I know the relationships between the ministers in the below problem can be represented by a $K_n$ graph since each one of the n ministers has connections to the n-1 other ministers/ministries they represent. However, I can't seem to figure out where to go next. Here is the problem:  In a certain government there are n cabinet ministers. Each of these has a senior aide. Exactly k of these n aides leak important information to the press. None of the ministers know whether their own aide leaks information or not. However,because of interdepartmental rivalries, every minister has many sources in the other ministries, and thus is aware of all the leaks outside of her ministry. There is an unspoken law in this government that as soon as any minister has irrefutable proof that their aide leaked information, they must re the aide at their first press conference. Every ministry has one press conference per day at exactly 8:00 A.M. For a long time no one is fired, because no minister is aware of the leaks, if any, made by their own aide. One afternoon a newspaper runs a story, and announces that news are leaked by some of the top aides. Exactly k days later, all the guilty aides are fired. What happened?
To be honest, I have no idea how to even start this problem. I'm sorry I don't have any work to show, but I'm just at a blank. Help?  Chapter 2: Problem 27:  University B, once boasted $17$ tenured professors of mathematics. Tradition prescribed that at their weekly luncheon meeting, faithfully attended by all $17$, any members who had discovered an error in their published work should make an announcement of this fact, and promptly resign. Such an announcement had never actually been made, because no professor was aware of any errors in her or his work. This is not to say that no errors existed, however. In fact, over the years, in the work of every member of the department at least one error had been found, by some other member of the department. This error had been mentioned to all other members of the department, but the actual author of the error had been kept ignorant of the fact, to forestall any resignations. One fateful year, the department was augmented by a visitor from another university, one Prof. X, who had come with hopes of being offered a permanent position at the end of the academic year. Naturally, he was apprised, by various members of the department, of the published errors which had been discovered. When the hoped-for appointment failed to materialize, Prof. X obtained his revenge at the last luncheon of the year. &quot;I have enjoyed my visit here very much&quot;, he said, &quot;but I feel that there is one thing that I have to tell you. At least one of you has published an incorrect result, which has been discovered by others in the department.&quot; What happened the next year?&quot;   Chapter 2: Problem 28:  After figuring out, or looking up, the answer to Problem 27, consider the following: Each member of the department already knew that Prof.X asserted, so how could his saying it change anything?
"Leave a comment" link even when you can't On  I disabled  comments for "USDCAD options vs CADUSD options arbitrage?", but it  still has a "Leave a Comment" link.   Clicking the link takes you to    where it, of course, shows that you can't leave a comment.   The timeline page correctly does not show the 'comment' link:    How to fix? Primary decomposition of a monomial ideal how long to cook a turkey per pound Can the present perfect continuous construct be used in passive voice? Can the passive voice be used for sentences in the present perfect continuous?  I understand the present perfect and its passive voice (have/has built, has been built). However, the passive voice of present perfect continuous is unclear:   Active: They have been building this house for years. Passive: ???   I have seen suggestions of “This house has been being built for years,” but it sounds awkward and I don't recall anywhere else where two be verbs follow one another. Is this the right form to use? As per passport my name is Manish Kumar Ahuja, but by mistake I book a ticket with the name of Manish Ahuja. My middle name is not on the ticket. Can I fly on this ticket? Because the airline can't add the name and they say that the ticket is non-refundable. Example of set which contains itself I am trying to understand   How can a set contain itself? Can you show example of set which is not a set of all sets and it contains itself. Computing sums of divisors in $O(\sqrt n)$ time? I have a sequence: $1,3,5,8,10,14,16,20,23,27,\dots$  I know that the recursive relation is:  $$p[i] := p[i-1] + \text{number of factors of $i$}, \quad \text{with $p[1]=1$.}$$  How do I solve this recursive relation in $O(n^{0.5})$ time? I have to write a program for it, and $n$ can be as large as $10^9$. So, some appropriate precomputation and prestorage is allowed, provided that they comply (memory and time wise) with the range of $n$. Can I connect a PC to a PoE switch? Okay, so I understand what calculus, linear algebra, combinatorics and even topology try to answer (update: this is not the case in hindsight), but why invent category theory? In Wikipedia it says it is to formalize. As far as I can tell, it sort of generalizes a bunch of fields in mathematics, like in topology, graphs and groups we have isomorphisms and automorphisms and homomorphisms, it seems to me like it just finds a bunch of results that are true to these functions in a bunch of disciplines. But why would generalizing that be of use? I mean, wouldn't it be easier to prove them specifically in each of the important disciplines? I'm installing a brand new motion activated security light to an existing box on my garage. After removing the old light, I notice it only has white/black wiring while the new light fixture has of course white/black/green (ground) wires to attach. What do I do if no ground wire is available with the old box? What are best practices for choosing and pairing fonts? I've always found font selection to be a little bit of a mystery. What factors should be considered when choosing a font? How does one successfully incorporate two fonts into a design?  I realize that some factors will include things like:   Legibility Scalability Appropriateness/Mood/Emotion   However, how does one objectively judge legibility or appropriateness? Are there objective principles that can be applied for font selection? If possible, please list resources or examples in your answers. Studies focused on the effectiveness of different fonts would also be extremely useful. Formal Proof that area of a rectangle is $ab$ I tried to prove that the area of a rectangle is $ab$ given side lengths $a$ and $b$.  The best I can do is the assume the area of a $1\times1$ square is $1$. Then not the number of $1\times1$ squares that fit in an $a\times b$ rectangle is $ab$. Therefore area is $a\cdot b$. This does not seem rigorous however. Is every sub-lattice of $\mathcal P(X)$ isomorphic to a sub-lattice of $\mathcal P(X')$ containing singleton sets? What is the meaning of the term "thread-safe"? Does it mean that two threads can't change the underlying data simultaneously? Or does it mean that the given code segment will run with predictable results when multiple threads are executing that code segment? Gaussian Integers and Quotient Rings Show that $\mathbb{Z}[i]/(2+3i)$ is a finite field.    By applying some isomorphisms and realizing $\mathbb{Z}[i]$ as the quotient ring of $\mathbb{Z}[x]/(x^{2}+1)$, I found that:  $\mathbb{Z}[i]/(2+3i) \cong \mathbb{Z}[x]/(x^2+1,2+3x)$  However, I run into trouble as I can't divide $x^{2}+1$ by $2+3x$ in $\mathbb{Z}[x]$. What would be another way to approach the problem?  Thanks for the help. Console flashes on launching, then closes (ms-windows) How to reinstall yum? What is the name of the default font in math mode? I draw figures in Inkscape. When I label elements within the figures with variable names that I have used in the underlying TeX document, I would like them to look exactly the same as in the document. (e.g. l does not look the same as $l$)  What is the name of the math mode font so I can select it correctly in the Inkscape font list?  If the exact font should not be available, what is a similar looking font that is present on most systems? My friend had a  on the board with 10 counters on it.  If I destroy it and he sacs it in response to gain the life, is there anything that can be done about that?  Like, can I put another destroy artifact on the stack to resolve before his sac resolves? Why has Steve Nash moved to the LA Lakers in sign-and-trade deal? Congratulations to Steve Nash on his move to Lakers !  His contract with the Suns has expired so I don't understand why he moved in a sign-and-trade deal.    Why resign and move in trade? Why didn't the Lakers sign Nash without any trade?