This question here: Interview riddle seems to have gotten a lot of views (perhaps by someone posting to reddit or something), with a lot of me too answers, leading to a protection. Such questions have no definite answer.

I think (i.e. my opinion) is that such questions are basically out of scope of this site.

I think we should be closing questions like these, not as off-topic, but as a duplicate of a Generalized Question (which is yet to be created, on the main site, tagged with (faq), please read the linked meta question for more details about the (faq) questions). This (faq) question will answer why these type of questions are typically pointless (perhaps not in those direct terms), and how one can mathematically and concretely justify any solution.

The reason I am posting this on meta is that this question seems really popular, and might lead to close/reopen wars (it has been reopened after one closure!). If there is sufficient support for/resistance to this proposal, we can direct the close/reopen war soldiers to this thread.

PS: I tried searching for a faq question already, but didn't find one.
PS1: Also could not seem to find an existing question which we be a good candidate for the generalized question.

  • $\begingroup$ It's closed now. $\endgroup$ – Daniel Fischer Jul 18 '14 at 18:42
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    $\begingroup$ I'm not sure it's out of scope. The question can be interpreted as asking for a "simple" function $f(x,y)$ such that $f(3,4)=8$, $f(4,5)=50$, etc. Yes, "simple" is somewhat subjective, but even computer algebra system have an idea of what it means to simplify an expression. To be clear: I won't miss "continue this pattern" questions if they are all closed, but I'm not convinced they deserve to be closed. $\endgroup$ – user147263 Jul 18 '14 at 18:44
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    $\begingroup$ Maybe it should be pointed out that you can't close a question on main as a duplicate of a meta thread. $\endgroup$ – Asaf Karagila Jul 18 '14 at 19:29
  • $\begingroup$ @AsafKaragila: The point was to create an answer on the main site, which answers a general question, and close it as a dupe of that. $\endgroup$ – Aryabhata Jul 18 '14 at 23:35
  • $\begingroup$ Oh, that wasn't clear, then. $\endgroup$ – Asaf Karagila Jul 18 '14 at 23:42
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    $\begingroup$ Why not just move this type of question to puzzling.stackexchange.com? It seems precisely on topic there. $\endgroup$ – Bamboo Jul 19 '14 at 19:22
  • $\begingroup$ @ariddle: I agree that that is another option. Not sure how easy it is to setup a migration path though (and how receptive that site will be). $\endgroup$ – Aryabhata Jul 19 '14 at 21:49
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    $\begingroup$ @ariddle (and Aryabhata, I guess): Since puzzling.SE is in beta, a migration path will not be set up. It is generally also strongly discouraged for moderators to migrate questions to beta sites. Whether or not such questions would even be acceptable on puzzling.SE is something that can be asked on Puzzling Meta. (I'm personally not certain how much of a future puzzling.SE has. The basic site stats seem to suggest that it's in trouble.) $\endgroup$ – user642796 Jul 20 '14 at 6:12
  • $\begingroup$ What is wrong with creating a tag and then ignoring said tag? $\endgroup$ – user1729 Jul 21 '14 at 11:42
  • $\begingroup$ Also, related question from the main site. The OP there asked quite a few questions about this subject, both here and on meta (they seemed to be trying to undermine a book), and is the OP in the post boywholived links to in their answer. $\endgroup$ – user1729 Jul 21 '14 at 11:43
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    $\begingroup$ Also also, I think such a faq-thread is a bad idea. It is likely to gain a lot of upvotes, and so a faq will be pushed to the top of the highest voted question list, which would be silly. $\endgroup$ – user1729 Jul 21 '14 at 11:45
  • $\begingroup$ @user1729: I don't see why a generalized question at the top of voted list would be silly. It would be a mathematical question on-topic on this site. In fact getting a (faq) question pushed up is a good thing. In any case, it is probably better than getting a 1000 of "guess what is next" type of questions being pushed to the top :-) $\endgroup$ – Aryabhata Jul 21 '14 at 17:45
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    $\begingroup$ If we need any more examples to think about, here's another one. $\endgroup$ – rschwieb Aug 20 '14 at 18:13

As some answers above have mentioned, the skill of recognizing pattern is important in mathematics, much like in many other fields. I agree with that. However, I also agree with the asker that those question are nothing more than "guess what I am thinking". In science, recognizing a pattern produce conjecture, but those conjecture can be tested in reality. If you are doing a mathematical problem, and you produce a sequence of number to see the behaviour of the structure at small size for example, then you could form a conjecture on that sequence, and your guess can be tested against the mathematical reality. In all these cases, the answer is unique, and can be checked. Further more, the context would also tell you what answer is reasonable. For example, if physicist are conjecturing the possible energy level of certain atom, then even if the absolute of the energy level in eV happened to match the index of first letter in words in the Spanish translation of Sun Tzu's The Art of War, they are not going to form a conjecture linking the two. Similarly, in mathematics, let's say you are trying to find the index of the center of a sequence of group, and the first few term happens to match the digit in expansion of $\pi$ in base $13$, you aren't going to form a conjecture on that, even if they matches rather well for a long time.

By contrast, the question such as the interview riddle above are completely out of context (well, technically, if we know what position and which company is the person interviewing for, it could provide a bit of clue, but that's hardly enough). Anything is a possible thing that is relevant to the pattern: ages of famous political figures, common isotopes of certain element, word counts in Shakespeare, relationship between zeta function and quarternion, number of wings of certain insects, GDP of various countries, etc. They are best for Puzzling site. Sure they might produce mathematically complicated answer here, but those answer are meaningless: complicated answer are unlikely to be the correct answer, but there are no contexts or standard to judge them. On the other hand, if it were to be in the hand of the Puzzling site, while they might not produce mathematically sophisticated answer, they can take into account a wide variety of possible pattern, mathematical or not.

EDIT: Also, just to illustrate the different between the two, here is my example of the 2 version of the possible question:

Bad question:

Find the next 10 values in the sequence:


Good question:

Suppose we have the sequence $a_{n}$ such that for any $n$ there exist an integer $r$ where $a_{n}\leq r\leq a_{n+1}$ and that $(a_{n+1}^{2}-a_{n+1}r-r^{2})^{2}=(r^{2}-ra_{n}-a_{n}^{2})^{2}$. The sequence start out as follow:


Find all the possible values of the next 10 values in the sequence.

Also, I found this older question here on Meta: Number-guessing, sum of all natural numbers and hot trend questions and an even older one linked from there Guess the next number/guess the relation etc

Oh, I already posted this in the comment, but I think this is too funny to not link to: http://spikedmath.com/492.html


I don't support closing such questions outright. Pattern recognition is a part of mathematics. The skill of recognizing a pattern in a sequence of numbers is more valuable than the skill of repeating the sentence "you can find a formula to fit any set of points". Some patterns have more mathematics underneath them than others, and we won't know at first glance.

Sure, repetitive questions like "what does $\odot$ mean in
$$2\odot 3 = 1, \quad 3\odot 5 = 6,\quad 5\odot 7 = 2, \quad 7\odot 11 = 12, \quad 11 \odot 13 = 6,\quad 13\odot 17 = 28"$$ would quickly get boring, but it's a self-correcting problem: when toys get boring, they are quickly abandoned (and then a question gets closed without much argument). Meanwhile, this particular toy brought over 11K page views to the site, most of which (I presume) come from outside the regular visitors. Chances are, some of them clicked more than one question.

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    $\begingroup$ Agree with boring part. Disagree with the abandoned part. There will be newer users who will see these older highly popular questions and will post new ones. The broken window theory applies here, and one the reasons I propose to have a generalized faq question (read this meta thread for what I am talking about: meta.math.stackexchange.com/questions/1868/…) which can serve as the parent question. This is a site for objective Q&A, and these questions are inherently subjective. In fact one could argue to close them as off-topic/localized. $\endgroup$ – Aryabhata Jul 19 '14 at 0:03
  • $\begingroup$ It because of the popularity that I even suggest putting some effort into a generalized question, and closing as a dupe, rather than closing as off-topic outright. $\endgroup$ – Aryabhata Jul 19 '14 at 0:05
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    $\begingroup$ @Aryabhata But what would the generalized Q&A be? A lecture on how one can fit a function to any set of points? The puzzle-like questions are of wide interest because they present a concrete, accessible challenge. This does not generalize. $\endgroup$ – user147263 Jul 19 '14 at 4:38
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    $\begingroup$ It obviously generalizes, and has at-least one nice answer backed by concrete and accessible mathematics. Surprisingly, it even has wide application. You either are unaware, or are forgetting the purpose of the stackexchange sites: Objective Q&A. As a corollary to this, not all questions (irrespective of whether you think they are of wide interest or have any "mathematical" content) are on-topic. These guessing questions definitely fall into that bracket. For such puzzles, maybe puzzling.stackexchange.com is more appropriate (and perhaps even welcome there). $\endgroup$ – Aryabhata Jul 19 '14 at 4:57
  • $\begingroup$ The problem with the original question is rather this: If we are not supposed to think of $\times$ as multiplication, are we supposed to think of $0,\ldots,9$ as decimal digits? Or as digits in a positional system? Or as digits at all? And why then should $=$ still denote equality? It is precisely this ambiguity which turns a "find a nice function" question to a "guess what I'm thinking" question. $\endgroup$ – Hagen von Eitzen Aug 1 '14 at 12:00

I have answered one such question here. Based on that experience I think that user who proposed the question believes that there is a unique answer to his question. I guess this 'belief' can be seen across all similar questions.

My answer to such question is to educate them any arbitrary number qualifies as an answer. From my experience, the questioner was visibly happy after knowing this.

On a different perspective.

I think that these questions does add value to the site. We should acknowledge the fact that "interview riddle" brought ~11k views which certainly is not a small feat. The society often conceives mathematician as in this quote "Mathematicians won the war. Mathematicians broke the Japanese codes..."(Beautiful Mind). In that regard, "The interview question" did bring forward many non-trivial answers like this (which is not the level of answer, you receive at Puzzling.SE) and is definitely harder than many 'on-topic' questions, thus qualifying as a better 'time pass' to many users. And closing them outright is a bad move.

So if you want a generalized question, I consider an answer that educates the user that there is more than one answer to this question. In that case this question is worth considering.

Quoting from this meta post:

For professional mathematicians, edge cases/non-obvious counterexamples can be interesting, as can the detailed hypotheses necessary to make certain statements true/false. Indeed, understanding such things is part of the pleasure of mastering a theory. But not all questions have to be answered from that vantage point.

I remember an early question I answered where the OP, coming from a quantum mechanics background, asked if commuting operators were necessarily simultenously diagonalizable. Several of the initial answers emphasized the edge cases that make this literally false; on the other hand, it is typically true, and is a basic principle of quantum mechanics, and I don't think focusing on the subtleties of why it wouldn't always hold was necessarily the best answer for the OP.

In general, I would hope that people are thoughtful about where an OP is coming from, and about what kind of answer they might be looking for. Let's try to encourage people's appreciations of mathematics. I hope that our site can show an enjoyment of mathematics as something wonderful, not just as something recondite and technical, doctrinaire, full of edge cases, counterexamples, and cautions against error.

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    $\begingroup$ Your answer is a good candidate for the generalized question. As to the amount of views, I would say that is a function of bikeshed + posting on reddit (or some traffic driving site), rather than the question type itself. For instance, consider this question: math.stackexchange.com/questions/33215/what-is-48%C3%B7293 which drove a lot of traffic from facebook: yet it was closed (rightly IMO), as a dupe of a general question which clarifies matters. Yes, every special case might have some surprising solution, but that would imply that we never close any questions as abstract dupes. $\endgroup$ – Aryabhata Jul 19 '14 at 15:01
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    $\begingroup$ I see that meta post you link as more as a "do not be hostile to such question asker" rather than "allow guess the number to be open". In fact, that meta post seems to agree that linking to a generic question with excellent answers is a good idea (and by implication, perhaps closing as dupe). Note that closing questions as dupe is definitely not an hostile act. I would also like to mention that guess the number questions which provide context (like: "I was counting the number of triangulation and got these for till n=5") etc, are perfectly valid questions. $\endgroup$ – Aryabhata Jul 19 '14 at 15:14
  • $\begingroup$ Saying that there is more than one answer is quite different from saying every answer is equally valid, and therefore the question is not worth thinking about. There is a -- partly subjective -- concept of complexity of a function, and the asker of such a question wants to fit a function of low complexity to the given data. Interpolating polynomial isn't a satisfactory answer unless it happens to have much fewer nonzero terms than one would expact (or some other special structure, like $(x+1)^n$). $\endgroup$ – user147263 Jul 19 '14 at 16:55
  • $\begingroup$ Imagine you are working on a combinatorial problem involving integer parameter $n$. You handled the cases $n=1,2,3,4,5$ and found the answers to be $1,3,7,15,31$. Do you say to yourself "alas, this tells me nothing about the problem, the answer for $n=6$ could be any number, because Lagrange polynomial"? Or do you say: "aha, I see what's going on"? $\endgroup$ – user147263 Jul 19 '14 at 16:58
  • $\begingroup$ @Thisismuchhealthier. The point I wished to make was: this problem does not have a unique solution. I don't want to close such questions either. $\endgroup$ – hrkrshnn Jul 19 '14 at 17:05
  • $\begingroup$ My point is that I don't think "any arbitrary number qualifies as an answer" is a satisfactory answer. Fitting a general polynomial of degree $(n-1)$ to $n$ data points is obvious overfitting, and unlikely to be a satisfactory model for the data. The search for a model with few parameters that can match the observations is a nontrivial task. It may be that there is no unique best model, but some will be better than others -- consequently, some answers will be better than others. $\endgroup$ – user147263 Jul 19 '14 at 20:46
  • $\begingroup$ @Thisismuchhealthier. Your example is irrelevant to the point of this meta thread. In your example, you are working on a combinatorial problem, so you have a context right there, and would actually make a fine question on this site. The problem is mainly with "Guess what I am thinking" kind of riddles/questions, which are entirely subjective (and of little value to this site). $\endgroup$ – Aryabhata Jul 19 '14 at 21:48
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    $\begingroup$ @Aryabhata The number of parameters in a model is not something subjective. Let's drop the context from my example: what is the next number in "$1,3,7,15,31,\dots$"? Someone answers: $63$, because it fits the function $2^n-1$. Another person says: $42$, because it fits the function $$\frac{-19}{120}n^5+\frac{59}{24}n^4 -\frac{335}{24}n^3+\frac{901}{24}n^2-\frac{2693}{60}n+20$$ Would you say that both answers are equally good, and the comparison of them is entirely subjective? $\endgroup$ – user147263 Jul 19 '14 at 21:53
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    $\begingroup$ @Thisismuchhealthier: Yes it is subjective. Coming from a CS background, I prefer polynomials :-). If you have anything which will make the question less subjective, add it to the question! For instance, here you can ask for a formula which can be evaluated in $O(\log n)$ time if you are angling for "simple" etc. $\endgroup$ – Aryabhata Jul 19 '14 at 21:57
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    $\begingroup$ I think this spikedmath.com/492.html would be a good addition to the discussion. I agree with Aryabhata here: it's all context. If the sequence come from some sort of Ramsey theory number, I might expect the number to increase much faster than exponentially, and believe that the apparent exponential growth is just an irregularity at small number. If I am doing counting on a structure with obvious self-similarity, an exponential growth could be expected. Integer sequences without context is something you can merely conjectured on, and there are no standards to judge the answer. $\endgroup$ – Gina Jul 20 '14 at 4:20
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    $\begingroup$ @Thisismuchhealthier. About the question I answered, OP did find a simple expression for the next term, but "BUT this is not the correct answer"(quoting OP) was his conclusion on finding that it didn't match the answer as per book. Don't you think in this case "any arbitrary answer qualifies as an answer" is satisfactory as this brings confidence to OP to accept that his relation also qualifies as an 'answer'? $\endgroup$ – hrkrshnn Jul 20 '14 at 6:07
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    $\begingroup$ @boywholived I think what you said was generally reasonable, and if OP was satisfied, all the better. I would not be satisfied with "any arbitrary number qualifies", because I expect an answer to be supported by a formula with low complexity relative to the size of data set. This requirement is basically an unwritten rule of all such puzzles. Depending on the quality of the puzzle, it may not identify a clear winner; still, some answers will be clearly better than others. Following $1,3,7,15,31$ with $10^{10}$ would take a contrived formula indeed. $\endgroup$ – user147263 Jul 20 '14 at 6:15

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