This thread is intended to help with closure of duplicates. The main use case are situations where a first attempt at duplicate closure was not successful.1 The creation of this thread was decided here.

Some basic rules:

• Do not use this thread for other types of closures, it is only for duplicates.
• Only use this thread if the usual means did not work. In particular, you should have cast already a vote or flag, the question already has been through the close votes review queue, and it was still not marked as duplicate.2
• Keep in mind that very short answers are automatically turned into comments. So your post should probably contain at least some brief explanation in addition to links to the duplicates. (Maybe short discussion why they are duplicates, if it is not clear that they are indeed duplicates. Or a brief explanation which of the two posts is more suitable as the duplicate target.)
• If it turns out that discussion grows long and that it is contentious whether the particular question is indeed a duplicate, the question might deserve a separate post on meta.
• After the question is closed, mark this clearly at the beginning of the post. This will help other users to see which request have already been resolved. (And if question is reopened again, this should be marked in the same place. In short, the system is the same as in the thread for reopening and undeleting.)

A more low-key way to draw attention to a possible duplicate is this chat room, which has been created for similar purposes. For example, if just one vote is missing you might consider posting there instead.

1 Other smaller coordination-tasks related to duplicate management, such as proposals for reversals of duplicate directions, could also happen here. We'll see how things evolve.

2 You can find review for a particular post - if the review is already finished - from the post timeline. Alternative ways are using this SEDE query by entering the question id. If the close votes are less than week old, it is possible that they are not in the last data dump. A more laborious alternative is to look in close votes review history, which however allows to track reviews in process. Or, just wait 24 hours after your vote; then it is very likely the post has passed review.

• Thanks to quid for helping with improving the wording of this post. Sep 10, 2016 at 13:10
• Another reason for starting this thread (apart from the discussion linked above) is the fact that we had recently two posts discussing why some duplicate close vote failed (here and here). I am not sure whether both of them would be good fit for this thread, but if there are going to be more posts of this nature, it might be better to concentrate them in one place. Sep 10, 2016 at 13:12
• @Martin Is my post at all like what you had in mind for this thread? I don't want us to use it at cross purposes. Oct 2, 2016 at 13:46
• Wow, I just noticed this post. Is it alright if I bump this thread with new requests?
– user279515
Jun 26, 2020 at 15:02
• @Brahadeesh I will strongly support that. Jul 25, 2020 at 16:28
• @ArcticChar Great, I'll consider posting new duplicate requests of mine here, then. :) I also had a conversation with Martin in his chatroom about this and he also thought that trying to revive this thread might be fine.
– user279515
Jul 25, 2020 at 16:49

NOTE: This post is about collecting links to different posts that I think should be closed as duplicates of this post explaining the basic techniques in modular arithmetic. Before acting I recommend that you also study this thread where several relevant points were raised in the answers.

OBSERVE: Votes on this answer may not reflect the popular opinion about individual questions on the list. This is because some votes were cast when the list was shorter, and it is natural that different voters feel differently about different items on the list.

• This question is about calculating the remainder of a high power of $3$ modulo $35$ (open). [Now closed as a duplicate, reopened by a single gold-badge-user vote, and then closed again as a duplicate and later deleted.].
• This question is about calculating the last digit of a high power of $3$ (open).
• And another one about the last digit of a power of seven (open).
• This underlines what is IMNSHO wrong about the profileration of the replicas - nine more or less identical answers within half an hour from a bunch of users picking low hanging fruits (open). Anyway, some saving grace comes from the fact that the modulus differs from a power of two by one, and that can be exploited.
• Status updates, additions etc by other users obviously welcome. Oct 2, 2016 at 13:48
• Please use one answer per question so that the votes can be seen for each question. Otherwise you are in effect "borrowing" votes from prior questions, and this might bias decision-making. Further, some users may wish to vote differently on different questions in this class. Dec 7, 2016 at 22:50
• @BillDubuque I see your point. Initially I wanted to use this to collect more data on the public opinion about these near-duplicates. To that end it surely makes sense to keep links to posts that I view as duplicates of the same target in one place. By that logic I would use another answer to collect opinions about a different dupe target. I will add a disclaimer. Dec 20, 2016 at 7:33
• More questions about last digit or last two digits and similar problem can be easily found. (This comment is mainly directed to Jyrki Lahtonen; so if you think it is not too useful for others, feel free to delete it after you see it.) Dec 21, 2016 at 8:56
• I do not want to repeat all I said here, but shortly: Shouldn't we treat the questions about last digit (or two, three, ... digits) slightly differently than questions on modular arithmetic. The question to find last digit is probably accessible to somebody who only learned basic arithmetic and knows nothing about congruences or divisibility. Dec 30, 2016 at 20:04

Closed

The question Help on Surjection, Injection, and Bijection basically consists of four different question, all of which have been answered on this site in the past. So it should be either closed for the reason the it contains multiple questions or as a duplicate. (I have even asked in chat which of the two possibilities is better, but I got no reply.)

I am aware that duplicate is not ideal choice, but it seemed to be less harsh on a new user and in a comment I have added an explanation that one post should not contain several questions together with link to the relevant discussion on meta.

I voted to close a duplicate. I hoped that other users will notice alternative links in comments and perhaps choose some other post as a duplicate target. (So that, ideally, after closing process the duplicate banner would contain links to some or all of the subquestions.) This did not happen, but at least the links are available in the comments and in the sidebar.

The post already has been through review queue and it has at the moment 4 close votes (as a duplicate), so one last vote is missing.

• Since this thread was around for two weeks, I thought that it might be a good idea to try what happens if it is used for the purpose for which it was created. Sep 25, 2016 at 8:52

Closed as duplicate and then merged

The question How to find $\sum_{n=0}^{\infty}\frac{1}{2^{2^n}}$? is, as far as I can say, a duplicate of Sum of Infinite Series $1 + 1/2 + 1/4 + 1/16 + \cdots$. Still after I voted to close as a duplicate, several users voted to leave open in the review.

I have also asked in chat whether or not it should be closed - the only feedback I got was this message confirming it is a duplicate (and that merging the two questions might be a reasonable idea).

Certainly I am open also to closing the questions in the other direction. (I simply voted to close the newer question as a duplicate of the older one.)

And since I am already a making a post on meta about this pair of questions, I will also ask what you think about merging them. (Of course, merging cannot be done by regular users, so the final decision will be up to the moderators.)

Multiple Exponentiation in Modular Arithmetic appears to be a duplicate of How do I compute $a^b\,\bmod c$ by hand? . The second question is intended to be the abstract duplicate question for all questions on that specific topic. The first question linked was closed as a duplicate and re-opened by a user with a gold tag badge.

• Dupe closures should not be abused by sending askers on possible wild goose chases (that may or may not be successful). That generic thread is a motley mix of unrelated techniques. It is far from complete, and (pedagogical) quality leaves much to be desired. If one has the (extensive) number theory background required to know for sure that some answer(s) there cover all the ways one would attack some question then one should specifically link to said answers (and elaborate on how they apply if need be). Mar 11, 2019 at 19:24
• Problems of this sort can be solved using all sorts of number theory, group theory, linear algebra. etc. They are posed not for the purpose of getting an answer but, rather, to expose students to the diverse techniques in number theory and algebra that can be used in their solution (similar to integration problems in calculus). One can use them to motivate and illustrate many major topics in number theory (e.g. euclidean algorithm, quadratic reciprocity, (cyclic) group theory, Euler, Fermat and Carmichael theorems, Chinese Remainder Theorem and lifting the exponent and related p-adic ideas... Mar 11, 2019 at 19:46
• Closed again, by vote of users, as a duplicate. Mar 12, 2019 at 1:46
• You've done an amazing amount of research for this, thanks. What action(s) do you recommend? Feb 4, 2022 at 22:40
• @hardmath Thank you for the kind words. It would be great if the questions in the first list could all be closed as a duplicate of the main thread. For the questions in the second list, I would recommend the same, but I understand if there are differences of opinion regarding whether some of those are duplicates. Certainly it wouldn’t do any harm to take a good look before voting to close as duplicate (or leave open) for the questions in the second list. Feb 4, 2022 at 22:47
• @TheAmplitwist Look like you flagged a large amount of those at a same day, and somehow those flags are invalidated. I suggest that next time when you flag a group of posts as duplicate, do post it in CURED. Sometimes those flags are just not handled (don't know why), and even if they get into the queue, it's easier to handle them at the same time instead of one-by-one in the close-vote queue. Feb 6, 2022 at 1:50
• I've tried flagging them one-by-one as well, to not much avail. My guess is that the high number of views for some of these questions causes the flags to be invalidated sooner. I was hoping that posting in this thread would allow for handling them all at once, but CURED is certainly another option. However, I am taking a break from chat, so I won't be able to post there. Feb 6, 2022 at 7:30

### Crazy old lady airplane Seat Probability Problem, with 100 passengers

The target post is above. Here are 41 duplicates. Just edit this post if you find more.

### Crazy old lady airplane Seat probability problem, generalized to n passengers.

The target post is above. Here are duplicates. Just edit this post if you find more.

Closed as duplicate

Please close this as a duplicate of this. The close-vote review is completed and still missing one vote.

• Now closed although it is now present on reopen queue
– user876009
Sep 12, 2021 at 13:45
• It would be nice to have the titles of these posts in your request itself, instead of the words "this" and "this". Just a suggestion :) Sep 12, 2021 at 13:47

Closed

Please consider closing this question as a duplicate of this one. The wording are quite different, but essentially both questions ask about the relationship between two definitions of smooth manifolds which are not compatible. Close vote review is completed earlier today.

Closed as duplicate

I think that Verifying some trigonometric identities: $\frac{\csc\theta}{\cot\theta}-\frac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$ and Proving that $\frac{\csc\theta}{\cot\theta}-\frac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$ are duplicates. The post with more answers seems to be a reasonable duplicate target. The post already went through review without being closed. (Merging does not seem to be suitable in this case, since the newer post has an answer which also comments on the mistake made by the OP.)

### Recommendations of first introductory books on Combinatorics for greenhorns and teenagers.

The above is the target post, already linked to many duplicates! Here are MORE duplicates. Just edit this post, if you find more. No need to seek permission to edit this post.

### Recommendations of first introductory books on Number Theory for greenhorns and teenagers.

The above is the target post, already linked to many duplicates! Here are MORE duplicates. Just edit this post, if you find more. No need to seek permission to edit this post.

• I have removed around half of them - those are already closed (all but one are closed as duplicate) Aug 20, 2021 at 13:06

The two posts have been merged

The posts $X$ is homeomorphic to $X\times X$ (TIFR GS $2014$) and What can be said about topological properties of a space $X$ that is homeomorphic to $X\times X$ $?$ are (in my opinion) duplicates of each other. They contain exactly the same question (probably from the same school or from the same type of exam), the difference between them is only in the part where the OP says what they were able to do.

I think they should be merged (I have already flagged but the flag is still pending) or at least marked as a duplicate. As far as the choice of duplicate target goes, if they are not merged then probably the newer question could be better, since the answers there are (in my opinion) a bit simpler. But otherwise there is not much of a difference between them. The post already went through close votes review.

No need to seek permission to edit any of my posts here. Just edit this, if you find more duplicates.

Please merge, or close as duplicates, these overabundant requests for Multivariable Calculus textbooks? Don't hesitate to add other duplicates to this post.

• Why downvote me?
– user53259
Jul 27, 2021 at 18:46
• There is a lot of works to merge two posts, let alone 10. There would be lots of duplicate answers, also the wording has to be changed. And the process is irreversible. It is more realistic to ask for closure as duplication instead. Jul 28, 2021 at 5:34
• @ArcticChar "It is more realistic to ask for closure as duplication instead." Done.
– user53259
Jul 29, 2021 at 7:05
• Which one of the above is the duplicate target? Jul 29, 2021 at 10:13
• @ArcticChar I think that only an older post can be targeted as a duplicate? But I edited my post to identify it now. OK?
– user53259
Jul 29, 2021 at 23:38
• On occasion, ugro, an older post has been closed as a duplicate of a newer one. It can be done. Jul 30, 2021 at 0:04
• As mentioned by Gerry Myerson, the age is not the primary factor when choosing the duplicate target. See: Original post marked as duplicate and other discussions linked there. Jul 30, 2021 at 5:39