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It's the 5th time this happens to me. Tried to solve some simple problem for about 20 minutes (four of them were just help-asking on passages of proofs) and decide to ask for some help/hint on Mathematics. Then, when I was elaborating the question, writing down the main problem and pointing the problematic part of the thing, I just discovered that something that can do the trick. Is that usual for you?

First question here, don't know if this is a suitable place.

And if there is another tag that can fit here, please edit.

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    $\begingroup$ Its why we encourage that questions contain plenty of context. This helps everyone out, including you! $\endgroup$
    – Simply Beautiful Art
    Commented Jul 29, 2017 at 23:52
  • $\begingroup$ @SimplyBeautifulArt yeah, that's true $\endgroup$
    – Filburt
    Commented Jul 30, 2017 at 0:00
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    $\begingroup$ Closely related: Formulating the problem helps solving it ... $\endgroup$
    – hardmath
    Commented Jul 30, 2017 at 3:10
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    $\begingroup$ As a tip for next time: write your problem on some scratch pad (e.g. a text editor or the sandbox), being very sure to keep explanations and your musings as clear as possible. Walk away for a few minutes, and proofread your draft again. If you still don't have a solution in mind after doing all that, then you post your question. $\endgroup$
    – J. M. ain't a mathematician
    Commented Jul 30, 2017 at 3:57
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    $\begingroup$ Yes, this is not just usual, this is the whole point of the Feynman algorithm for solving problems. $\endgroup$
    – Asaf Karagila Mod
    Commented Jul 30, 2017 at 8:35
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    $\begingroup$ Yes, 70% times when my friend calls for my help he starts describing the problem and then he says "nevermind" and hangs up. $\endgroup$
    – kingW3
    Commented Jul 30, 2017 at 9:54
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    $\begingroup$ This happens to me all the time. Formulating a problem and your progress in a way to communicate it to someone else frequently puts things in a perspective that solves the problem. This is the whole reason that preemptively answering "what have you tried" is a healthy and useful exercise. $\endgroup$
    – rschwieb
    Commented Jul 30, 2017 at 18:38
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    $\begingroup$ It is so common it's almost an industry standard (albeit in a different industry) to exploit it. See rubber duck debugging in software development. $\endgroup$
    – Arthur
    Commented Jul 31, 2017 at 7:20
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    $\begingroup$ As a side note, another approach that helps me solve problems is to sleep on them. This does not mean you should think about the problem all night. In fact, you should stop thinking about the problem for the rest of the day and resume tomorrow. There have been countless times when I've solved problems in under 5 minutes after waking up which I could not manage just yesterday. $\endgroup$
    – Simply Beautiful Art
    Commented Aug 1, 2017 at 21:51
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    $\begingroup$ A couple of years ago I blogged: “As often happens, I found the answer myself shortly after I asked the question. I wonder if the reason for this is that my time to come up with the answer is Poisson-distributed. Then if I set a time threshold for how long I'll work on the problem before asking about it, I am likely to find the answer to almost any question that exceeds the threshold shortly after I exceed the threshold. But if I set the threshold higher, this would still be true, so there is no way to win this particular game.” $\endgroup$
    – MJD
    Commented Aug 2, 2017 at 18:28
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    $\begingroup$ This was so awesome, I decided to link it in How to ask a good question: Provide Context: Include your work. Just as a heads up. $\endgroup$
    – Simply Beautiful Art
    Commented Oct 26, 2017 at 0:44

8 Answers 8

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Yes, this is actually rather common!

It's easy to get tunnel vision, where you're too focused on things that don't work or have a mistake stuck in your head. The mere act of switching gears to post the problem can be enough to widen your view.

Furthermore, writing out the question and elaborating on what the difficulty is can make one think more carefully about the problem than one had previously, at least when one intends to write well. For many problems, "simple" problems in particular, simply being careful about what you're doing may be enough to solve the problem.

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  • $\begingroup$ In the last case of mine, the problem was a passage on a proof. A simple trick, but with that lot of context I couldn't see it. When I elaborated the statement to fit Mathematics (with the minimum hypothesis possible), I just looked again at the problem and ... voilà $\endgroup$
    – Filburt
    Commented Jul 30, 2017 at 13:10
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    $\begingroup$ @Filburt (the good ones) I write 3 or 4 questions per day, I post only one every month. $\endgroup$
    – reuns
    Commented Aug 11, 2017 at 5:16
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Let me post a comment here.

I am a quite old physicist with a lot of problems and I need to express them. The problem is that my wife does not understand anything in physics, mathematics, chemistry.

So, to explain (better say, to describe) my problem, I need to shift it in rather simple terms and words in order that she, at least, looks to be understanding something.

Doing so, I solved more than a dozen of problems of mine (but in almost 60 years) !

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    $\begingroup$ "You don't actually understand something unless you can explain it to a five-year old", or so the old aphorism goes... $\endgroup$
    – J. M. ain't a mathematician
    Commented Jul 30, 2017 at 17:26
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It's actually super common, you'll find that when explaining a problem you understand that it needs to be broken down into it's smaller problematic parts, a problem is just a collation of sub problems and putting time into thinking about those sub problems (like you do when you explain the problem to someone) is a useful trick to understand the problem better as a whole and therefore gain context and insight into a more workable solution - it's actually used as a method for solving problems. We programmers tend to call it rubber duck programming because we often explain our problem to a rubber duck and just by the act of explaining it - it helps us come up with the solution.

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    $\begingroup$ Do you have to supply your own rubber duck, or are they provided by your employer? $\endgroup$
    – Gerry Myerson
    Commented Jul 31, 2017 at 13:05
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    $\begingroup$ Solving problems by breaking them into smaller parts is a extremely powerful tool, in programming one of such tools is dynamic programming. $\endgroup$
    – kingW3
    Commented Jul 31, 2017 at 13:09
  • $\begingroup$ @GerryMyerson Depends on the company! I'm pretty sure google supply rubber ducks :D could be one of those google myths though... $\endgroup$
    – Eujinks
    Commented Jul 31, 2017 at 13:13
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I found this and couldn't help but share! Quacky!

Rubber Duck Debugging:

The name is a reference to a story in the book The Pragmatic Programmer in which a programmer would carry around a rubber duck and debug their code by forcing themselves to explain it, line-by-line, to the duck.

As it turns out, it's actually put into practice in programming, and it probably appears under other names as well.

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    $\begingroup$ But, what's in it for the duck? $\endgroup$
    – Gerry Myerson
    Commented Jun 27, 2021 at 6:32
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    $\begingroup$ A friend perhaps? :-) $\endgroup$
    – Simply Beautiful Art
    Commented Jun 27, 2021 at 12:29
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I use this technique extensively. After looking at a new problem or concept I like to take the dogs for a walk or go for a bike ride and "talk to myself."

I say, "Ok Dave, What do we know? What do we want to know? What are the links between what we know and what we want to know?"

Focusing on articulating the problem without worrying about solving the problem often brings new insight.

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I think the process of explaining a problem to someone else can lead you to finding a solution that you hadn’t thought of before.

Here is an example that happened to me. A university professor told me about a special property of the number 9. He noticed that 9 has three positive integer divisors {1,3,9} and that $\sqrt{9} = 3$. He wondered if there were any other positive integers $n$ such $\sqrt{n}$ = the number of positive integer divisors of $n$.

I quickly came up with a trivial example, the number 1. I tried for a while to come with another example but did not succeed. In fact, at this point, my intuition was that no other positive integer satisfied the desired property. But I had no proof. I wrote a computer program in Maple that examined all the positive integers up to $1,000,000,000,000$ and the program only found 1 and 9 satisfied the desired property. This result supported my conjecture but did not prove it.

After a few days without solving this problem, I met a Math student by chance and started to explain the problem to her. During that talk I discovered that my conjecture was true (1 and 9 are the only positive integers that satisfy the desired property) and in that talk I figured out a proof of this.

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It is very common in software development to show a problem to a colleague / co-worker and start to explain why you think that your program should work but, before they even say anything, you spot the problem yourself. Somehow, needing to explain the problem to another has enabled you to see what you could not spot alone. We have sometimes considered using a mannequin for this purpose. Composing a good question for this site may achieve the same.

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Situated cognition is a theory in cognitive science that explains this phenomenon: Situated cognition states that much of cognition, especially the 'higher' forms of cognition, including reasoning, decision-making, planning, and problem-solving, happens not solely inside your head, but comes about through interactions with your environment, particularly when this environment includes a symbolic language, whether this be natural language (like English), or a specialised language (math, diagrams, etc).

So, whereas we often think that our symbolic expressions are merely expressing our already completed thoughts about something, situated cognition says that these expressions are much more than that: they actually enable you to think, or at least think further, than you could without that language. Language is a powerful tool to help us think, and help us solve problems.

Indeed, imagine how much math you would be able to accomplish without the language of math ... not much at all! And think about this: why is it that when faced with a tough problem, you often find yourself talking to yourself? If all the thinking is happening inside our heads, what use is it to express those thoughts to yourself? But situated cognition says: expressing these ideas to yourself is useful, as you are engaging yourself in a dialogue, and dialogues are yet another way to interact with your environment; even if you have a dialogue with yourself, it is the act of hearing those expressions that will cue certain responses, and thus further the reasoning process.

So yes, the act of carefully formulating and describing the problem is part and parcel of the very act of problem-solving, so what happened to you is not at all surprising, and is in fact very common, and has a perfectly good cognitive explanation.

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  • $\begingroup$ Do you really talk to yourself when solving math problems? That's something I've never done. It would greatly slow me down since I can reason mentally much quicker than verbally. $\endgroup$
    – Bill Dubuque
    Commented Aug 8, 2017 at 3:07
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    $\begingroup$ I talk to myself internally, if that counts. I don't see the need to disturb neighbors while wrestling a tough problem. $\endgroup$
    – J. M. ain't a mathematician
    Commented Aug 8, 2017 at 3:19
  • $\begingroup$ @BillDubuque Sometimes I do, yes. But then, I am not a very good mathematician. And I should be clear that I am not saying that all thinking has to go through external language, or even using an inner voice .... but sometimes it does help. $\endgroup$
    – Bram28
    Commented Aug 8, 2017 at 3:19
  • $\begingroup$ @J.M.isn'tamathematician Yes, if you use an inner voice, situated cognition says that that is an internalization of something that used to happen through actual speech and hearing ... presumably the brain figured out the shortcut straight from the speech production centers to the speech processing centers, but the process is still regarded as a dialogue. $\endgroup$
    – Bram28
    Commented Aug 8, 2017 at 3:21

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