Here's a classic puzzle (I think Martin Gardner talks about it somewhere, though I'm not sure exactly where):     Alice and Bob are co-conspirators. Alice is dealt five random cards from a standard deck; she looks at the cards, and then shows four of them to Bob in some chosen order. Using only the four cards and the order they are shown in, Bob must determine the identity of the fifth card.   The solution that I know of is:      Alice and Bob agree on an ordering of the $52$ cards, and a bijection between $S_3$ to $\{1,\dots,6\}$. With this information, an ordering of any three cards gives an integer between $1$ and $6$ inclusive. Now, Alice uses the pigeonhole principle to find two cards of the same suit. Without loss of generality, their ranks are $r_1$ and $r_2$ where $r_1+a \equiv r_2 \pmod{13}$ and $1 \leq a \leq 6$ (if they weren't originally, swap $r_1$ and $r_2$). Alice puts the card of rank $r_1$ first (naming the suit) and then orders the other three cards in a way that tells Bob $a$ (naming the rank).   Notice that $52$ cards is the maximum possible number that this solution works for, since it depends heavily on there being only four suits and thirteen cards in each suit. However, it seems like it wastes a fair amount of information whenever     the suit distribution isn't 2-1-1-1.   Is there a method to do this which works on a deck containing more than $52$ cards? The obvious upper bound on deck size is that Bob sees a list of four ordered cards and can deduce from it the original five cards Alice was dealt, so it it works for a deck of size $n$ we must have $$ 4!\binom{n}{4} \geq \binom {n}{5} $$ or $n \leq 124$. But that still leaves quite a gap...
Long ago someone demonstrated the following card trick with a standard 52-card deck:  (1) A volunteer selects 5 cards from a shuffled deck, which the performer does not see.  (2) The assistant puts exactly 4 of those in a pile and passes it to the performer.  (3) The performer can then identify the 5th card.  There are a few ways to execute this trick, one of which goes as follows.  (1) The assistant finds a suit common to 2 of the 5 cards, with values $x$ and $y$ modulo $13$.  (2) He then removes one of them, say $x$, such that $x-y \in [1..6]$ modulo $13$.  (3) He then puts the other one at the top of the pile.  (4) And he puts the 3 other cards after that in a permutation corresponding to $x-y$ modulo $13$.  Since (1) works by pigeonhole, (2) works because $\frac{13-1}{2} = 6$, and (4) works because there are 6 possible permutations of 3 cards, the trick succeeds. My question is whether the same trick can work with more than 52 cards. It is not obvious to me how to do so, nor compute the maximum number of (pairwise distinct) cards possible. Also, it seems to me that even if the trick can be made to work for $n$ cards, it may not work for $n-1$ cards! Any ideas?
Deleted comments are persisting in the global inbox I realise that there is other questions out there about the  (First open one I found is ), but this is marked .  I often get the below issue where it will notify me about comments on my  (in this case what appears to be the user re-writing after deleting their comment) that no longer exist.  Also, occasionally, I see comment snippets with text that doesn't match what is on the page when I view it.  Let me know if I should be trying to edit that one to remove the tag indicating that it has been fixed, and add the above as another answer. Accepted way to abbreviate cities that have "Saint" in their name My first question here, so I'm not sure if this OT or not, so please bare with me.  Background: I don't live in US. I'm working with some data that I'm pulling from . Sometimes, I'm getting a city name as St. John and sometimes as Saint Ann.   So my question is, is there an accepted way on how people write a US city names which starts with Saint or is this just some random thing (or just specific to AdWords maybe)?  I'm asking this, because if for example I'll get both St. John and Saint John in the same file, should I consider them the same or should I be suspicious? Loop doesn't see value changed by other thread without a print statement On my Toshiba Encore 8 Tablet I noticed that the CPU usage is very high. In Taskmgr / ProcessExplorer, I can see that OneDriveSetup.exe is hogging 1 CPU cores heavily for hours.    How an I solve this hang and finish the OneDrive Update? Display sheet name in Google Spreadsheets Is there any way to display a sheet name as the result of a function in a Google Spreadsheets (exactly like , except in Google Sheets)? How to disable phone number linking in Mobile Safari? I have a div with two images and an h1. All of them need to be vertically aligned within the div, next to each other. One of the images needs to be absolute positioned within the div. What is the CSS needed for this to work on all common browsers? &lt;div id=&quot;header&quot;&gt;   &lt;img src=&quot;..&quot; &gt;&lt;/img&gt;   &lt;h1&gt;testing...&lt;/h1&gt;   &lt;img src=&quot;...&quot;&gt;&lt;/img&gt; &lt;/div&gt; I've recently read the Yahoo manifesto . They recommend to put the JavaScript inclusion at the bottom of the HTML code when we can.  But where exactly and when?  Should we put it before closing &lt;/html&gt; or after ? And above all, when should we still put it in the &lt;head&gt; section? Is it okay to use the phrase "try and"? Is it grammatically correct? e.g. "I will try and answer ..." I read the following sentence in book (App).     I will try and answer any questions you ask.   Why don't they use the following? (it is much common):     I will try to answer any questions you ask.   Is there any difference in meaning?  Update:  I noticed that a lot of books and news websites are using this phrase:        Although Katty will try and answer as many questions as she can.    He puller would have a chance to try and answer the question on her own ... Earlier today I posted  which yielded a .  I'd like to attract attention to this answer in the hope that the answer will get upvoted, as a kind of reward for the quality of the answer.  Is there anything that I can do to achieve this? Forgot which email I used, can't login I forgot what email I used for my other ploxtic id: how do you find out what my email was?   The account is user:2078229. Singular values of the data matrix and eigenvalues of the covariance matrix I changed my username and now I cannot play online I don't understand this and I'm now worried as I have donated to a lot of servers and I don't want to lose this account. On the launcher it says my new username but when I log on it says "Account not Authenticated with minecraft.net" and "Non-Valid Username".  How can I regain the ability to use my account online? Scaling problem with fillbetween in pgfplots I had a heck of time while editing my  question. But, I guess, I discovered a new weird problem in the editor while formatting hyperlink text next to  tag.  To illustrate this problem better, I would like to give the following examples:  Has problem with one or none white-space in between:  [meta-tag:status-planned] [details page reads:][2] [meta-tag:status-planned][details page reads:][2]     Gives weird output as:   [details page reads:]   [details page reads:]   But, its okay with two white-spaces in between:  [meta-tag:status-planned]  [details page reads:][2]   Gives the correct output as:      Something is strange here. I don't understand why do I have to insert two white-spaces instead of just one? Are the two white-spaces are mandatory when using [meta-tag:some-tag] with hyperlink text adjacent to it? Or, is it one of those  thing?  But wait, it's get even further complicated if I do like:   1. No white-spaces in between:  [meta-tag:status-planned][meta-tag:status-completed][Moderator-only Tags][3]     2. One white-space in between:  [meta-tag:status-planned] [meta-tag:status-completed] [Moderator-only Tags][3]       3. Two white-spaces in between:  [meta-tag:status-planned]  [meta-tag:status-completed]  [Moderator-only Tags][3]         They all seems to work perfectly fine. Now, thats very strange! Why would the two [meta-tag:some-tag] works like a charm with or without requiring to insert any white-spaces? Can anyone explain this in a better way. I don't want to jump on the conclusion by calling this as a bug yet, so I used  tag instead of  tag. Please share your thoughts. Could a constant function be considered increasing? Given the definition of an increasing function as a function $f:E\to\mathbb{R}$, where $E$ is nonempty such that $x_1,x_2\in E$ and $x_1&lt;x_2$ imply $f(x_1)\leq f(x_2)$, then couldn't, for example, $f(x)=-1$ considered increasing? Since the inequality is not a strict inequality. Finding $\frac {d(\tan \theta)}{d\theta}$ I always used to open a new VLC window; go to "open file"...etcetera when I wanted to watch something because when I click on a file in Windows Explorer and chose "open with" or just clicked on the file or right-clicked to choose "play with VLC" it always opened in an old already used VLC window.  Is there any way to get VLC to use a new one without always resorting to: Open VLC, "open file", find file etcetera? How to make hair material different from the plane itself? In Cycles Render, I added hair to a plane (as I was making a grass field) and I don't seem to know how to make the actual flat plane's material different from the hairs themselves, because I'd like to make the plane have an image texture and the hairs have a yellow-like color to them, but of course, if I attempt to make the material yellow-ish, the plane AND the hairs will become yellow. I'm done with the plane, I just want to control the hairs' materials. Does RSA work for any message M? I decided to read the original RSA paper A Method for Obtaining Digital Signatures and Public-Key Cryptosystem because of a question I had about RSA (which is not the question I'm about to ask, but may be a question on here at some time) and came across something I had never seen before in any textbook description of RSA (Note: this comes from the beginning of ).  for any integer (message) $M$ which is relatively prime to $n$, $M^{\phi(n)} \equiv 1\pmod{n}$  The part I don't recall ever hearing in class, or coming across in a text book, or seeing in any description of RSA I've ever read is that $M$ and $n$ must be relatively prime. So, I decided to play around with a toy example of RSA to see what would happen. I use $p=13$, $q=31$, $e=7$ and $M=2p=26$ (for completeness: $d=103$, $\phi(n)=360$, $n=403$). I observed the following using my toy instance of RSA: $M^e\equiv M\pmod{n}$, $M^e\equiv 0\pmod{p}$ and $M^e\equiv M\pmod{q}$. Do these equivalences hold for all $M,p,q$ when $M$ and $n$ are not relatively prime? How would I prove this? What is the real-world effect of this?