Let us say we have a mighty man crossing a bridge, carrying 4 bags of concrete, each of which weighs 50 pounds.    Let us say, for the sake of the argument, that the mighty man himself weighs 300 pounds.  The support capacity of the bridge $x$ is less than the 500 pounds they weigh together - therefore, the mighty man cannot carry the concrete in a stack across the bridge - it will fall.  However, the mighty man does have the ability to juggle all the concrete bags so that he is only in contact with one of them at a time.    Is there such a value of $x$ possible that he can cross the bridge juggling all four bags of concrete at once, or will the bridge necessarily exceed its support capacity?
A friend offered me a brain teaser to which the solution involves a $195$ pound man juggling two $3$-pound balls to traverse a bridge having a maximum capacity of only $200$ pounds. He explained that since the man only ever holds one $3$-pound object at a time, the maximum combined weight at any given moment is only $195 + 3=198$ pounds, and the bridge would hold.  I corrected him by explaining that the acts of throwing up and catching the ball temporarily make you 'heavier' (an additional force is exerted by the ball to me and by me onto the bridge due to the change in momentum when throwing up or catching the ball), but admitted that gentle tosses/catches (less acceleration) might offer a situation in which the force on the bridge never reaches the combined weight of the man and both balls.  Can the bridge withstand the man and his balls?
This is a very simple question but I believe it's nontrivial.  I would like to know if the following is true:      If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and $S$ are isomorphic.    Thanks!  If there isn't a proof (or disproof) of the general result, I would be interested to know if there are particular cases when this claim is true. Prove that $g$ is tangent to the graph of $f$ So I had these two functions and the following exercise:  $f(x)=x^3-2x^2$ $g_p(x)=px$ Prove that $g_{-1}$ is a tangent line to the graph of $f$. How many points do $g_{-1}$ and $f(x)$ have in common?  So yeah, the points in common are fairly simple to figure out:  Points in common: $x=1$ and $x=0$  So to see if it's actually tangent to the graph, I took the derivative of both functions at the points where they meet. The result:  $f'(x)=3x^2-4x$ $g'_{-1}(x)=-1$  So $f'(1)=-1$ and $f'(0)=0$ So I'm not sure why I get this answer? By the question, we are supposed to prove that its tangent to the graph of $f$ but by my calculation, it seems like only $f(1)$ is tangent with $g_{-1}(1)$. Help is appreciated! (Please avoid hint answers). What is a "static" function in C? The question was about plain  functions, not  static methods, as clarified in comments.  I understand what a static variable is, but what is a static function?  And why is it that if I declare a function, let's say void print_matrix, in let's say a.c (WITHOUT a.h) and include "a.c" -  I get "print_matrix@@....) already defined in a.obj", BUT if I declare it as static void print_matrix then it compiles?  UPDATE Just to clear things up - I know that including .c is bad, as many of you pointed out. I just do it to temporarily clear space in main.c until I have a better idea of how to group all those functions into proper .h and .c files. Just a temporary, quick solution. RX 590 IN Blender and Cycles Render I don't know why, but I cant render anything in CUDA and OptiX but OpenCL work fine. I cant use Cycles render because it's very laggy. I have newest  version of Blender and an RX 590 Red Dragon 8gb. Why does that happen?  How can I turn on my gpu? Suppose $\sqrt6 = \frac pq$ where $p$ and $q$ have no common factors.  $$6 = \frac {p^2}{q^2}$$  $$6p^2 = q^2$$  So $q^2$ and therefore $q$ is divisible by $6$.   $$p^2 = \frac {q^2}{6}$$  So $p^2$ and therefore $p$ is divisible by $6$.   S0, $p$ and $q$ have a common factor $6$. Contradiction.  Therefore $\sqrt6$ is irrational.  Does the proof work? UTF-8 -> ASCII in C language Row colour gaps in tabularx with \aboverulesep and \{} \documentclass[]{article} \usepackage[table,dvipsnames]{xcolor} \usepackage{multicol,booktabs,tabularx}  % Table settings \renewcommand{\aboverulesep}{1pt} \renewcommand{\belowrulesep}{1pt}  \begin{document}     \begin{tabularx}{\textwidth}{@{}X l@{}}         This is Header 1 &amp; This is Header 2 \\         \toprule         \rowcolor{Apricot}         This is Text 1 &amp; This is Text 2 \\\midrule         This is Text 1 &amp; This is Text 2 \\\midrule         This is Text 1 &amp; This is Text 2 \\         \bottomrule     \end{tabularx} \end{document}   I have the above table MWE using tabularx, and I am trying to colour the row from the table. However, due to the use of my \aboverulesep and \belowrulesep, I now have gaps above and below my table row that is not coloured.   Also, because of the use of my @{} on either side of the table to 'remove' the excess padding on the sides of the table, the row colour is being applied in those padding as well.   Qn 1: How can I colour the aforementioned gaps (arising due to the \aboverulesep) in, while maintaining the rule separation?   Qn 2: How to remove the colour from the two sides of the table?     EDIT  This is in response to Zarko's answer. I want to maintain the use of @{} on either side to remove the padding. If I were to adapt this into your answer, like so:  % @Zarko's answer \documentclass{article} \usepackage[table,dvipsnames]{xcolor} \usepackage{booktabs, cellspace, tabularx}  % Table settings \renewcommand{\aboverulesep}{0pt} \renewcommand{\belowrulesep}{0pt} \setlength\cellspacetoplimit{5pt} \setlength\cellspacebottomlimit{5pt}  \begin{document}     \begin{tabularx}{\textwidth}{@{}SX Sl@{}}% &lt;-- S is append for activate additional vertical space          This is Header 1 &amp; This is Header 2 \\         \toprule         \rowcolor{Apricot}         This is Text 1 &amp; This is Text 2 \\\midrule         This is Text 1 &amp; This is Text 2 \\\midrule         This is Text 1 &amp; This is Text 2 \\         \bottomrule     \end{tabularx} \end{document}   I obtain this:   and so my second question (Qn2) is still unresolved.. For Qn 1, I prefer to use the solution by @Skillmon with \renewcommand{\arraystretch}{1.15} as I need not add a new package. Why do votes get locked? I downvoted  and now really can't say why, so I wanted to remove my downvote. However, I get the message:  You last voted on this question yesterday Your vote is now locked in unless this question is edited  Why is this? I don't think I voted on it multiple times, and even when I did, I can change my mind, can't I? Will I be able to play my Destiny Characters if I buy Legendary Edition (PS3) How do I calculate the closest power of 2 or 10 a number is? What is the most efficient way to cacluate the closest power of a 2 or 10 to another number? e.g.  3.5 would return 4 for power of 2 and 1 for power of 10  123 would return 128 for power of 2 and 100 for power of 10  0.24 would return 0.25 for power of 2 and 0.1 for power of 10  I'm just looking for the algorithm and don't mind the language. Why shouldn't I care what model/make/year my bicycle is? We often get questions on bicycles.stackexchange about identifying a bicycles manufacturer, model, make, and year.  Often these questions are downvoted or closed -- usually with a comment attached to them that it's not necessary to know the exact model/make/year of a bicycle if you want to fix, repair, or ride your bike. Why should owners not care about the model/make/year of their bicycle? Especially when it comes to less expensive bikes (aka BSOs) and BMX bicycles? This is intended as a canonical question that we can point closed questions to. Also see      (generally not helpful for identifying a brand/model) I have a multiboot system set up. The system has three drives. Multiboot is configured with Windows XP, Windows 7, and Ubuntu - all on the first drive. I had a lot of unpartitioned space left on the drive and was reserving it for adding other OSes and for storing files there in the future.  One day I went ahead and downloaded Partition Wizard and created a logical NTFS partition from within Windows 7, still some unpartitioned space left over. Everything worked fine, until I rebooted the computer a few days later.  Now I'm getting:    error: unknown filesystem.   grub rescue&gt;   First of all I was surprised not to find any kind of help command, by trying:    help, ?, man, --help, -h, bash, cmd, etc.  Now I'm stuck with non-bootable system. I have started researching the issue and finding that people usually recommend to boot to a Live CD and fix the issue from there. Is there a way to fix this issue from within grub rescue without the need for Live CD?  UPDATE  By following the steps from , I was able to boot to initramfs prompt. But not anywhere further than that.  So far from reading the manual on , I was able to see my drives and partitions using ls command. For the first hard drive I see the following: (hd0) (hd0,msdos6) (hd0,msdos5) (hd0,msdos2) (hd0,msdos1)  I now know that (hd0,msdos6) contains Linux on it, since ls (hd0,msdos6)/ lists directories. Others will give "error: unknown filesystem."  UPDATE 2  After the following commands I am now getting to the boot menu and can boot into Windows 7 and Ubuntu, but upon reboot I have to repeat these steps.    ls ls (hd0,msdos6)/ set root=(hd0,msdos6) ls / set prefix=(hd0,msdos6)/boot/grub insmod /boot/grub/linux.mod normal   UPDATE 3  Thanks Shashank Singh, with your instructions I have simplified my steps to the following. I have learned from you that I can replace msdos6 with just a 6 and that I can just do insmod normal instead of insmod /boot/grub/linux.mod. Now I just need to figure out how to save this settings from within grub itself, without booting into any OS.  set root=(hd0,6) set prefix=(hd0,6)/boot/grub insmod normal normal   UPDATE 4  Well, it seems like it is a requirement to boot into Linux. After booting into Ubuntu I have performed the following steps described in the :  sudo update-grub sudo grub-install /dev/sda   This did not resolve the issue. I still get the grub rescue prompt. What do I need to do to permanently fix it?  I have also learned that drive numbers as in hd0 need to be translated to drive letters as in /dev/sda for some commands. hd1 would be sdb, hd2 would be sdc, and so on. Partitions listed in grub as (hd0,msdos6) would be translated to /dev/sda6.  UPDATE 5  I could not figure out why the following did not fix grub:  sudo update-grub sudo grub-install /dev/sda   So I downloaded  based on an answer from  post. That seemed to do the trick after I picked the "Recommended Repair (repairs most frequent problems)" option. How do I make Div Container 100% width only on homepage of site? Can't boot without Flash Drive plugged in CPU time out limit was being hit while processing child records on the inline opportunity visualforce page, to handle this i wrote a remoteActionFunction on the page that invokes controller method that's taking long to process and when i get CPU LIMIT exception in the result, i am invoking javascript call back function that invokes a action function which handles the same code asynchronously(@future).  Question: by setting the timeout (Visualforce.remoting.timeout = 120000;) can't we overcome CPU TIME out limit of 10,000 ms. Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$ I'm having issues figuring out how to approach this problem:     Conclude that if $ad-bc = \pm 1$, then   $$\gcd(x,y) = \gcd(ax +by, cx + dy)$$   The fact that $\gcd(x,y) = \gcd(x+ky, y)$ is a very special case of this exercise   I believe there is a property that if $a = bq_1 + 0$ where 0 is r, then b divides a, and $b=\gcd(a,b)$  Is that at all relevant in this case? Let $a,b,c,d,p$, and $q$ be natural numbers such that $ad-bc = 1$ and $\frac{a}{b} &gt; \frac{p}{q} &gt; \frac{c}{d}$.  How prove that  $q \geq b+d$? How to explain the different forms of the Hamilton-Jacobi equation? In Arnold's Mathematical Methods of Classical Mechanics, he derives the Hamilton-Jacobi equation (HJE) using a generating function $S_1(Q, q)$ to get  $$ H\left(\frac{\partial S_1(Q, q)}{\partial q}, q, t \right) ~=~ K(Q, t). $$  However, this is different from what I've seen in other physics texts. For example, Goldstein uses the generating function $S_2(q, P, t)$ to get the equation  $$ H\left(\frac{\partial S_2(q, P, t)}{\partial q}, q, t\right) ~=~ - \frac{\partial S_2(q, P, t)}{\partial t}. $$  Why is there this difference? Are the two equations saying the same thing? How can I remove duplicate values from an array in PHP? Casting a function pointer to another type Let's say I have a function that accepts a void (*)(void*) function pointer for use as a callback:  void do_stuff(void (*callback_fp)(void*), void* callback_arg);   Now, if I have a function like this:  void my_callback_function(struct my_struct* arg);   Can I do this safely?  do_stuff((void (*)(void*)) &amp;my_callback_function, NULL);   I've looked at  and I've looked at some C standards which say you can cast to 'compatible function pointers', but I cannot find a definition of what 'compatible function pointer' means.