Here's an expression I am struggling to evaluate: $$\LARGE   {\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\:\cdot^{\:\cdot^{\:\cdot}}}}}} $$  The value turns out be $2$, but I don't understand how do we get it. Can anyone give the solution?  EDIT: The original problem is as follows:   If $y(x)= { x }^{ { x }^{ { x }^{ { x }^{.  } } } }$, then evaluate $y(\sqrt { 2 })$.
Find $x$ in   $$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$   A trick to solve this is to see that $$\large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} \quad\implies\quad 2 = x^{\Big(x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}\Big)} = x^2 \quad\implies\quad x = \pm \sqrt{2} $$  Are these solutions correct? If not, why? If yes, are there other solutions?    PS: An extension of this discussion can be found in
Prove that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides Prove,by vector method,that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides.    My Attempt: Let the trapezium be $OABC$ and that the O is a origin and the position vectors of $A,B,C$ be $\vec{a},\vec{b},\vec{c}$.Then the equation of $OB$ diagonal is $\vec{r}=\vec{0}+\lambda \vec{b}................(1)$ And equation of $AC$ diagonal is $\vec{r}=\vec{a}+\mu(\vec{c}-\vec{a}).......(2)$ And the equation of the line joining the mid points of $OA$ i.e.$\frac{\vec{a}}{2}$ and $BC$ i.e. $\frac{\vec{b}+\vec{c}}{2}$ is $\vec{r}=\frac{\vec{a}}{2}+t(\frac{\vec{b}+\vec{c}}{2}-\frac{\vec{a}}{2}).........(3)$. Here $\lambda,\mu,t$ are scalars. I do not know how to solve $(1)$ and $(2)$ and put into $(3)$ to prove the desired result. Please help me.Thanks. What is god for religious people? In order to discuss with religious friends about god, I would like to know if there's some accepted objective definitions of God for main religions, even if belief is subjective.   For example, is he "a person who..."? In that case I would address features of a person, as capabilities and limitations. Is he "an entity..."? In that case I would use systems theory.   I feel lost on this, being agnostic from a religious family, where I was taught that God is a mess of things, including a beautiful bearded face, omnipotence and a mix of human defects and metaphysical powers. "contribute to investigating" or "contribute to investigate" I'm trying to find, which of the following sentences is correct?  A) We try to highlight how they contribute to investigate their importance. B) We try to highlight how they contribute to investigating their importance.  I found that both cases were used frequently in formal writing. Edited: Sources for each case: For A: &quot;Their multifaceted nature means that there are very many routes to investigating their importance for understanding social change&quot; in &quot;A lineage for contemporary inter-city studies&quot; book. B: &quot;More factors are analysed to investigate their importance in scoring from shots and shot probabilities&quot; in &quot;Applications of logistic regression to shots at goal at association football&quot; book. I have a series of ranges with start dates and end dates. I want to check to see if a date is within that range.  Date.before() and Date.after() seem to be a little awkward to use. What I really need is something like this pseudocode:  boolean isWithinRange(Date testDate) {     return testDate &gt;= startDate &amp;&amp; testDate &lt;= endDate; }   Not sure if it's relevant, but the dates I'm pulling from the database have timestamps. Where should I put my bash scripts Why can't light escape from a classical black hole? How do I stop Windows 10 from updating my graphics driver? Every time Windows 10 runs updates (which I see no way to control), it updates my graphics driver.  I like the old one because it is faster.  So, I have to reinstall it.  This is getting to be a pain.  Is there any way to stop Windows 10 from updating it again? I want to install the latest Python tarball on Ubuntu, downloaded from .  Is this is a correct way to install?  ./configure make make install   If not, how do I do that? I've just started playing with Guice, and a use-case I can think of is that in a test I just want to override a single binding.  I think I'd like to use the rest of the production level bindings to ensure everything is setup correctly and to avoid duplication.  So imagine I have the following Module  public class ProductionModule implements Module {     public void configure(Binder binder) {         binder.bind(InterfaceA.class).to(ConcreteA.class);         binder.bind(InterfaceB.class).to(ConcreteB.class);         binder.bind(InterfaceC.class).to(ConcreteC.class);     } }   And in my test I only want to override InterfaceC, while keeping InterfaceA and InterfaceB in tact, so I'd want something like:  Module testModule = new Module() {     public void configure(Binder binder) {         binder.bind(InterfaceC.class).to(MockC.class);     } }; Guice.createInjector(new ProductionModule(), testModule);   I've also tried the following, with no luck:  Module testModule = new ProductionModule() {     public void configure(Binder binder) {         super.configure(binder);         binder.bind(InterfaceC.class).to(MockC.class);     } }; Guice.createInjector(testModule);   Does anyone know if it's possible to do what I want or am I completely barking up the wrong tree??  --- Follow up: It would seem I can achieve what I want if I make use of the @ImplementedBy tag on the interface and then just provide a binding in the test case, which works nicely when there is a 1-1 mapping between the interface and implementation.  Also, after discussing this with a colleague it would seem we'd head down the road of overriding an entire module and ensuring we have our modules defined correctly.  This seems like it might cause a problem though where a binding is misplaced in a module and needs to be moved, thus possibly breaking a load of tests as bindings may no longer be available to be overriden. Polynomial irreducible - maximal ideal I have a couple of ideals which I wonder if I correctly classify as maximal/prime ideal.  $I_1 = \langle 2x^2 + 9x -3\rangle$, $I_2 = \langle x - 1\rangle$  $\mathbf 1)$ Is $I_1$ a maximal ideal in $\mathbb{Q}[x]$?  Yes, since $I_1$ is irreducible with $p=3$ using Eisenstein's criterion, thus maximal ideal.  $\mathbf 2)$ Is $I_2$ a prime ideal in $\mathbb{Q}[x]$?  Yes, since $I_2$ is obviously irreducible, and thus a maximal ideal, and every maximal ideal is a prime ideal.  $\mathbf 3)$ Is $I_2$ a maximal ideal in $\mathbb{Z}[x]$?  Yes, $I_2$ is obviously irreducible, and thus a maximal ideal.  $\mathbf{Edit:}$ No, as it is not a field. $$ $$  Am I right in my conclusions?  Appreciate any help. When is "EDIT"/"UPDATE" appropriate in a post? I often see questions edited into this form:     Lorem ipsum dolor sit amet blah blah blah paragraph paragraph original question.            UPDATE: The Lorem disappeared and left this message: "new information".   I find this format unaesthetic (since Stack Exchange posts are supposed to be useful in the long term and update boundaries are artifacts of the creation rather than the content) and feel that new information should, whenever possible, be incorporated into the original text “as if it had always been that way” if practical, and pointing out an update should be done mostly in comments.  Is this an appropriate writing style, and if so under what circumstances? If not, should it be removed or discouraged? Broken blog link on pages with lists of moderators The page  has a link like this:  which doesn't seem to exist anymore. This is the same on  And also on Why is variance squared? Possible Duplicate:          Consider the field extension $\mathbb{Q}(\sqrt2)$. I want to show that $\sqrt5 \notin \mathbb{Q}(\sqrt2)$. If this were not the case, then we could write $\sqrt5 = a + b\sqrt2$ for $a,b\in\mathbb{Q}$. However, I do not see the contradiction here. Is there a better/easier way to prove this? What is CUPS server and how to share a printer (Locally or over a network) The title tells you all, except that it's about sharing printers.   How do I connect to a shared printer on the network through CUPS or terminal? There are numerous (, , ) questions concerning this problem. I realize this is a borderline duplicate, but I'll re-iterate the futility of this particular filter and why it should be outright removed from Stack Overflow.  Do you know how effective this is in preventing users who just don't care from posting questions with bad titles? 100% ineffective. What do these users do?   Replace it with a synonym: "issue," "trouble," "predicament," "botheration" or what have you. Or worse, they simply work around the filter by deliberately misspelling the word or inserting space or punctuation:    () ?    and the coup de grâce:      I mean, even Jeff . This adds zero value to the title. In fact, it looks worse. Why, oh why are people  to scrap a bad feature? All this does is:   Prevent people who have a  to use the word "problem" in the title from posting questions with an appropriate title. Break the search feature because of workarounds or the need to use less descriptive titles than the error itself   This is an extraordinarily useless (and harmful) feature. Please, for the love of Cthulhu,  remove, eliminate, abolish, eradicate, extirpate this "feature" from Stack Overflow.    Bonus:     So I can't mention the Halting Problem in the title? – Keith Thompson Nov 1 at 19:05      @KeithThompson Maybe it's a "Halting Issue". Or "Need help with Halting". Or the "Halting thing in Computability Theory". – slhck Nov 1 at 19:14 Suppose $ T \subset \mathbb{C} $. Show that the corresponding set $ S \subset \Sigma $ is  a. a circle if $ T $ is a circle. b. a circle minus (0, 0, 1) if $ T $ is a line.  Here we are defining $ \Sigma $ to be the Riemann sphere, given by the set: $$ \Sigma = \left \{(\xi, \eta, \zeta) : \xi^{2} + \eta^{2} + (\zeta - \frac{1}{2})^{2} = \frac{1}{4} \right \} $$  To take a point from $ \mathbb{C} $ to $ \Sigma $ we can use the following:  $$ \xi = \frac{x}{x^{2} + y^{2} + 1}; \eta = \frac{y}{x^{2} + y^{2} + 1}; \zeta = \frac{x^{2} + y^{2}}{x^{2} + y^{2} + 1} $$  We define a circle on $ \Sigma $ to be the intersection of a plane of the form $ A\xi + B\eta + C\zeta = D $ with $ \Sigma $. We also know the converse of this problem is true, that the intersection above yeilds a set in $ \mathbb{C} $ with the following property:  $ (C - D)(x^{2} + y^{2}) + Ax + By = D $. As you can see, when C = D, then an equation for a line is yeilded, otherwise it is a circle.   I really am at a  loss about how to solve this problem. The only thing I can think to do is to pick 3 points on a circle or radius $ r $ with center $ z_{0} $, use these points to find two vectors in $ \Sigma $, take their cross product to get a normal vector, use this normal vector to get a plane. Once I have the plane in form $ A\xi + B\eta + C\zeta = D $ then I could prove that the circle I had chosen corresponds exactly with $ (C - D)(x^{2} + y^{2}) + Ax + By = D $. Is there not an easier, less computation way to do this? SHOWPLAN permission denied when including execution plan in Data Explorer If I select the "Include execution plan" checkbox in the Data Explorer, the query fails:     SHOWPLAN permission denied in database 'StackOverflow'.     I see that  following maintenance. I don't know whether there was any recent maintenance. It's been several days since the data was updated and longer since an update was deployed, if that's related. Asking distant cousins for DNA? In my family we have a DNA contradiction – two cousins (at about the 4th cousin level) took a Y-DNA test, but did not match and had two completely different haplogroups.   Since then, I have traced living cousins with the same surname – the only ones left living that I am aware of – and I would love to pay to have them tested to see if we can figure out where the non-paternity event (NPE) or other event happened (i.e. this new cousin could help us determine which line the NPE occurred in depending on which cousin he matches).   Does anyone have a sample letter, or know the best way to go about asking a distant cousin for DNA?   I have their address and am planning to write a letter, but I can't imagine receiving such a letter, so I am not quite sure how to ask without scaring them off. Euclidean distance vs Squared