Hitting Walls in Proofs

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Hitting Walls in Proofs

Postby z4lis » Fri Jun 05, 2009 12:30 am UTC

What do you do when you've exhausted everything you can think to throw at a problem? Do you buckle down and force yourself to keep churning out ideas, scribbling on some paper? Do you go outside and do something completely different, and come back to it later, refreshed? Or perhaps you just sleep on it and the solution comes to you the next morning? I was just curious what most mathematicians do when dealing with this inevitable situation.

For me, I'd say I'm inclined to bashing at it until I grow tired and can't think about it productively anymore, and then force myself to work on other things. Only once has the "sleep" bit happened to me, and it was rather exciting. :D Evidently, I also create forum posts.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

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Re: Hitting Walls in Proofs

Postby t0rajir0u » Fri Jun 05, 2009 12:36 am UTC

I find that doing something different helps. Your brain unconsciously works on things even when you don't pay attention to them, or something like that. Sleep helps too, as does eating.

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Re: Hitting Walls in Proofs

Postby Buttons » Fri Jun 05, 2009 12:41 am UTC

I once had a professor who measured how long we should work on a problem set by the number of showers we should take. I also endorse train rides.

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Re: Hitting Walls in Proofs

Postby antonfire » Fri Jun 05, 2009 12:43 am UTC

I find that the "go do something else" thing happens to me naturally. Perhaps more often than it should.
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Re: Hitting Walls in Proofs

Postby NathanielJ » Fri Jun 05, 2009 12:47 am UTC

My routine with difficult problems is generally:

1. Try to build a counterexample, and observe why I can't.
2. Use experience from (1) to build on the proof.
3. Take a break once I get stuck again.

Repeat as needed.
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Re: Hitting Walls in Proofs

Postby Cosmologicon » Fri Jun 05, 2009 1:08 am UTC

Change your point of view.

That video won't tell you anything you don't already know, but it's still fun.

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Re: Hitting Walls in Proofs

Postby skeptical scientist » Fri Jun 05, 2009 1:16 am UTC

I try to do something completely different, but sometimes that approach backfires. I've gotten stuck on a problem and decided to read a book, only to realize that I haven't absorbed anything I've read in the last 5 pages because I'm still thinking about that stupid problem I'm stuck on. If that happens, I decide I wasn't ready to let it go, so I go back and think about it more until I'm really frustrated (at that point I can usually go back to reading and actually read).

When I am working on a problem and get stuck, I also do the "try and build a counterexample" thing. So I generally only get frustrated and stop when I'm stuck on both directions - proving the theorem and trying to find a counterexample.
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Re: Hitting Walls in Proofs

Postby Qoppa » Fri Jun 05, 2009 3:35 am UTC

When I get stuck on a problem, I usually end up getting up and doing something else for a bit to keep myself from trying the same approach(es) again and again. Even while doing something else though, I'll generally keep thinking about it in the back of my mind, and often times I'll figure it out or at least get another idea to try. I obsess about problems I can't get, especially if I know I should be able to get them.

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Re: Hitting Walls in Proofs

Postby angelfire » Fri Jun 05, 2009 4:14 am UTC

if its homework of sorts, maybe try working backwards from the answer?
you might come to a point where the proof looks similar to the part you got stuck, then flip the portion and voila, qed

but i suppose this wont work for eh, more advanced stuff ;p

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Re: Hitting Walls in Proofs

Postby intertubes » Fri Jun 05, 2009 4:18 am UTC

I've found that just taking a break and coming back to it will help me most of the time; usually I made a mistake that I'll catch when I leave and come back.
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Re: Hitting Walls in Proofs

Postby achan1058 » Fri Jun 05, 2009 4:37 am UTC

angelfire wrote:but i suppose this wont work for eh, more advanced stuff ;p
As far as I know, it still does, though not always.

What I do myself is usually to take a walk or something.

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Re: Hitting Walls in Proofs

Postby LLCoolDave » Fri Jun 05, 2009 6:06 am UTC

angelfire wrote:if its homework of sorts, maybe try working backwards from the answer?
you might come to a point where the proof looks similar to the part you got stuck, then flip the portion and voila, qed

but i suppose this wont work for eh, more advanced stuff ;p

It may not work if you aren't quite clear on how the answer should look like :P

I think it's most frustrating when you can intuitively see something to be true, yet are completely unable to find some sort of formal reasoning that even supports your idea with a bit of handwaving that could be closed by more thought.
Sometimes it helps just working through the definitions and basic theorems concerning the problem you're tackling again, there may be something in there you are missing or that leads to a new train of thought.

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Re: Hitting Walls in Proofs

Postby pingu » Sun Jun 07, 2009 1:19 am UTC

In an exam, as soon as I hit the slightest hurdle, I go on to the next question. My brain thinks very passively I guess? So I can frequently be seen flipping through the pages, working on several problems at once.

Outside of an exam, if I can't see a way to do something, I give it a small moment, and then work on the next problem (basically the same thing, but less tendency to change what I'm working on). When I get back to it, usually I'll have thought of a different way of looking at it or something. Otherwise rinse and repeat. If it's the last problem, I'll work on something else entirely (even something random, not related to the subject).

So apparently my one trick is using my brain like a multi-core processor.

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Re: Hitting Walls in Proofs

Postby quintopia » Sun Jun 07, 2009 7:43 am UTC

What works the best for me: Talk it out with someone who works on similar but somehow different things. Also, just someone who knows more (or at least as much) about the broader subject area than you. If you can hash it out with someone else on a whiteboard, get them caught up on everything you've got so far, sometimes they'll see a technique from their own work that might be worth trying to apply.

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Re: Hitting Walls in Proofs

Postby acb » Sun Jun 07, 2009 9:53 am UTC

Talking it through with someone is really helpful. It is useful to have someone there to shoot down any ideas that won't work which means they I am more likely to throw them out there in the first place. Another good one is to try and prove/find in your notes something that is similar to what you are stuck on, and apply any techniques learnt to your problem.

My favourite is to determine what the simplest case is and prove it for that, then see if the proof can be generalised or if there is anything in the proof that will help.

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Re: Hitting Walls in Proofs

Postby Certhas » Sun Jun 07, 2009 12:32 pm UTC

I let my mind wander, look out of the window, go for a walk.

I'm actually really unproductive when sat in front of a desk with a pencil. But then the next morning under the shower, or while walking to the office I'll see an idea, or think of something useful to read up on.

The bad part about this is that if I'm really stuck I don't have any ways to push for the solution. You can't go for more vigorous walks.

Then it becomes invaluable to talk to other people. I am in mathematical physics, from what I have seen that is more collaborative then pure maths, so often I'll be working on something with other people, and I've found it best to work with people who are complementary to my approach, who throw the kitchen sink at the problem. It often means we can unstuck each other.
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