Find all $a \in \{1, 2, \ldots, 8, 9\}$ such that $\exists n \in \mathbb{N}$ and the digits of $\dfrac{n(n+1)}{2}$ are all $a$.   In other words, we have to find the values of $a$ such that $\dfrac{n(n+1)}{2} = a \cdot (1111\ldots1111)$ where $a$ is a Numerical Digit of Base $10$, i.e. $a \in \{1, 2, \ldots, 8, 9\}$.  I made the following progress:  Let's take $(1111\ldots1111) = R$ for convenience. So we get the following equation: $n^2 + n - 2aR = 0$. Taking the roots of $n$ by using the Quadratic Formula, we get $n = \dfrac{-1\pm\sqrt{1+8aR}}{2}$ .  As $n \in \mathbb{N}$, $-1\pm\sqrt{1+8aR}$ must be even, so $\sqrt{1+8aR}$ must be odd and hence $1+8aR$ must also be odd. We also know that $1+8aR$ must be a perfect square.  So, our problem reduces down to $1+8aR = f^2$ where $f \in \mathbb{N}$.  I couldn't proceed ahead from here.     also talks about pretty much the same problem. However, no conclusive solution was reached in that thread. I want to do the above question with an analytical approach. Hence, I beleive that my question is not a duplicate.  [Source: As far as I know, this problem is from IMOTC India]
For which $x$ exists sum $1 + 2 + 3 + \ldots + n$, where $n &gt; 3$, which has notation $xxx\ldots x$?   In other words, I am looking for a sum of natural numbers which gives a result which has all same digits, e.g. $5555555$ or $22222222222222222$, or whatever else.  How to find this series and how to proof it for all possible series?  I have found the following examples:   $1 + 2 + 3 + \ldots + 10 = 55$ $1 + 2 + 3 + \ldots + 11 = 66$ $1 + 2 + 3 + \ldots + 36 = 666$     And what about next series which ends with (different) repeating $x$? Hot to find and proof it?
I made a plot but it's not at all what I want. There are two things I'm having trouble with: 1.) The bars should be at the bottom, but there's a space between them and the x-axis. 2.) E1/E4, E2/E5 and E3/E6 should have the same color.  Here's a picture of how it is and how I want it below:   And here's my code:  \documentclass{standalone} \usepackage{pgfplots,pgfplotstable} \begin{document} \begin{tikzpicture} \begin{axis}[ symbolic x coords={E1, E4 (dyn.), E2, E5 (dyn.), E3, E6 (dyn.)}, xtick=data, width=\linewidth, height=6cm ] \addplot[ybar,fill=blue, bar width = 1cm] coordinates {     (E1,   8)     (E2,  15)     (E3,   50)     (E4 (dyn.), 4)     (E5 (dyn.), 17)     (E6 (dyn.), 40)  }; \end{axis} \end{tikzpicture} \end{document}   I hope you can help me. Thanks. Preventing Brute Force Logins on Websites As a response to the recent  and , what is the best way to secure your website against brute force login attacks?  Jeff's post suggests putting in an increasing delay for each attempted login, and a suggestion in the comments is to add a captcha after the 2nd failed attempt.  Both these seem like good ideas, but how do you know what "attempt number" it is?  You can't rely on a session ID (because an attacker could change it each time) or an IP address (better, but vulnerable to botnets).  Simply logging it against the username could, using the delay method, lock out a legitimate user (or at least make the login process very slow for them).  Thoughts? Suggestions? I am in a situation where when I get an HTTP 400 code from the server, it is a completely legal way of the server telling me what was wrong with my request (using a message in the HTTP response content)  However, the .NET HttpWebRequest raises an exception when the status code is 400.   How do I handle this? For me a 400 is completely legal, and rather helpful. The HTTP content has some important information but the exception throws me off my path. Answering beyond the question's scope Is answering beyond the scope of the question a valid thing? Or should we just stick to the question asked and ignore any potential bugs which might be present (of course OP is unaware of that, as of now).  To make myself more clear, please have a look at this .  There are 3 answers which talk about the issue asked and that's it. If you look at my answer, I've added an additional point about a potential bug which the OP has missed now (may find out later). So is it a good thing to point out such potential bugs or not? Should this practice be encouraged? Array#each vs. Array#map How many questions on meta have asked, "Why can't I ask this question when that question is open"? I read this book a few years back (around 5?). It was part of a multibook series but I have only read one book.  It's about this teenager who I think had been programmed to be an assassin but then somehow retained his conscience and was out to get the people responsible for that.  I briefly remember a girl, and this one part where he realizes he can ride a horse and then there's a chase too. I think the guy's name was Jim or Jimmy, but not quite. I'm also quite sure he was either part computer, or had a chip or something in his brain which had been programmed to give him a host of mental and physical abilities which he retained and used as well as the series progressed. For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives. Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset $K\subseteq \Omega$. Then by Cauchy Theorem and Morera Theorem, $f$ is holomorphic. Let $f_n$ and $f'$ be the derivatives of $f_n$ and $f$ respectively. Prove that $f_n'\to f'$ uniformly on any compact subset $K\subseteq \Omega$.   How to prove? Full file date (without GNU utilities) I am trying to obtain the full date (created or modified) of a particular file for passing to another program. I have tried variations of options with the ls command but none provide a full date for files less than 6 months old and I have limit usage of the options. When I try certain options I have seen trying to research this I get the following message:  usage: ls [-1ACFHLNRabcdefgilmnopqrstuxEUX] [File...]   None of which seems to provide what I needs as far as I can tell so I tried to use the stat command but it is not available to me.  I am using the Korn shell on AIX 5.3 which has limited commands available, can anyone suggest another way that I might be able to obtain a file's created or modified date as a full date (either dd/mm/yyyy or yyyy/mm/dd). How to effectively use the man pages? I am looking forward to mastering linux. I have been repeatedly told to refer to man pages when you are in doubt.   Though I am very comfortable with the commandline, I find it very difficult to quickly and efficiently use man to find help with commands. Hence I end up googling up how to get things done. It is even more painful when the system I am using does not have internet access and I have to walk back to my office to get help from the internet.  How does one master man pages? What are the most common tricks that can be used to quickly get whatever you want from the man page? What are the most commonly used keyboard shortcuts? In windows we have an option to view recently opened media files as my recent documents, I was  wondering does there exists something like that in Ubuntu?  via terminal or via GUI? Accumulation Points for $S = \{(-1)^n + \frac1n \mid n \in \mathbb{N}\}$ I was recently asked to find the accumulation points of the set $$S = \{(-1)^n + \frac{1}{n} \mid n \in \mathbb{N}\}$$  I answered that the accumulation points are $\{-1,1\}$, because despite the fact that $\frac{1}{n}$ diverges, we can still use $\frac{1}{n}$ to find an element arbitrarily close to either $1$ or $-1$, i.e. to show that there is an element in an arbitrarily small neighborhood of $1$ or $-1$. (No proof was required for this question --- this is my intuition.)  Am I right at all? Are commands complete sentences? seems to indirectly ask the question, but the upvoted answer says &quot;the actual sentence is...&quot;. So my question is this: Are commands considered to be grammatically sound, complete sentences in the English language? Some examples:  Go. Add a note. Do work. Finding the inverse of the arc length function I create landing pages in order to launch PPC campaigns for my customers. However, I would like to host them all together under a sub-domain of MY website.   For example:  I want that someone who visits www.clientnumber2.com/landingpage could see the landing page and still the same URL www.clientnumber2.com/landingpage, but actually host it myself.  The same for client3, client4, and so on... How to do "Files to store without compression" in 7-zip? Is there any way to selectively prevent certain file types from compression in 7-zip, like in WinRAR?  I'm trying to compress a huge folder (many gigabytes), and I don't want the already-compressed files (e.g. ZIP files) to be re-compressed -- they should be stored without compression. How to find the determinant of this $3 \times 3$ Hankel matrix? Today, at my linear algebra exam, there was this question that I couldn't solve.   Prove that $$\det \begin{bmatrix}  n^{2} &amp; (n+1)^{2} &amp;(n+2)^{2} \\  (n+1)^{2} &amp;(n+2)^{2}  &amp; (n+3)^{2}\\  (n+2)^{2} &amp; (n+3)^{2} &amp; (n+4)^{2} \end{bmatrix} = -8$$   Clearly, calculating the determinant, with the matrix as it is, wasn't the right way. The calculations went on and on. But I couldn't think of any other way to solve it. Is there any way to simplify $A$, so as to calculate the determinant? Attributive nouns vs. of-genitive I would be very grateful for some advice on how to decide whether to use an attributive noun or the "of genitive" (periphrastic genitive).  It seems to me that an attributive noun is generally preferable, except when it leads to ambiguity, often due to the combination of several modifiers with mixed dependent relationships.  Additionally, the of-genitive, when used stand-alone, seems to impart a more formal tone, a higher register.  Simple examples (preferred first, less-preferred second):  The department secretary vs. the secretary of the department.  But:  The excellent secretary of the department vs. the excellent department secretary.  Any additional guidelines would be most welcome. Whenever my cheese gets too old and has a bit of blue/white mold on the side, I'm not sure whether I should throw it completely out or not. Is it safe to eat if I cut off the edge? Should I cut a bit off of all the sides just to be safe?  I'm particularly interested in the answer for cheddar cheese, but answers about other cheese are welcome. I've been modeling a hermit crab and I don't know why but some areas of the mesh are really dark on some spots, like the legs and the claws.   I hope you can help me fix it.