How close a match we need to close a recurring question as a duplicate?

Question

An hour ago I closed a question as a duplicate of an, in my opinion appropriate, generic question. Two gold badge holders have since expressed disagreement with my decision, and the question has since gone thru a full reopen/reclose cycle.

There is always room for interpretation, and because I have a history of also fighting everything that smells of rep farming, I should not rule out the possibility that I have crossed some line myself in an overzealous effort to find more duplicates.

Another example of a question I closed as a duplicate yesterday. This time there has been one gold badge owner in disagreement, and the question was reopened overnight (in my timezone of GMT+2), but has since been reclosed by another gold-badge owner. In my opinion the answers to the duplicate target cover this question more than adequately. The late parts in Arturo's answer feel particularly fitting.

So time to collect some feedback from you.

How close a match to a question from the list of generalizations of commonly asked questions do I/we need to instaclose a new incarnation?

Answer 1

The OP asked in original version (now substantially rewritten a further from a rant): "Am I the only one concerned about duplicates?" My answer to this would be certainly not, as witnessed by many people suggesting duplicates, voting to close as duplicates, discussions in chat about duplicates (both in the main chatroom and in c-r-u-d-e), etc.

As already said by several other users, it is very difficult to give a definitive answer to the question "What is a duplicate?" and there will always be some grey area. Let me just make some points related to duplicates and also illustrate my point of view on some examples. This is just my personal opinion. It is natural that in a such diverse community as this site, there will be many different opinions on this.

---

When you vote to close as a duplicate, the text you choose from dialogue is: This question has been asked before and already has an answer. The formulation (emphasis mine) "this question has been asked before" suggests that this should be exactly the same question.

On the other hand, after the question is closed, the duplicate banner says: This question already has an answer here. This formulation suggests that it may be a completely different question, but the answers in some way contain answer to this one. (And I have seen some duplicate disputes where this phrasing was used as an argument.)

If we take this interpretation then, for example:

• A question could be closed as a duplicate of another question which is more general than the given question.
• A question could be closed as a duplicate of another question which is a special case of this question, but some of the answerers of the duplicate target provided a proof for a more general case.
• A question could be closed as a duplicate of another question which seems different, but after some non-trivial effort it can be seen that they are asking basically the same thing.
• A question could be closed as a duplicate of another question which is entirely different, but one of the answers contains an answer to this question as an auxiliary lemma.

I disagree with both these interpretations, the first one is very narrow the second one is extremely broad.

Two questions do not need to be exactly the same to be considered duplicates. If the difference is small enough so that we can reasonable expect from anybody, who has the background sufficient to understand the two questions, to be able to transform one of them to the other one, then one of them probably should be closed as a duplicate.

On the other hand, if we close something as a duplicate of a much more general question, this might be contraproductive. For example, consider the question about reverse triangle inequality for absolute value. If this is closed as a duplicate of questions about reverse triangle inequality in metric spaces, then this duplicate would be useless for many users. Somebody who just started to learn about absolute value will probably not be able to grasp what the more abstract notation $d(x,y)$ means. (This problem is less prominent for reverse triangle inequality in normed spaces, since the notation is very similar. But I would still consider these two questions to be different.)

---

Let us have look at some examples

My personal opinion is that question should be closed as a duplicate if they are the same or very similar to each other.

If we want to close a question as a duplicate of a more general question, we should check whether we are losing some answers. By this I mean that if a special case can be answered in a way which is very difficult to generalize, I would not vote to close. If it is difficult to think of any other method of answering the special case which is not already covered in the more general question, then go ahead and close as a duplicate.

Example 1 - sums. The following example is, in my opinion, a good illustration of what I have in mind.

• Question 1: $\sum_{k=0}^n \frac1{2^k}$
• Question 2: $\sum_{k=1}^n \frac1{2^k}$
• Question 3: $\sum_{k=0}^\infty \frac1{2^k}$
• Question 4: $\sum_{k=0}^\infty \frac1{4^k}$
• Question 5: $\sum_{k=0}^\infty \frac{5^k}{7^k}$
• Question 6: $\sum_{k=0}^\infty x^k$ for $|x|<1$ (This question is one of the questions among abstract duplicates: Value of $\sum\limits_n x^n$ You can see which questions were closed as duplicates are among linked questions.)

In this case I would consider Question 1 and Question 2 duplicates. (Anybody who has learned enough mathematics to be asked this question should be able to see how omitting one term changes the sum.)

But I would keep the finite sums and infinite sums separate. (It is true that knowing answer to Question 1 and taking the limit answers Question 3. But to answer Question 3 we do not necessarily need to know the formula for partial sums.)

I would not close Questions 3 and 4 as a duplicate of Question 6. My argument is that we lose some approaches which are difficult to generalize. For example, Questions 3 and 4 can be answered by writing the summands in binary. And there are also picture proofs for $x=\frac12$ of $x=\frac14$, which are probably difficult to generalize for other quotients:

(Both pictures are taken from Wikipedia, from the articles about this series and about geometric series in general. I will also add links to Wikimedia commons: https://commons.wikimedia.org/wiki/File:Eye_of_Horus_square.png and https://commons.wikimedia.org/wiki/File:GeometricSquares.svg.)

However, as I do not see anything special about $x=\frac57$ and any answer there would probably be the same as if we worked with arbitrary $x$, in this case the close vote seems to be reasonable.

Example 2 - modular arithmetic.

I will also comment on this, since this is one of the closures mentioned by the OP.

Let us have a look also at questions about modular arithmetic. We have the question How do I compute $a^b\,\bmod c$ by hand? This question does a good job in listing some common approaches and is a good candidate for a duplicate. But again, if there are some clever tricks which can be used for this special case but which are difficult to generalize (and are not mentioned in the general question), I would not close it as a duplicate.

• Question A (generalization/abstract duplicate): How do I compute $a^b\,\bmod c$ by hand? (Questions closed as duplicates of this one can be found among these questions.)
• Question B: $234^{476} \bmod 333=?$
• Question C: $255^{123} \bmod 256=?$
• Question D: $5^{51} \bmod 103=?$
• Question E: $2^{83} \bmod 167=?$
• Question F: $2^{32} \equiv -1 \pmod{641}$

I do not see a problem in closing question B a a duplicate of Question A. But I would keep question C separate, since this can be solved simply by using $255^{123} \equiv (-1)^{123} \pmod{256}$. (This trick is not mentioned in the general question. Even if it was, it would be probably difficult to find it. This suggests an idea whether we could perhaps add CW answer to Question A listing some common tricks which can simplify such computations done by hand. In this case, replacing $a$ by $c-a$ when computing $a^b \bmod c$ could be mentioned, together with some examples. I suppose there are several other tricks which can often be useful.)

I would keep also Question D separate. If we notice that this is exactly the congruence from Euler's criterion for $p=103$. So this can be computed using quadratic residues. This cannot be used in general (for arbitrary exponents).

Question E can be considered a question of similar type. However, it is also related to Mersenne numbers and Sophie Germain primes, so some results about these numbers could be used here as well. So in this case there are even more reasons to keep this question separate from the all-encompassing abstract duplicate. Similarly, Question F is related to the smallest composite Fermat number, which is a result interesting in itself. So I think that such question should not be closed as a duplicate. (An indeed, we have a question on this on the main site. And it is probably not surprising that it was not closed as a duplicate of the general question about $a^b\bmod c$.)

I would have no objections to closing Question D and other similar questions as a duplicate of Question A, if Question A had answer using very similar trick. In this case it would be good idea to add a link to the specific answer in the comments. (For example, if we add an answer showing that quadratic reciprocity can be used to solve question D as an answer or if we choose some question of this type as "canonical duplicate target", I do not have problem with closing other questions of this type as duplicates. Of the two possibilities, making one separate question of this type seems preferable to me; since this would give an opportunity to post several answers with several different techniques for this type of question.)

Similarly, if Question A had an answer listing some useful tricks for this type of computation, I would have no objections to closing Question C as a duplicate.

---

Let me also remind two points which seem to be sometimes forgotten by users voting on duplicates:

• In the choice of duplicate target the age of the question does not matter. What matters is quality of the question and the answers. See here: http://meta.math.stackexchange.com/questions/16417/original-post-marked-as-duplicate/16418#16418.
• The proof-verification and solution-verification questions should be treated a bit differently (both when answering and when deciding whether they are duplicates). Perhaps a bit later, if the OP received sufficient feedback on their proof, the question can be closed as a duplicate. This was suggested here: http://meta.math.stackexchange.com/questions/12864/closing-as-a-duplicate-if-the-post-contains-ops-own-proof-solution/12867#12867 (and gained some support).

Answer 2

I had posted a Comment on Remainder when dividing $13^{3530}$ with $12348$ suggesting one (potential) duplicate/closely related Question. I'll share my thought about where the line might be drawn in treating such problems as duplicates.

Executive Summary

• The topic here is more intricate than some for "abstract duplicate" treatment, needing more than the minimal boilerplate to cover all useful variations.

• The Question invites us to point out both general methods and concrete examples, and likely the poster was not aware of any inconsistency in these directions.

---

As a general preference, I would close as duplicate rather than "lack of context" when the subject matter is fully addressed by an older Question & Answer. This seems expedient to me.

In the particular type of "how do I find the residue of $a^b$ mod $c$" problems, there are a few variants worth delineating. One has to do with primality of $c$ and the application of Fermat's little theorem directly, versus the cases where $c$ is composite and we need to concern ourselves with coprimality of $a$ to $c$ before turning to Euler's generalization, and possibly with Chinese remainder theorem if there is a common factor. So that's one axis of variation.

Another axis of variation is the size of the operands. Many of the Questions posted in this vein have modest operands that one can (with patience) handle via pencil and paper. Occasionally one gets posted with operands that suggest either some calculator or CAS help might be welcome, and thus opens an opportunity to discuss binary/modular exponentiation.

I looked through available duplicates without coming to a decision about immediately closing this as a duplicate (heh, no gold badge for me!). But I was inclined to give the OP some room to read up on past posts and edit if some additional clarification was desired. To me the particular phrasing:

How do I solve these type of exercises? I know there's some algorithm for solving them, I just haven't found a concrete example. Could anyone point me in the right direction?

(which quotes more than half the body of the Question) simultaneously leads in both directions: toward instruction in a general method (but how general?) and toward a concrete example (but possibly not the one broached in the title and opening sentence).

I'd be happy with now closing this Question as a duplicate of something, as it doesn't cover new ground or originate a new technique, particularly as the OP has by now had ample opportunity to see there are a lot of related previous Questions.

Answer 3

The following is just my personal opinion

The question is: What makes a duplicate?

I don't think it is easy to provide a clear cut answer. There will be some disagreements and grey area questions and I don't pretend to be able to provide a consistent policy suggestion.

General principle: I would in general side with leaving questions open if there is a (articulable) disagreement.

Of all the closing reasons, I think the duplicate is the one I care about the least. I see it as not much of an offence to the general purpose of Stack Exchange. That said, I think there is a place for closing duplicates and I would be against eliminating the closing reason. It makes sense to have the reason to collect all the answers to the same question in one place and it encouraged people to search before they ask. I have often started writing an answer and then before I hit post the question is closed. It is slightly irritating when I have put time into writing what I think is a nice answer.

So what is a duplicate?

I think that there are clear duplicates like computing the exact same integral, computing the exact same derivative, and proving the same result.

I can see a situation where someone asks a question that basically has been asked before. It is exactly the same integral that has been asked before, but the "context" is different. Maybe someone is soliciting feedback on a certain solution. Maybe someone is asking for a hint and doesn't want to consult a full solution. I think questions like that should be left open.

If it is "just" a question with an attempt that is an exact duplicate of another question I think it should be closed.

About the questions that you came across: I think Remainder when dividing $13^{3530}$ with $12348$ should be closed because of lack of context. But I also think that it can be closed as a duplicate of How do I compute $a^b\,\bmod c$ by hand?. I think this because the answers to the duplicate question could equally well be given to the first question. And this is maybe a good metric for deciding whether a question should be closed as a duplicate of another question (with answers)

If an/the answer(s) to another question could equally well be posted as answers to the question, then it should be closed as a duplicate of this question.

I don't like closing questions as abstract duplicates. I don't like this because it assumes the questioner will be able to apply apply the general case to his/her specific situation. I have many times been told that a certain theorem proves something that I want to prove. But at my level I don't see the connection. I can show one of my students an example of how to prove something and then give them something that is (truly very) similar, but I am met with a I-have-absolutely-no-idea-I-have-never-seen-anything-like-this-before look.

Concretely I don't think this question How to integrate $\cos^2x$? should have been closed as a duplicate of Evaluating $\int P(\sin x, \cos x) \text{d}x$.

--

I also wanted to add to the debate that all of this shows the importance for the questioner to adequately

• research before asking
• show context.

If you do not show context then your question is easily closed as (not having context or) a duplicate of the same question elsewhere. But if you provide context with a link to a possible duplicate question explaining why the answer(s) given there do not help, then it is much easier to help.

--

There is a related discussion about what to do when there is disagreements. In particular I could imagine that moderators have a hard job when they disagree with actions of (in-particular) high-rep users. I would like to see this discussion happen in another thread.

Answer 4

A summary of what I have learned/observed from the responses and the discussion in this thread (thanks all) up to June 22nd:

1. Many users have argued against the use of so called abstract duplicates as duplicate targets. The most common reason seems to be that while informative, the answers to a more general version of a question may not serve the asker well (if at all).
2. Many posters comment on the difficulty of giving a working definition of a "duplicate". Acknowledged. We need to judge case-by-case (not surprisingly that is nearly always the conclusion, when the userbase is polled about where to draw the line). This sounds like a healthy approach.
3. Many posters used illuminating examples about how they try and decide whether to close one question as a duplicate of another. Good reading!
4. There was a lot more variety in the responses to my actions in the chosen two specific occasions. I was delighted to see that I'm not at an extreme end of the spectrum. Still, I interpret the discussion as a kind of warning that it is not clear whether my actions were palatable to a majority.

The last point means that I need to fine-tune my approach here. In the future I will refrain from closing as a duplicate questions like this as the first voter (which was the case here). This is in line with our tradition of moderators not leading the rush to the barricades.

What remains unanswered:

1. Exactly how close a match do we need? Ok, so we judge case-by-case, and use abstract duplicates sparingly. But there is a whole spectrum between exact duplicate and abstract duplicate. When the latter term was coined, somebody suggested minor variant as a better term. My chosen examples don't give enough material to really poll this. May be the following? After this meta thread started these math gems have been reopened. Related to my other example case: Martin found this duplicate target. That is, the same integral apart from a factor $4$. If there is a reason not to close the example case as a duplicate of the latter, I would like to hear it.
2. What to do with minor variants? As comments to questions similar to the current topic I often see lines like surely this is a duplicate of something... May be such commenters are more welcoming of closures? As abstract duplicates if nothing better is available. I propose that we continue to add links from all fitting questions to their respective abstract duplicates. In addition to helping the asker, this makes it easier to find better duplicate matches (when judged prudent), and keeps the lovingly maintained list of generalizations of common questions alive.

My (thank you, Capt'n Obvious) observation: Not unexpectedly the attitudes of the users to the use of abstract duplicates reflect how they view the site. Those who think of the site as primarily a teaching arena are often very much against, those who view the site mostly as a repository of knowledge also emphasize the enhanced searchability and better organization.

Answer 5

My opinion:

$A$ should be closed a duplicate of $B$ iff answers to $B$ are complete and appropriate answers to the question asked in $A$ (ignoring everything else).

So in particular:

• Most forms of context (the OP's attempted proof, the OP's level of mathematical sophistication, what the OP knows, etc.) should be ignored when considering whether to close as duplicate.

  Of course, it's helpful and constructive to make answers specific to the OP's context, BUT the ability to write such answers is less important than getting duplicates linked properly. Otherwise, we would have lots of copies of the same question, only with different attempted proofs, different background levels, etc. which is not in the spirit of mathSE.

• Abstract duplicates may or may not be valid duplicate targets, depending on whether the abstraction obscures or modifies the question too much.

  Specifically, if $B$ is a more general question than $A$, but not so far removed that answers to $B$ are not great answers to $A$, then $A$ should be closed as a duplicate of $B$. So (to answer the OP's case), computing $13^{3530} \mod 12348$ is a duplicate of $a^b \mod c$. On the other hand, "Why is addition of integers commutative?" isn't a duplicate of a question about category theory or about ordinal numbers.