothesis applies to this set. Thus, all the horses in this set are the same color.   Since $\{H_2, H_3, ... , H_{n+1}\}$ is also a set of $n$ horses, the induction step likewise holds for this set. Thus, all the horses in this set are the same color too.         Therefore, all $n +1$ horses in $\{H_1, H_2, H_3, ... , H_n , H_{n+1}\}$ are the same color. QED.   The issue the instructor was trying to point out is clearly valid. For the case $n = 1$, there is only ${H_1}$. So this case says nothing about possible overlapping elements of each set of $(n + 1)$, for instance $H_2$ in the above proof.    But it was proposed in the class discussion that this was the only problem. Had you been able to prove $P(2)$ true, then a proof of the above format would have been fine.    My interpretation is that yes, you could prove all horses are the same color, if you can prove that any set of two horses will be the same color. But this format would not work.  Why not? The problem I see is that the above proof is for the existence of at least one particular pair of sets of horses of sizes $n$ and $n + 1$, such that in each set, all horses are the same color.  Particularly when the set of size $n$ is a subset of the set of size $n + 1$.  In order to prove the induction step, don't you need to prove that sets of sizes $n$ and $n + 1$, do not necessarily contain the same, overlapping elements?  You could prove that any horse can be added to a set of 2 horses.  Take the last two, and they must be the same color, and so on.  Wwhatever color the first two happen to be, all other horses must thus be the same color.    Am I misinterpreting the example, or am I making a logical error? Thanks in advance. I posted a question and a couple of respondents asked for the source code. I could only post portions of the code.  If attachments were allowed I could have uploaded a .zip file with the whole application.  Will this be a feature in SO? I seem to have clicked on something that changed my display preferences. The main site is now very minimalistic, with fewer controls and different fonts.  I want to change it back, but where is it? Sound of a next scene starts one moment earlier than video I saw this several times in different movies, but I don't understand what is this supposed to mean. The main idea is when the scene is about to change, for one second or so we still see the picture from the old scene, but the sound is already coming from from the upcoming scene. Why do they do this? What's the purpose of this technique? Why would a single celled organism evolve to be multi-celled? I read a story this week on Richard Lenski who has been 'evolving' E. coli for more than 50,000 generations now.  One comment I read was from someone who doesn't accept Evolution who pointed out that we haven't seen a single celled organism 'evolve' into a multi-celled organism.  Another person responded and said that a bacteria is not going to evolve into something that isn't a bacteria.  So, if Evolution created single celled organisms and then multi-celled organisms how might that change have happened?  And is it possible to recreate that set of driving forces to make a bacteria something other than a bacteria?  To that end, what advantage does being multi-cellular have over being unicellular (if that's even a word)? Why can't we initialize members inside a structure? Why can't we initialize members inside a structure ?  example:  struct s {    int i = 10; }; Double dual mappings $\newcommand{\Hom}{\operatorname{Hom}}$Let $U,V$ be vector spaces. Denote the dual space of $U$ with $U^\intercal$ and the dual mapping of a linear map $\Phi$ be $\Phi^\intercal$. Define the double dual embedding on $U$ to be $\iota_U:U \to U^{\intercal\intercal}$. ($u \in U, f \in U^\intercal$) $$ \iota_U(u)(f) = f(u) $$     If $\Phi \in \Hom(V,U)$, then $\Phi^{\intercal\intercal} \circ \iota_V = \iota_U \circ \Phi:V \to U^{\intercal\intercal}$   Proof: Let $v \in V, f \in U^\intercal$. Then $$ \iota_U \circ \Phi(v)(f) = f(\Phi(v)) $$ but how do I show that $$ \Phi^{\intercal\intercal} \circ \iota_V(v)(f) = f(\Phi(v)) $$ ? Can a git submodule be created retroactively? I want to emphasize that I only have 5 dollars, so can I use the sentence:     Five dollars is all I have.   Or must it be:      Five dollars are all I have.   Please give the explanation as well. Why $\gcd(qb+r,b)=\gcd(b,r)$? Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so?    Addendum by LePressentiment on 11/29/2013: (in the interest of  and averting a duplicate)  What's the intuition behind this result? I only recognise the proof and examples solely due to algebraic properties and formal definitions; I'd like to apprehend the result naturally. I accidentally committed an unwanted file (filename.orig while resolving a merge) to my repository several commits ago, without me noticing it until now. I want to completely delete the file from the repository history.   Is it possible to rewrite the change history such that filename.orig was never added to the repository in the first place? Wireless USB Adapter 0bda:c811 Realtek Semiconductor Corp iPhone 5S, LTE does not work What packages will let me use Cyrillic characters in math mode? Note I'm not actually planning on writing any, for instance, Russian prose or even any Russian words. It's just that, on paper, I've always had the habit of denoting sets by uppercase roman letters, sets of sets by script letters and, when the need arises, sets of sets of sets by uppercase Cyrillic letters. Is there some package I can load which will allow me to typeset (in LaTeX math mode) Cyrillic letters much as I would typeset Greek letters? ie. $\Zhe$ to produce an uppercase zhe. Thanks! Say we have two integers $n$ and $k$, such that $0 &lt; k &lt; n$ and $gcd(n, k) = 1$.  Why is it that if we take every integer $x\in\{0,\dots, n-1\}$ and compute $k\cdot x \ (mod \ n)$, we get as remainders all numbers from $0$ to $n-1$, with no repeats?  In other words, why is $\ f:\{0,\dots,n-1\}\rightarrow\{0,\dots,n-1\}$, $\ f(x)=k\cdot x\ (mod\ n)$ a bijection for $k,n\in\mathbb{N},\ gcd(k,n)=1$? First, I'd like to apologize for the awkwardly-phrased title.   Instead of trying to explain the structure I'm thinking of, I'd like to give some context and then an example or two. I've used and seen this construction used conversationally, and if my memory and intuition serve, conversationally only. I'm curious to know if there is a formal term for it.  Ex 1  "If you don't like what I'm offering, the door is that way."  A purely logical interpretation would make little sense with this example. In this context, the inference is that the speaker is indicating to whomever he is speaking that they should leave, which suggests that there might be an elision, e.g.  "If you don't like what I'm offering, ... [then you may leave. T]...he door is that way."  Ex 2  "If you're hungry, I have food."  This is a similar context. The implication is that the speaker will offer food to whom she is speaking if they are hungry. A (strictly) logical interpretation would, again, make little sense (if you are hungry, then I have food doesn't convey any intent to provide it, nor does it make sense; the speaker always has food, she just will offer it if her guest is hungry).  I recognize that this might be too specific to have a term associated with it. I'm just curious to know what it is if it does exist. Evaluating the integral $\int \frac{x^2+x}{(e^x+x+1)^2}dx$ Some vertices don't move along with the rig \phantom causes a spacing issue between paragraghs