A few minutes ago I started answering one of those "how does this sequence continue" question to which the standard answer is, essentially, "any way you like". The question was deleted while I was composing this answer. Perhaps there should be some generic such answer we can point to each time this kind of question recurs.

Edit in response to comments and answers.

The answer below is indeed too snarky, standing alone. In any particular case I would first try to guess the intended context and address that, then append the discussion pointing out that without more information there really are infinitely many answers.

There are many questions in math classes and on this site and in math riddles that present a finite sequence of integers and ask you determine how the sequence continues.

The only correct mathematical answer is that any continuation is possible.

Often the asker (perhaps a teacher) does not know that there is no single correct answer. There may be some context that helps you figure out what the asker had in mind.

One natural first step is to look at the differences, and then the differences of the differences. When these become constant there is a polynomial that generates the finite sequence you start with. Perhaps that's the continuation the poser had in mind.

When I was growing up in New York the classic riddle was to extend the sequence $$ 8,14,23,34, \ldots $$ The answer is $42$ - the next stop on the Manhattan branch of the Brighton Beach subway line.

If you ask at the Online Encyclopedia of Integer Sequences how $$ 1,2,3,4,5,… $$ continues you find 6257 entries. The first is the one you'd expect, of course.

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    $\begingroup$ This "however you want" answer often strikes me as not ideal, and not really appropriate. (It depends on the details though.) Either don't answer, or answer in the way presumably as intended (with a sidenote that it is not really a well defined question). $\endgroup$ – quid Jan 1 '19 at 16:15
  • $\begingroup$ @quid Answering any particular question I might try to guess what was intended, and suggest that - while noting that without more information the question has many answers. In this particular deleted question the OP asked for a method that would work on any sequence, with no example provided. $\endgroup$ – Ethan Bolker Jan 1 '19 at 16:21
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    $\begingroup$ Sounds like the question was rightfully deleted then. $\endgroup$ – Eevee Trainer Jan 1 '19 at 16:23
  • $\begingroup$ Since the post addresses "how does this sequence continue" in a rather generic fashion, might it not be added to another Question in this genre? The notion of abstract duplicate might apply (see List of Generalizations of Common Questions), if you believe this is really a "standard answer", though others may feel there's a substantial amount of variation involved. $\endgroup$ – hardmath Jan 1 '19 at 17:08
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    $\begingroup$ I think there was a previous attempt to handle this kind of question: What's wrong with asking for the next term in a sequence?... well, maybe many attempts if following the duplicate and linked questions. $\endgroup$ – Andrew T. Jan 2 '19 at 2:39
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    $\begingroup$ A few more past discussions related to this kind of questions (especially David E Speyer's answer is interesting): Guess the next number/guess the relation etc and What to do about question that ask about the next term in the sequence? $\endgroup$ – Martin Sleziak Jan 2 '19 at 5:53
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    $\begingroup$ A FAQ item for this type of questions was also discussed here: Do we need a FAQ question for “Guess what I am thinking” type of questions? Around that time the tag-info for the pattern-recognition tag was edited to include some basic info (OEIS, possibility of various solutions, etc.) $\endgroup$ – Martin Sleziak Jan 2 '19 at 6:38
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    $\begingroup$ It might be also worth checking the frequent tab of this tag to see whether some questions similar to the FAQ you suggest already exists. $\endgroup$ – Martin Sleziak Jan 2 '19 at 6:38
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    $\begingroup$ My classic is “5, 10, 20, 30, 36”, which has a unique entry in OEIS. $\endgroup$ – egreg Jan 2 '19 at 14:19
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    $\begingroup$ Classic indeed. :-) I had had to look it up. Not being fluent in Italian, the actual numbers never registered with me. $\endgroup$ – quid Jan 2 '19 at 15:45
  • $\begingroup$ I still don't understand what you are asking in the title. What is the "Orphan answer (worth saving)", to which you refer, in your title? $\endgroup$ – amWhy Jan 2 '19 at 21:19
  • $\begingroup$ @amWhy "Orphan" because by the time I finished the answer the question had been closed. But I wanted to post the answer anyway ... $\endgroup$ – Ethan Bolker Jan 2 '19 at 21:40
  • $\begingroup$ @EthanBolker. Thanks for clarifying. I would answer "worth saving?", with a resounding "No*, for reasons you acknowledge, and others point out in comments and answers. $\endgroup$ – amWhy Jan 2 '19 at 21:54
  • $\begingroup$ 42 is a very famous answer, but in this case, shouldn't the answer be Times Square? $\endgroup$ – Gerry Myerson Jan 2 '19 at 22:21

A few minutes ago I started answering one of those "how does this sequence continue" question to which the standard answer is, essentially, "any way you like".

That is not the "standard answer."

It's a snarky answer, usually coming from people who are - rightfully - tired of PSQs or problems otherwise posted without context. Anyone who posts that thinking it's a genuine, helpful answer is a bit out of their minds, as it takes an iota of common sense to realize that such questions come in a context of there being an underlying rule to such sequences, the question being, of course, what rule.

Yes, "any way you like" is technically correct, but let's not delude ourselves by pretending it's the most helpful or correct answer that the OP of any such question wants. You're not helping anyone by responding in so unhelpful of a manner and hiding behind "it's technically correct" while simultaneously knowing there is some underlying rule.

We could similarly do the same for most questions here: since probably 99% of the questions here do not explicitly define their various operations - they do not go into formal definitions of what $+$ means, what $\times$ means, what $\mathbb{Z}$ means, what $\int$ means. The statement of the continuation to a sequence being "whatever you want," is no more helpful than saying "well, technically, since these operations, these notations, these terminologies were not formally defined in the question, we can impose our own arbitrary meanings on them so they all have whatever answer you want."

$2+2 = \text{fish}$ for no other reason that I did not define what $+$ means.

See how pointless mathematics becomes in that light? The sheer futility and pointlessness in such an exercise alone should encapsulate why "any continuation you want" is a laughably poor answer. It spits on the very spirit of mathematics.

Any page made to emphasize this fact should also emphasize the importance that the context of the question is also posted, and that "any way you like" should not be encouraged as an answer on the premise that the sequence has an underlying rule or only one valid continuation.

And personally, no such page or post should be created. Maybe it could address a certain class of sequences, but to claim that making a page about all number sequences just to say "any continuation is valid" is self-defeating for the prescribed reasons.

If anything, this is a statement that questions posted with no context or attempts - posts such as PSQs - simply need to be swiftly closed. The closure message already provides details on what makes a proper post. So we already have the framework to address the issue.

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    $\begingroup$ +1 I think you're right, and have edited my question accordingly. $\endgroup$ – Ethan Bolker Jan 1 '19 at 16:42
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    $\begingroup$ -1 "See how pointless mathematics becomes in that light?" I reject the notion that sequence continuation has anything to do with mathematics, as much as with "logical thinking" (and a fair bit of reverse psychology) which is a wholly different discipline that can not really be taught via StackExchange format. Therefore also the comparison with the absurd definition of $+$ does not make sense. $\endgroup$ – Lord_Farin Jan 2 '19 at 13:19
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    $\begingroup$ The symbol $+$ has a well-defined meaning in mathematics (or, really, a collection of well-defined meanings, where the context usually makes clear what is going on; though ambiguity is possible: if $A$ and $B$ are sets, what is $A+B$? the union? the disjoint union? the Minkowski sum? without further context, any of the above could be meant). On the other hand, the question "What is next in the sequence?" does not have a well-defined or broadly understood "correct" answer. It is very much up to the asker of the question to make clear what is intended. $\endgroup$ – Xander Henderson Jan 2 '19 at 13:34
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    $\begingroup$ I don't find @EthanBolker's answer to be at all snarky. At worst, he is answering with a "frame challenge": he challenges the framing of the question, then provides an answer which explains why that framing is bad. It is a way of taking a question which is not about mathematics, and turning it into an opportunity to teach the asker something which is about mathematics. That being said, this basic answer has already appeared all over MSE, and the question should probably be closed as a duplicate and not answered again. $\endgroup$ – Xander Henderson Jan 2 '19 at 13:38
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    $\begingroup$ " It is very much up to the asker of the question to make clear what is intended." -- Which is precisely the point I'm getting at. Close questions that do not include appropriate context or explanation. This can apply to any question, including "continue the sequence" sort of questions. $\endgroup$ – Eevee Trainer Jan 2 '19 at 18:37
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    $\begingroup$ @XanderHenderson I disagree. The answer starts out snarky, but never explains clearly that there is not enough information to determine a unique "rule", nor do they explicitly point out that ' "What is next in the sequence?" does not have a well-defined or broadly understood "correct" answer." ' $\endgroup$ – amWhy Jan 2 '19 at 21:58

This subject comes up from time to time, see for example part of this answer Number-guessing, sum of all natural numbers and hot trend questions. One thing I'd want to stress is that not all questions of this type are the same.

I agree that in a strict sense almost all are presented in a way that makes them ill-posed and/or not really mathematics, but to quite varying degrees.

However, that is not that uncommon. Many word problems also suffer from this problem.

If $1$ painter takes $12$ hours to paint the house, then how long do $4$ painters take.

While I don't like this type of question all that much, it is still common and I think it's not quite to the point to just say the question is ill-posed as there is an implicit assumption.

Of course there are other types of word problems that are more a playful. Like one that starts "Assume you are the captain of a ship. {Some long story involving passengers getting on an of the ship.} How old is the captain?" or the other where somebody is walking to some place X, they meet a largish group that is described in a convoluted way that actually goes in the other direction and then the question is how many people are walking to X.

I feel that something similar is true for such sequence-continuation problems. What can be tricky is to keep them apart, but usually context that is provided can help to tell them apart.

I think it can still be a good idea to comment on the fact that there is an implicit assumption made there (and to make this assumption precise might be non-trivial). But nevertheless for questions that originate from a math-context (like a course), I think it's appropriate to address them as intended.

If somebody would post a Q&A that explains the difficulties with such questions, it could be a useful, but I think it should be more substantive than "every continuation is possible." That's a bit of a simplistic answer. There is a lot of interesting things to be said about such problems, see, e.g., https://arxiv.org/abs/math/0702086

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    $\begingroup$ I agree with this, but I believe comments to the asker, when no context is provided (I'm currently studying [this topic] in [this course]. We've already covered [this content]), need to encourage the asker to provide such context, without which the no-context-question is "unclear" or "lacking context". I think answers based on a "guess" (when no context, such as I list) are premature. In this particular case, I understand what Ethan was trying to communicate, but he should have expressed it in a comment, + request for more context. As an answer, it's unhelpful. $\endgroup$ – amWhy Jan 3 '19 at 21:28
  • $\begingroup$ Yes, I agree with that too. $\endgroup$ – quid Jan 3 '19 at 22:11

After writing the comment to Eevee Trainer's answer, I decided that it deserved elaboration as an answer.

The issue with "how does this sequence continue" is that it is not a mathematical question.

These questions are a mixture of common sense, pattern recognition, and reverse psychology -- especially when they are not multiple choice.

Because of this I contend that these questions fall in the same category as numerical addition and multiplication: it is not a question about mathematics, but rather a question which applies mathematical concepts.

Since the question does not aim to understand the underlying patterns (i.e. the mathematics), it is therefore not a good fit for Maths.SE and should be closed. It would probably be helpful to have a reference available where this difference is explained, so as to not leave the OP confused.

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    $\begingroup$ That's just not true like this. I'll write an answer to the meta question later, but there are mathematicians that expand considerable effort to "guess" the next term in a sequence. (And to avoid a misconception, I am talking about sequences of a few integers that arise in combinatorics for instance, not forecasting stock prizes or some such thing.) $\endgroup$ – quid Jan 2 '19 at 15:31

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