We have a $10×10$ grid with white blocks. First we color 9 blocks red, and after that in each step we color a red block, black and a white block, red. We continue this process until we have a black grid with 9 red blocks. Prove that there exits a step in which a black block is adjacent to a white block( two blocks are adjacent when the have a common edge. ) There are always 9 red blocks and at some point 9 red blocks won't be enough to cover all the edges of black blocks. But I don't know how to prove this without explaining a process of coloring.
Given a $10×10$ grid with $9$ red blocks and $91$ white blocks, in each step we color one red block black and after that one white block red until there are $91$ black blocks and $9$ red blocks. Prove that there exits a step in which a black block shares an edge with a white block. I ran a few trials and couldn't get past the fifth step without adjacency. But I don't know how to build a proof.
Why does centering not work with calligra? 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? \centering doesn't seem to center my text I am trying to center this page but it will not work.  All is get is for the first line to be indented about one "tab" space to the right, but the second line will not shift.  However, the first line's indent does not appear to be centered either.  Is it possible that the binding offset or the indentfirst package is interacting with this negatively?  All of the other pages seem to work just fine. Any help would be appreciated.  \documentclass[12pt, a4paper, twoside]{article} \usepackage[margin=1in,bindingoffset=15.5mm,heightrounded]{geometry} \usepackage[T1]{fontenc} \usepackage{indentfirst} \begin{document} \pagestyle{empty} \vspace*{36 pt} {\centering \textbf{\emph{\Huge{Since I Saw You Last,\\There's a Change Upon You}}} } \end{document} Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$ I'm trying to solve  proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following:     Prove that for $n \in \mathbb{N}^{\ge 1}$: $$ \sqrt{n} \le  \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1 $$   My first approach has been to convert the summation into a function in terms of $n$ so that I could find the maximum and minimum values in the interval. However, there doesn't seem to be an easy way to express the summation as a function. Thanks in advance! How do you get \mathbb{1} to work (characteristic function of a set)? Neural networks output probability estimates? Suppose my training data contains ~100 variables, and each example is tagged as "success" or "failure".  I understand how a neural network can be used to try and predict success vs failure based on the variables.  However I am interested in the neural network outputting the posterior probability rather than success or failure. In fact, I evaluate the efficacy of the NN based on how accurate the probabilities are (eg. AUROC over the entire dataset) rather than % of cases with correct prediction.  Are NN's the right tool for the job here and if so how do you structure the NN to output this?  (NOTE: I'm a ML newbie!) Selection of logical connectives {¬,∨,∧,⇒,⇔} in set theory? I am doing an analysis with 2 groups of patients (control and disease) whose blood pressures have been measured continuously for 24 hours. My first goal is to test the effect of GROUP on BP. In the beginning, I built the mixed effect model with TIME as random effect variable and group as fixed effect model. The model I used is listed below:  mod &lt;- lme(BP ~ factor(GROUP), random=~1+ TIME| ID, data=gdata)   However, the experimenter wants to know the effects of GROUP on the slope of blood pressure changing per hour. In another words, we need to have blood pressure slope, which is the change of blood pressure by time, as response variable in the model. And I generated another model as bellow:  mod &lt;- lme(BP ~ factor(GROUP)*TIME, random=~1+ TIME| ID , data=gdata)   However, in the new model, the time works both as random effect variable and fixed effect variable. I am not sure if this is allowed in the mixed effect model. Where is the usage of Ubuntu cloud guest image on OpenStack documented? Find the Galois group of $x^3-5$ over $\mathbb{Q}$. In this case, the roots of $x^3-5$ are $\{\sqrt[3]{5},\omega\sqrt[3]{5},\omega^2\sqrt[3]{5}\}.$ I think $\mathbb{Q}(\sqrt[3]{5},\omega\sqrt[3]{5})$ is the splitting field of $x^3-5.$ Then, $[\mathbb{Q}(\sqrt[3]{5},\omega\sqrt[3]{5}):\mathbb{Q}]=3,$ I think the automorphisms are: $$id$$ $$\sigma:\sigma(\sqrt[3]{5})=-\sqrt[3]{5}$$ $$\tau:\tau(\omega\sqrt[3]{5})=-\omega\sqrt[3]{5}$$ I'm not sure about what I have done. But how to determine the Galois group based on this? And how to determine the solvability of this polynomial? I accidently ran  sudo chmod 755 -R /   instead of  sudo chmod 755 -R ./   I stopped it after few seconds, but now there is some problems such as  sudo: must be setuid root   How can I revert permissions back? Can't run a shell script So I have this script:  #!/bin/bash # Flips the screen (hopefully)  syntax_error=0 orientation=0  current_orientation="$(xrandr -q --verbose | grep 'connected' | egrep -o  '\) (normal|left|inverted|right) \(' | egrep -o '(normal|left|inverted|right)')" case $current_orientation in         normal)                 current_orientation=0         ;;         left)                 current_orientation=1         ;;         inverted)                 current_orientation=2         ;;         right)                 current_orientation=3         ;; esac  if [ $current_orientation -eq 0 ]; then         orientation=2 fi  if [ $current_orientation -eq 2 ]; then         orientation=0 fi method=evdev  # LENOVO S10-3t CHANGE ==&gt; Hard Coded my device number to 11!!!!!!!!  device=11  swap=0 invert_x=0 invert_y=0 real_topx=0 real_topy=0 real_bottomx=4020 real_bottomy=4020  case $orientation in         0)                 swap=0                 invert_x=0                 invert_y=0                 topx=$real_topx                 topy=$real_topy                 bottomx=$real_bottomx                 bottomy=$real_bottomy         ;;         1)                 swap=1                 invert_x=1                 invert_y=0                 topx=$real_topx                 topy=$real_topy                 bottomx=$real_bottomy                 bottomy=$real_bottomx         ;;         2 )                 swap=0                 invert_x=1                 invert_y=1                 topx=$real_topx                 topy=$real_topy                 bottomx=$real_bottomx                 bottomy=$real_bottomy         ;;         3 )                 swap=1                 invert_x=0                 invert_y=1                 topx=$real_topx                 topy=$real_topy                 bottomx=$real_bottomy                 bottomy=$real_bottomx         ;; esac  if [ $method = "evdev" ]; then         xinput set-prop "$device" "Evdev Axes Swap" $swap         xinput set-prop "$device" "Evdev Axes Swap" $swap         xinput set-prop "$device" "Evdev Axis Inversion" $invert_x $invert_y         xinput set-prop "$device" "Evdev Axis Calibration" $topx $bottomx $topy $bottomy         if [ $orientation = 2 ]; then                            xrandr -o inverted         fi         if [ $orientation = 0 ]; then                 xrandr -o normal         fi fi  #   It's for flipping screen in my Lenovo S10-3t. I copied it from the netbook's wiki page, and added the #!/bin/bash at the top. The filename is flipscreen.sh. How can I make it work? Proof that any self-complementary graph has to have $4k$ or $4k+1$ vertex, for some $k \in \mathbb N$ C# regex pattern to extract urls from given string - not full html urls but bare links as well How to enable default file manager(nautilus) to handle desktop icons Combinatorics identity question I encountered this particular combinatorial identity :$$\begin{align}&amp;1\dbinom{n-1}{r-1}+2\dbinom{n-2}{r-1}+\cdots+(n+1-r)\dbinom{r-1}{r-1}\\=&amp;\dbinom{n}{r}+\dbinom{n-1}{r}+\cdots+\dbinom{r}{r}\\=&amp; \dbinom{n+1}{r+1}\end{align}$$How does one figure it out? Novel, group transported to a fantasy world, wild/chaos magic I was reading  and was strongly reminded of a series I remember reading in my teens.  This would have been a series of novels (possibly just one), published in English, and likely before 1980 - I read it in the 90s and it was fairly old, a used book.  I do not think it is the same as the linked question, which is why I'm asking this separately.  It was definitely a fantasy, not sci-fi.  Plot:  At least two girls (and possibly also some boys/other girls) are transported from modern Earth into a fantasy world.  I think they're a group of friends, possibly with some romantic pairings, and are in the 16-25 age range.  The first significant scene I remember is two of the girls ending up in a city, where they are taken in by an older female artist (might have been a different creative type, like a sculptor/actress).  She's lesbian/bisexual.  At some point she attempts to give one of the girls a love potion, but they reverse it on her, so instead the artist falls in love with the girl and accompanies her for the rest of the books.  Twist: It's later revealed that the artist's friends recognized the love potion and removed it.  She pretended to be in love and went along because she thought it would be interesting.  At some point at least one of the girls ends up basically being a magical sex slave/prostitute.  I remember her being tattooed or painted - I think in blue - and also possibly given some magical jewellery/piercings that ensure her obedience/love of her owner.  I do not remember much of the over-arching plot, just a few disjointed pieces:  The world has chaos/wild magic storms.  The books might have called them something else.  Anyone or anything caught by these storms is transformed into something else entirely, so different objects or species.  These show up throughout as minor plot encounters.  I think they can be defended against with a special material.  Near the very end, a large storm engulfs a major city, giving it entirely new architecture and transforming the inhabitants into green, hermaphrodites with a centaurish body structure.  I am not 100% sure this was the same novel, but I recall a secondary character with a secret - at night he turns into a woman (not just his body, but an entirely different person).  At some point it's revealed that the woman was his lover, and he saved her life by sharing his body with her - of course, now they can never meet/touch again. Which tag is used in the title of a question?   It does not seem to be the most popular It does not seem to be the first in the tagging order:   This question   is prefixed with NGINX (which is actually completely fine)  And it is tagged with "php nginx fastcgi magento" in the order of popularity.  Can we choose the tag to use in the title manually? What original Dr Who episodes are left when not watching unfinished stories? If shuffle-playing playlist &times;100 resulted in [10 13 10  3  2  2] different songs being repeated [1 2 3 4 5 6] times, what is the estimate for the total number of songs? (assuming shuffle play was completely random)  Update: (R code)  k &lt;- 50 # k number of songs on the disk indexed 1:k n &lt;- 100 # n number of random song selections m &lt;- 20 # m number of repeat experiments colnum &lt;- 10 mat &lt;- matrix(data=NA,nrow=m,ncol=colnum) df &lt;- as.data.frame(mat) for(i in 1:m){   played &lt;- 1+floor(k*runif(n)) # actual song indices (1:k) selected    freq &lt;- sapply(1:k,function(x){sum(played==x)})   # = number of times song with index x is being played   histo &lt;- sapply(1:colnum,function(x){sum(freq==x)});   for(j in 1:colnum){    df[i,j] &lt;- histo[j]   } } df   Resulting in: e.g. 20 distributions (V1=number of single plays, V2=number of double plays, etc):     V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 1  15 13 11  1  2  2  0  0  0   0 2  15 12 10  1  4  0  1  0  0   0 3  12 14  7  6  3  0  0  0  0   0 4  17 16  6  4  2  0  1  0  0   0 5  17 10 12  5  0  0  1  0  0   0 6  13 15 11  6  0  0  0  0  0   0 7  10 14  9  3  2  1  1  0  0   0 8  12 17  5  6  3  0  0  0  0   0 9   9 19  8  3  1  2  0  0  0   0 10 13  9 11  6  1  0  1  0  0   0 11 16  9 12  5  2  0  0  0  0   0 12 15  9 11  6  2  0  0  0  0   0 13 19  9  7  4  4  1  0  0  0   0 14 17 11  4  7  3  1  0  0  0   0 15 11 20  8  1  3  1  0  0  0   0 16 14 12 10  5  0  2  0  0  0   0 17  9 12  8  7  3  0  0  0  0   0 18 10 15  9  4  2  0  1  0  0   0 19 14 11 12  7  0  0  0  0  0   0 20 16 14 11  3  1  1  0  0  0   0   Now I need to get from here to the Poisson modelling--my R is a bit rusty (?lmer)...--Any help would be appreciated...  Attempted Poisson modelling: disappointing fit?!  plot(1:colnum,df[1,1:colnum],ylim=c(0,30),    type="l",xlab="repeats",ylab="count") for(i in 1:m){   clr &lt;- rainbow(m)[i]   lines(1:colnum,df[i,1:colnum],type="l",col=clr)   points(1:colnum,df[i,1:colnum],col=clr) }  df.lambda=data.frame(lambda=seq(1,5,0.1),ssq=c(NA));df.lambda for(ii in 1:dim(df.lambda)[1]){   l &lt;- df.lambda$lambda[ii]       ssq &lt;- 0       for(i in 1:20){         for(j in 1:10){           ssq &lt;- ssq + (df[i,j] - n*dpois(j,l))^2         }       }       print(ssq)       df.lambda$lambda[ii] &lt;- l   df.lambda$ssq[ii] &lt;- ssq     }     df.lambda     lambda.est &lt;- df.lambda$lambda[which.min(df.lambda$ssq)] lambda.est # 2.4 points(x &lt;- 1:10, n*dpois(1:10,lambda.est),type="l",lwd=2)  100*dpois(1:10,3) n/lambda.est   the estimated lambda stays around 2.3, with an n estimate of around 43; the fitted curve seems very discrepant, and seems to worsen with rising n !?  Doesn't this have to do with the fact, that our repeats are different from the 'classical' Poisson distributions: it's not just ONE event that repeats itself x number of times, but the sum of repeats of different items (songs)?!