This is the question I'm referring to from the title &quot;Martha and Janet are making cupcakes for a party. They have a 3x3 board to arrange them on and they have 3 indistinguishable blue cupcakes, 3 indistinguishable red cupcakes, and 3 indistinguishable orange cupcakes. In how many ways can they arrange these cupcakes such that no two cupcakes of the same color are adjacent horizontally or vertically.&quot; I was able to get the answer by just writing it out by hand and counting up the total number of arrangements, but wanted to know if there was any shortcut that would get the answer way faster.
I need some help with the exercise I was assigned in math class. We were given a 3x3 square made of unit squares, and we wanted to paint all the squares. We had 3 sets of paint, red, blue, and green. Each paint was distinct from each other, but two unit squares with the same color were the same. Additionally, there's only enough paint of each color to color 3 unit squares, and all the paint has to be used. The goal of the exercise was to find the total amount of distinct possible ways such that no two unit squares that share an edge (that is to say they are next to each other vertically or horizontally, having same color across a diagonal is fine). I've started by trying to take a look at each possible arrangement of the bottom 3 unit squares, but I can't seem to find a pattern between it. Is there a better way than continuing by brute force? Any ideas on how I should proceed from here?
Google map only loading half of itself What is the difference between ++i and i++? I have really no idea about this:  Problem: Show that there exists a function $f:[0,1]\rightarrow\mathbb{R}$ such that:  $f$ is discontinuous in all $x\in \mathbb Q$. $f$ is increasing in $[0,1]$. $f$ is integrable.   EDIT: Sorry, it is not discontinuous in all $x\in \mathbb R \setminus \mathbb Q$, just in $\mathbb Q$. Does $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mn}^3=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}a_{mn}^3$ occur here? Let $\{a_{mn}\},~m,n \in \Bbb{N}$, be an arbitrary double sequence of real numbers then does the following equality occur always?   $$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mn}^3=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}a_{mn}^3$$   I think there is no reason for the equality of this iterated series in general but I cannot figure out how to construct a counter example. Can anyone please help me how to create a example of $\{a_{mn}\}$ such that the given equality does not hold...!! Positioning of decorations on parallel lines I'm displayed three lines on a pgfplot. I want to show a marker for the slope of each line and the difference in height between lines.     I'm showing the slopes by adding triangles under the lines  (at a manual position) and placing an annotation inside the triangle (at a manual position). I'm showing the difference between the height of the lines by adding a brace manually as well.   How can I place the braces on the plot and anchor them at a certain point on each line? The triangles can be added at the end of each line declaration, but doing it like this is ok. The braces are put in place manually, which is not sustainable as I create more plots.   I want to avoid using xshift, yshiftand manual adjustment to put the braces,traingles and annotations in the right place. How can I do this?      \documentclass{article} \usepackage{pgfplots}  \pgfplotsset{compat=1.9}  \begin{document}   \begin{tikzpicture}[scale=0.45,trim axis left, trim axis right]     \begin{axis}[ylabel={$\hat{S(4)}$},xlabel={$S{^*}_{breakeven}$},ylabel style={rotate=-90},domain=1:25, legend style={legend pos=north west,font=\scriptsize}, legend cell align=left]  \addplot+[samples=200,mark=none]                                    {6.26 + 0.28*x} coordinate[pos=0.5](x) coordinate[pos=0.9] (y) (x)-| (y) node [midway,xshift=-0.6cm,yshift=0.3cm] {\footnotesize $Slope X$};;  \addplot+[samples=200,mark=none, dotted, thick]             {2.66 + 0.28*x}  coordinate[pos=0.3](a) coordinate[pos=0.6] (b) (a)-| (b) node [midway,xshift=-0.5cm,yshift=0.2cm] {\footnotesize $Slope 1$};  \draw [decorate,decoration={brace,amplitude=5pt,raise=4pt},xshift=4pt]  (225,107) -- (225,73)  node [midway,xshift=0.6cm] {\footnotesize $\beta_1$};  \end{axis} \end{tikzpicture} \end{document} This one-liner removes duplicate lines from text input without pre-sorting.  For example:  $ cat &gt;f q w e w r $ awk '!a[$0]++' &lt;f q w e r $    The original code I have found on the internets read:  awk '!_[$0]++'  This was even more perplexing to me as I took _ to have a special meaning in awk, like in Perl, but it turned out to be just a name of an array.  Now, I understand the logic behind the one-liner:  each input line is used as a key in a hash array, thus, upon completion, the hash contains unique lines in the order of arrival.  What I would like to learn is how exactly this notation is interpreted by awk. E.g. what the bang sign (!) means and the other elements of this code snippet.  How does it work? Highlight a word with jQuery I basically need to highlight a particular word in a block of text. For example, pretend I wanted to highlight the word &quot;dolor&quot; in this text: &lt;p&gt;     Lorem ipsum dolor sit amet, consectetuer adipiscing elit. &lt;/p&gt; &lt;p&gt;     Quisque bibendum sem ut lacus. Integer dolor ullamcorper libero.     Aliquam rhoncus eros at augue. Suspendisse vitae mauris. &lt;/p&gt;  How do I convert the above to something like this: &lt;p&gt;     Lorem ipsum &lt;span class=&quot;myClass&quot;&gt;dolor&lt;/span&gt; sit amet, consectetuer adipiscing elit. &lt;/p&gt; &lt;p&gt;     Quisque bibendum sem ut lacus. Integer &lt;span class=&quot;myClass&quot;&gt;dolor&lt;/span&gt; ullamcorper     libero. Aliquam rhoncus eros at augue. Suspendisse vitae mauris. &lt;/p&gt;  Is this possible with jQuery? Edit: As  , this is quite possible without jQuery - but I was hoping there might be a special method of jQuery which would let you do selectors on the text itself. I'm already using jQuery heavily on this site, so keeping everything wrapped up in jQuery would make things perhaps a bit more tidy. How do I update my graphics drivers? I am using a laptop and ubuntu 11.04. I would like the easiest way possible because I am new to linux.  here is my display information:  *-display              description: VGA compatible controller              product: Core Processor Integrated Graphics Controller              vendor: Intel Corporation              physical id: 2              bus info: pci@0000:00:02.0              version: 18              width: 64 bits              clock: 33MHz              capabilities: msi pm vga_controller bus_master cap_list rom              configuration: driver=i915 latency=0              resources: irq:41 memory:d0000000-d03fffff memory:c0000000-cfffffff ioport:e080(size=8)         *-communication UNCLAIMED              description: Communication controller How to Expand , Resize "extended partition"? iOS Calendar Access Permission Dialog, force it to appear? Smoke Generator does not work When I try to run the smoke simulator in Blender 2.62 I get no smoke at all. When I attempt to render, the only output is a black screen. What am I doing wrong here that is making the smoke simulation not render?  Is it a mistake I made, or a problem with Blender version 2.62? What is batch size in neural network? I'm using Python Keras package for neural network. This is the . Is batch_size equals to number of test samples? From Wikipedia we have  information:     However, in other cases, evaluating the sum-gradient may require   expensive evaluations of the gradients from all summand functions.   When the training set is enormous and no simple formulas exist,   evaluating the sums of gradients becomes very expensive, because   evaluating the gradient requires evaluating all the summand functions'   gradients. To economize on the computational cost at every iteration,   stochastic gradient descent samples a subset of summand functions at   every step. This is very effective in the case of large-scale machine   learning problems.   Above information is describing test data? Is this same as batch_size in keras (Number of samples per gradient update)? For many of my datasets I have the same imported data duplicated numerous times, each iteration displaying the data values for a different column (in my case it's fish catch weight per gridded cell per year, so lots of iterations of the same grid, with a different pattern of values per year).  For presentation purposes, it strikes me as helpful to use the same scale for each year, so I go to the original data and divide the maximum value into (arbitrarily) 10 classes and manually input the values and labels after setting the desired colour ramp and input data field:   This is extremely tedious, not just the manual excel calculation part, but also the input interface for the values &amp; labels*, hence my intention to make use of the 'save style' option. HOWEVER: when I save the style as QML and load it for the next layer, it saves everything - including the data field (in my case, year). When I change this field, QGis automatically generates the values &amp; labels based on the data for that field, ditching all of the values &amp; labels I set.   Is there any way to avoid this happening &amp; thereby duplicate class number, values and labels for each field by simply loading a style? (somewhat meta) Am I going around this the wrong way, OR, is there a different face I could use to climb this mountain? I am developing firefox extension and wanted to count total number of text box of webpage so how to count text box? Spring Data JPA Update @Query not updating? CSS two divs next to each other I want to put two &lt;div&gt;s next to each other. The right &lt;div&gt; is about 200px; and the left &lt;div&gt; must fill up the rest of the screen width? How can I do this? When a user without editing privileges suggests an edit to the post, a banner gets displayed until it is reviewed. Can the banner also provide a link to the Suggested edit? For example, I made an edit to a post on Unix&amp;Linux.SE. Can the banner also link to my ?     I know I can access the link through my Profile → Activity → Suggestions. But linking it directly on the face of the post would have immense benefits:  I have been a prolific editor on Ask Ubuntu (my first encounter with Stack Exchange Network) with a lot of Rep derived through Suggestions. I didn't get to know about these suggested edit links until very late (I guess I already had got about 400 edit suggestions approved by the time this came to my knowledge).  This means I didn't know about  when it mattered the most.  I never got to know why my suggestions got rejected because of my ignorance.  This would help reduce the barrier between Reviewers and Editors.  This would also help in case of Anonymous edits. Right now, there is no way an anonymous editor can access it unless the edit gets approved and he comes back to go through the Revision page of that particular post in order to access the Suggested Edit Review link. If the edit gets rejected - it's the end of the road for that anonymous editor. I just made an anonymous edit on one of the SE sites where I don't have an account and already feel like it's the end of the road for me since I just can't access the review link unless it gets approved.    Possibly also expand the banner to mention that they can also access the link to the suggestion via Profile → Activity → Suggestions. Where can I ask about Software Licenses/Licensing? How to simplify the summation kk! without using induction? Connect to Linux by name rather than IP