Find all $z$ such that $$(z+1)^6 = z^6.$$  I have moved $z + 1$ to the other side such that $$\left(\frac{z}{z+1}\right)^6 = 1,$$ which then makes it so   $$\frac{z}{z+1} = 1 = \frac{z}{z+1} = e^{ik\pi/3}.$$  I know $k \neq 0$ where $k = 0, 1, 2,\dotsc ,5$, but I don't know where to go from here.
Find all $x$ such that $$x^6=(x+1)^6.$$   So far, I have found the real solution $x= -\frac{1}{2}$, and the complex solution $x = -\sqrt[3]{-1}$.  Are there more, and if so what/how would be the most efficient find all the solutions to this problem? I am struggling to find the rest of the solutions.
Javascript - Change html tag type Measuring draw time of layer in ArcMap? I'd like to quantitatively measure the display performance of mosaic datasets, image service layers, raster layers, feature layers, etc. in ArcMap 10.  How can I measure the draw time of a particular layer (or the complete drawing process if that is the only option) in ArcMap? What are the interfaces, objects and events involved? Is the  class in .NET suitable for accurately measuring this type of operation?  Has anyone already created an ArcMap add-in or extension for this purpose? How to calculate a confidence interval for estimating a proportion when the population is very small? I plan to execute a little study on whether dentists tend to grant privately insured patients (PRIV) a sooner appointment compared with publicly insured patients (PUB). I want to restrict this study to the dentists of a single town in Germany.  In this town there are about 500 dental practices and I am willing to call maximum 100 practices (twice, once as a PUB and once as PRIV). (A sample without replacement of size up to 100)  So the result of this examination would be a list of up to 100 practices with an associated value of 1 - if the appointment for the PRIV was more than two days sooner - and a 0 - if not. The proportion of 1s is what I want to estimate.  Now the sources I have been reading always assumed a much larger population with a finite population correction close to 1. This is why I doubt that blindly using a conservative or approximate confidence interval would lead to a statistically meaningful result.   So my first question is, of course, whether it is reasonable at all trying to estimate the proportion using a confidence interval in this case? And if yes, then what statistical technology/method (leading to a CI) would be a good choice for this use case?   One idea that comes to mind is using a sample size of 25 f.x., in that case the FPC would be 0.95 - "close enough" to 1. But it seems stupid trying to improve the quality of an estimation by reducing the information used for the estimation. Using a Sprite image with input How to filter a dictionary according to an arbitrary condition function? I’m trying to solve a logic puzzle that goes like this:     The police have three suspects for the murder of Mr. Cooper: Mr. Smith, Mr. Jones, and Mr. Williams. Smith, Jones, and Williams each declare that they did not kill Cooper. Smith also states that Cooper was a friend of Jones and that Williams disliked him. Jones also states that he did not know Cooper and that he was out of town the day Cooper was killed. Williams also states that he saw   both Smith and Jones with Cooper the day of the killing and that either Smith or Jones must have killed him. Can you determine who the murderer was if one of the three men is guilty, the two innocent men are telling the truth, but the statements of the guilty man may or may not be true?   When i was young, i used to solve such puzzles graphically, drawing vertices and arrows and excluding impossible cases. But now i want to do that formally. How can i express this puzzle definition using proposition calculus notation, inference and truth tables? Or, in other words, when you stumble upon a logical puzzle and you want to solve it formally, what is your algorithm?  What i tried. I extracted several propositions from the definition:   S: didn't kill C S: J knew C S: W disliked C J: didn't kill C J: J didn't know C J: out of town W: didn't kill C W: S with C W: J with C   There are 3 groups of propositions, two of them are always true, the third may contain either true or false statements. Let be $A = 1 \land 2 \land 3$, $B = 4 \land 5 \land 6$, $C = 7 \land 8 \land 9$. Then from  by using truth tables and Karnaugh maps i deduct that the statement “some two of A, B, C are true” looks like $(A \land B \land \neg C) \lor (A \land \neg B \land C) \lor (\neg A \land B \land C)$. But wait, “may or may not be true” means that all statements might be true, not that at least one is false (hence the NAND is not appropriate here). So the sentence should be reformulated as “at least some two of A, B, C are true”. Am i going in the right direction? I need help with this exercise. What I've done so far is prove the exercise when $n=1$. So:  $$n=1$$ $$1!\ge2^{(1-1)}$$ $$1\ge2^0$$ $$1\ge1$$ Which is true  Therefore, now that I assume that the assumption is correct, I want to prove that with $n+1$, it will also be true. So, what I've done now is:  $$P(n) \implies P(n+1) $$ $$n!+n+1\ge 2^{(n-1)}...$$ And my  problem is that I do not know how to add the $n+1$ in the right side of the equation, therefore I'm not been able to finish the exercise.  Thank you so much for your help. If something's not very clear, please let me know. I'll try to be clearer next time :) Why do bridge non-Captain officers commonly go on away missions? Ever since the days of Kirk, Spock, and McCoy, bridge officers in Starfleet have always had a prominent role in away team missions.  However, it would seem more logical to me that such hazardous (and, generally, specialized) duties should fall to lower-ranking and/or more dedicated teams.  Of course, it wouldn't have made for great TV if every away team was comprised of a bunch of red-shirts.  And it would definitely increase the show's budget to have an extra group of main characters whose sole purpose was to handle the away missions.  Still, I'm curious to know if there's ever been a real in-universe explanation for this practice?  I know it's been brought up a few times, in a few series/movies, that the Captain should not be taking part in hazardous away team missions.  But nothing is ever said (that I'm aware of) of the rest of the still valuable bridge crew or other senior officers. I am planning an hour and a half ride with my 1 month old baby girl. I heard that being in the car seat for a long time is not good for newborns.  Is an hour and a half too long? Should we do a stop in the middle of the trip? I'm new in this computer terms. So anyone can explain to me what "virtual cores" and "hyperthreading" are at greatest detail a layman can understand?  People told me that virtual cores are imaginary cores acting like a physical core. If that is so, then can I make my AMD v105 processor single core a dual core processor?   Is it also possible to hyperthread my CPU? Are these functions built in? When I signed up I received a welcome message specifically indicating I should take the tour to earn my first badge:     Clicking the message takes me directly to the .  However, as   , the 'Informed' badge is not active here on meta.  Shouldn't the wording of the welcome message be altered if a user signs up on a site which does not use this badge? A man condemned to serve his sentence in other times Should we sign-then-encrypt, or encrypt-then-sign? Frequently, we want to send messages that are (a) encrypted, so passive attackers can't discover the plaintext of the message, and (b) signed with a private-key digital signature, so active attackers can't make Alice think that a message came from Bob when it didn't. Is it better to (a) generate the digital signature from the (hashed) plaintext, and then encrypt a file containing both the plaintext message and the digital signature? Or is it better to (b) encrypt the message first, and then generate a digital signature from the (hashed) encrypted file? Or (c) combine encryption and public-key digital signatures in some other way? A closely related earlier question (  ) seems to focus on symmetric-key MAC authentication. As , Do the same issues with (symmetric-key) MAC-then-encrypt apply to (public-key) sign-then-encrypt? How to use resistors? I have a power supply of 5 volts @ 1A. I want to reduce the current from 1A to 250mA. Which resistor should be use?   I tried this equation(Ohm's Law):  R = V / I  = 5/(0.75)  = 6.66 Ohms.  I tried the 6.66 Ohm resistor but it only reduce few mili Amps. Getting a collection of index values using a LINQ query Is there a better way to do this?  string[] s = {"zero", "one", "two", "three", "four", "five"};  var x =  s .Select((a,i) =&gt; new {Value = a, Index = i}) .Where(b =&gt; b.Value.StartsWith("t")) .Select(c =&gt; c.Index);   i.e. I'm looking for a more efficient or more elegant way to get the positions of the items matching the criteria. arXiv is smart enough to detect whether your PDF is generated by LaTeX. If it is, arXiv asks for the sources of the PDF instead of the PDF file itself. This is really annoying and inconvenient.  I ask a way to fool the arXiv and avoid PDFs being detected as generated by LaTeX (even if they were indeed).  I have tried this: : "create a pdf from word and then simply insert pages from your article, and delete the word's page".  And it did not work. How do I enable the "Universe" repository? Inhumans vs. Mutants in the Marvel Universe What's the difference between those two? Aren't Inhumans supposed to be an evolution of the human race "forced" by the Kree? Which is almost the same as Mutants, the natural evolution of the human race...  What exactly am I missing? When did The Doctor start running so much? Thanks to Netflix Streaming, I've been watching some random classic Doctor Who stories.  One thing that strikes me as different between the modern Who and old Who was that in modern Who, they do an awful lot of running.  It's become a sort of running gag that they've referenced in-universe.  In the classic Who I've seen, he tends to drive around quite a bit instead.  In addition, The Doctor in modern Who is incredibly reliant on his sonic screwdriver, which I often don't see at all in classic Who stories.  Did the Doctor running and the sonic screwdriver get introduced over time in classic Who, or were they both added as part of the modern Who revival? "The resource cannot be found." error when there is a "dot" at the end of the url