
Dealing with Hard Problems (2015) - mkagenius
https://artofproblemsolving.com/articles/hard-problems
======
joshe
Polya's little "How to solve it" booklet has great advice.

It's great because it's addressed to math classes but based on the practices
of a research mathematician. So it's not just solving the problems in the
book, it's also about solving new unsolved problems. It's specific to math,
but really useful for so many other questions.

There is much more to the book but one of my favorite parts is the Heuristics
and their simple explanations. Wikipedia has a good table of them

    
    
      Analogy         Can you find a problem analogous to your problem and solve that?
      Generalization  Can you find a problem more general than your problem?
      Induction       Can you solve your problem by deriving a generalization from some examples?
      Specialization  Can you find a problem more specialized?
      Draw a Figure   Can you draw a picture of the problem?
      ...
    

[https://en.wikipedia.org/wiki/How_to_Solve_It](https://en.wikipedia.org/wiki/How_to_Solve_It)

The calm, deliberate attitude of the book is infectious, it's great for
getting unstuck from tough problems. So do read the book itself in addition to
the wikipedia article.

~~~
ascar
Thanks for sharing this! The steps in Wikipedia are so clear and concise in
describing how I attempted to track down different bugs.

I'll definitely also read the book and use it as a guideline on what to try
next instead of randomly asking myself "what else can I try?".

------
akyu
I got a great deal of insight on problem solving from being a university
teaching assistant for 2 years. Every couple of weeks or so, there would be a
particularly difficult assignment, and I would get many visitors during office
hours that week.

The first thing I would always do is have the student walk me through their
current understanding of the problem. What surprised me was how incredibly
wide the range of mental models students used was. Also it was interesting
that quite often students make their mental models far more complicated than
the actual problem.

My job was really just to help the student simply and clarify their mental
model, and from there they usually solved the problem on their own.

Smart data structures are better than smart code. I think this applies to your
mind as well as code.

~~~
crispyambulance
Kudos for doing that.

Such techniques involve asking pointed questions to the student and actually
listening to what they say. This the basis for the Socratic Method and it
fucking works.

Sadly, we have become so accustomed to "searching" for answers with queries
(eg "googling it") that we're forgetting about really effective ways of
learning and solving problems. We're even forgetting that learning and solving
problems are the really part of the same activity.

The anti-pattern emerging is the "stackoverflow" style of problem-solving
where it's all about the learner asking "the right questions" and getting
smacked down for "the wrong questions."

~~~
toxik
I think this anti-newbie culture has been around far far longer than SO. It's
probably just human nature to deride the rookies.

------
yantrams
Taking a break from sitting at the desk and going for a walk and thinking
about the problem has worked surprisingly well for me many times. It was
something I first read about in a Poincare's article on how Mathematical ideas
are generated.

Coming to general methods of problem solving in recreational math, I always
look for symmetry and transforming the problem from one problem space to
another and induction techniques before attacking it. For a research/modelling
problem, my approach is to start working on the most simplified version of it
and gradually progress towards the original problem.

There is an excellent numberphile video on Josephus problem that illustrates
how examining specific instances to observe patterns can help solve the
general case.

[https://www.youtube.com/watch?v=uCsD3ZGzMgE](https://www.youtube.com/watch?v=uCsD3ZGzMgE)

Edit: Formatting

~~~
kumarvvr
I am a developer and almost all the solutions to my software engineering
(architectural) problems have occurred to me when I was walking. Even sitting
down quietly dosen't work. I have to keep walking.

It has gotten so absurd that I do a lot of walking at my work, usually in an
empty conference room and people have many a time wondered if I do any work at
all.

But, I love it. The challenge, the walk and the solution is exhilarating to
me.

~~~
blinkingled
Good for you :) - for me, laying down with eyes shut is when I will have
attractive solutions to hard problems! Happens sometimes standing in the
shower but those all tend to be strategic ones unlike the tactical ones that
get solved laying down.

Crazy stuff!

~~~
kumarvvr
Reminds me of a chapter from the book "Surely you must be joking Mr. Feynman"
(as far as I remember).

The chapter describes how Feynman came across how his colleague visualizes a
task. IIRC Feynman does it visually (imagines something visual in his mind)
and thus can do voice related activities while he is thinking, but his
colleague does it by visualizing sound (he thinks tick-tock sounds in his
mind), so can do visual tasks while he is thinking.

Really fascinating how a human mind works.

Also reminds me of the quote by Einstein, something about judging the ability
of a fish to climb a tree.

Everyone has a unique and effective way to navigate the world intellectually.

~~~
0xcde4c3db
> Also reminds me of the quote by Einstein, something about judging the
> ability of a fish to climb a tree.

Almost certainly not by Einstein, for what it's worth. It was probably made up
by motivational speaker Matthew Kelly or someone close to him, as the earliest
verifiable occurrence of the quotation is in his 2004 book _The Rhythm of
Life: Living Every Day with Passion and Purpose_ [1].

[1] [https://quoteinvestigator.com/2013/04/06/fish-
climb/](https://quoteinvestigator.com/2013/04/06/fish-climb/)

~~~
krallja
I wonder if it’s derived from the feminist slogan “a woman without a man is
like a fish without a bicycle”?

~~~
yantrams
Wow. Thanks for sharing that piece of trivia! Now I get what this song from
1981 - The fish needs a bike by Blurt - was referring to. I always thought it
was a reference to some surrealist work or something :)

[https://www.discogs.com/Blurt-The-Fish-Needs-A-
Bike/release/...](https://www.discogs.com/Blurt-The-Fish-Needs-A-
Bike/release/717696)

------
jordansmithnz
The article touches on something that helped me excel at university: I
consistently scored high grades, but compared to others around me I seemed put
in less time and effort. However, I don’t think I was significantly more
capable or brighter than the majority. After spending several years studying
alongside classmates, I do think the way I studied and ‘learnt to learn’ was
quite different to the status quo.

Like the article mentions, being able to answer every question means that
you’re not pushing yourself. I spent the majority of my time attempting
questions that I didn’t think I’d be able to do, while many classmates would
spend their time attempting questions they thought they would know. It forced
me to constantly ‘solve hard problems’, and become better at doing so. A lot
of the time I couldn’t successfully answer the questions I attempted, but in
doing so I’d learn a lot anyway.

The article content is great, but I’d add that practice is an essential part
of learning to solve hard problems. If you’re studying too, I’d really
recommend the learning technique I used, at least to some degree - it will
help you become a quick learner and good problem solver.

~~~
applecrazy
Thanks for the advice. I go to a top500 HS and all my peers seem to spend half
the effort to get higher grades than me, and it seems I need to push myself
harder in terms of the difficulty of the mathematics practice I do.

Side note: I used to think that the solution was expensive tutoring, but after
attending a few tutoring sessions with little benefit, I slowly realized that
tutoring (at least in my area) is for the unmotivated student who needs to be
pushed by someone else. Just wanted to share my experience and say that it’s
almost never the expensive prep classes that make a successful student.

~~~
CBLT
Funny thing about how I became a math major: I skipped precalculus in HS. All
my peers that were as bright (or more) in my school were turned off by the
year-long deathmarch of pointless bullshit[0] while I was taking more
intuitive classes like Calculus and Linear Algebra. We both challenged
ourselves, I just took the more fun challenges.

[0] [https://socratic.org/questions/how-do-you-evaluate-sin-
pi-5](https://socratic.org/questions/how-do-you-evaluate-sin-pi-5)

~~~
applecrazy
Wow, this is exactly what I’m thinking. The advanced precalculus course at my
school is actually harder than the college level calculus course and has much
lower enrollment to the point where the course is being phased out.

------
jakecrouch
It's worth noting that this basically gives you competitive tips - how to be
more effective at solving a problem after it's been given to you. I think it's
much more valuable to be good at recognizing important problems that others
don't believe are important.

~~~
johnchristopher
How do you deal with being the company's Cassandra (I am in that predicament
at the moment) ?

~~~
jakecrouch
It's nowhere near as bad or pervasive as the competition to get into student
debt. But I'm not really sure what psychotherapy would work. Maybe convince
people to read about mimetic theory.

------
sueders101
These types strategies, along with persistence are what got me through a math
degree that I felt highly unqualified for, especially when I started. They’ve
served me well professionally as well. As an “analyst” techniques have served
me well in producing actionable/appreciated insights. For anyone
aspiring:struggling in depths of mathematics I would suggest reading up on the
lives of historical mathematicians.(I’m A big fan of Galois) It’s important to
remember that we are truly humble in the face of ignorance. mathematics, and
only through the endeavors of struggle and confusion do we chip away at the
moutain of ignorance. Good luck to all those to venture into the darkness.

------
alexpetralia
Claude Shannon wrote a great piece on methods of problem-solving:
[http://www1.ece.neu.edu/~naderi/Claude%20Shannon.html](http://www1.ece.neu.edu/~naderi/Claude%20Shannon.html)

~~~
yantrams
Wonderful write up. Thanks for the share.

------
man2525
I tried looking into ways to improve my problem-solving a few years ago. I
think problem-solving changes depending on whether or not you know the end
goal. Problem solving without knowing the end result seems to involve
separating parts of the problem into systems, fitting said systems in your
head space, mentally making them run, and then noticing new problems. There
are bookshelves of books for what to do when you have a specific goal in mind.
Mainly work backwards and remove obstacles. I figure most of the people on
this site probably work between the two extremes, and do simulation/testing as
well. Careful thinking seems to improve overall speed, but that's what i'm
worst at.

------
wool_gather
> Reflect on successes. You've solved lots of problems. Some of them were even
> hard problems!

Good advice. Trying to learn something new, staring at a blank page or a list
of requirements that you don't understand, it's easy to get into the impostor
syndrome doldrums of "this is so hard, I must just be too dumb to do it".

You see the wall, right in front of you, and you wonder how to get over it.
But the fact that your path has even led you up to the wall is a strong
indication that you _can_ figure it out. If you understand the problem well
enough to see just how difficult it is, you're already on your way to solving
it. That doesn't mean it will be easy, of course. But _possible_.

------
uptownfunk
I grew up when the first math circle in San Diego was being launched. Richard
R. was an amazing teacher.. helped me qual for USAMO twice when I otherwise
would have never had a chance.. that too when my calc bc teacher couldn’t
solve many of the usamo probs himself even my linear alg Prof at the local jc
couldn’t crack many of the problems.. what a great resource for kids..
recommend it to anyone whose kid has an inclination to maths

------
lhuser123
> We ask hard questions because so many of the problems worth solving in life
> are hard. If they were easy, someone else would have solved them before you
> got to them.

This is so true. And the strategies explained in the article can be adapted to
deal with almost any problem in life. But you have to keep in mind that the
bigger the problem, the longer it will take to solve. Some will take days,
others years and even decades.

------
Myrmornis
How do people (parents?) use AOPS? In terms of logistics is it like after-
school math tutoring? All in-person at their campuses or is there online?

~~~
sincerely
I took some AoPS classes in middle school, they were entirely online - 2 (?
maybe 1 or 3?) hour online lesson once a week with weekly problem sets as well
(10 questions, the last few very hard). This was almost a decade ago so things
may have changed

~~~
saagarjha
They were doing essentially the same thing five years ago when I was taking
them. Some of my later classes moved away from the weekly problem set to
assigning problems on Alcumus.

------
signa11
'how to solve it' by george-pólya is also on the same lines, and _really_ very
good.

------
yeleti
While i agree with most of them, i would strongly recommend against “Work
backwards”. While this has helped me initially in mathematics, it took extreme
effort to get rid of this habit.

~~~
infinityplus1
Explain more why this is bad. I think it's very useful. In school, I used to
say I reverse engineered a problem by finding way back from solution to the
problem. Solving a problem from both ends is a reasonable way to get new
ideas.

------
kumarvvr
And if nothing else works, try the Feynman Algorithm to solve really hard
problems. :-)

~~~
HiroshiSan
Step 1: Be Feynman

------
RickJWagner
Hacker New paydirt. There's good stuff in this one.

