6 ms·

How to Solve It (1945)

203 points by GamerUncle 3y ago

furyofantares 3y ago
I remember this book as being written for teachers and was about about prompting students as you help them solve problems. I read it as a student, and felt it was somewhat formative even though I don't think I ever explicitly applied anything it said.
sverona 3y ago
Mathematics and Plausible Reasoning, in two volumes also by Polya, is the closest thing I've ever found to an explanation of how a working mathematician goes about her business. It has exercises, too. Great fun when I was in undergrad.
- 3abiton 3y ago
  Any recommendations of similar books?
  - i_am_a_peasant 3y ago
    How to Prove it is pretty nice
    - vkazanov 3y ago
      I have both the book discussed here, and How to prove it, and back in the day the Proof book was a revelation, unlike Polya's one, which was interesting but that's about it How to prove it just made proofs click, and I wish I understood the material earlier, by high school the latest. I am convinced that all the half intuitive half ("kind of like") mechanical ("just remember this") explanations we get in school and undergrad years are a huge waste of everyone's time.
      - i_am_a_peasant 3y ago
        There are other books in the literature that mainly focus on how to go about proving things rather than proving specific things. Some titles that come up are: Mathematical Proofs: A transition to Advanced Mathematics by Chartrand, Polimeni and Zhang. Book of Proof: Richard Hammack Jay Cumming’s Proofs I remember I once had a book that advertised itself as a book on Discrete Mathematics but it turned out to be more in line with the titles above, was a gem, if I find it I'll let you know. But it's nice to see I'm not the only guy who gets a kick out of this stuff. I wouldn't bother with things from the foundations of mathematics area, like Russel or Frege and whoever else came from that era, back when people still wrote an 800 page treatise on what a "number" even is. A lot of the material coming from that period seems to have been thrown down the drain.
        sverona 3y ago
        > I remember I once had a book that advertised itself as a book on Discrete Mathematics but it turned out to be more in line with the titles above, was a gem, if I find it I'll let you know. Every discrete math book I've ever seen has a strong proof component; it's supposed to be an undergraduate's introduction to rigor. The classes taught out of those books may not actually be that rigorous, depending on whether the students are going to be expected to rigorously prove things later, or if it's just Leetcode combinatorics. But for a guess, maybe Kenneth Rosen?
        i_am_a_peasant 3y ago
        Ah yes, I think this was it, I see a note that I have the 7th edition and I wanted to get a hold of the 8th.
    - retrocryptid 3y ago
      +1
dingosity 3y ago
Weird. I was just talking about this book to my offspring. You're reading my mind, @GamerUncle.
js2 3y ago
Ah yes, just four steps: 1. Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Look back. --- Compare with the Fenyman Algorithm: 1. Write down the problem. 2. Think real hard. 3. Write down the solution. https://wiki.c2.com/?FeynmanAlgorithm https://wiki.c2.com/?FeynmanAlgorithm (The discussion on FeynmanAlgorithm links back to Polya's book since not everyone is Feynman.)
- bonoboTP 3y ago
  Or his steps to do science: 1. Guess 2. Compute consequences 3. Compare with experiment - if they disagree, the guess was wrong. https://www.youtube.com/watch?v=OL6-x0modwY https://www.youtube.com/watch?v=OL6-x0modwY
- morkalork 3y ago
  >Write down the problem Sometimes this step is enough to get you 80% of the way to solving a problem.
- jiggawatts 3y ago
  Reminds me of the official Microsoft guidance for big projects like migrating Exchange to the cloud, merging an Active Directory, or whatever. They’re all verbatim the same, except for the product name. 1. Gather requirements. 2. Devise plan. 3. Run a proof of concept 4. Run production migration …etc. It’s about as helpful as someone telling you to “do your job”.
  - keyle 3y ago
    Classic: How to draw an owl: 1. carefully darw a circle as the head of the owl 2. draw the rest of the owl
- jodrellblank 3y ago
  You say 'ah yes' as if there's no value in these steps. Compare with more common steps: 1. Don't try to understand the problem, just stare at the given information blankly. 2. Instead of devising a plan or thinking real hard, try to "guess the teacher's password" and parrot any relevant sounding words for this ... or any related thing - https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-the-teacher-s-password https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-t... and the linked https://www.lesswrong.com/posts/48WeP7oTec3kBEada/two-more-things-to-unlearn-from-school https://www.lesswrong.com/posts/48WeP7oTec3kBEada/two-more-t... - as if the universe will accept the words (map) as the solution (territory). 3. Carry out incantations handed down from above, regardless of whether they could be applicable to this problem; they are not understood which leaves room for them being potentially powerful and relevant. 4. Give up. If the incantations of power didn't help, what else is there to do? Don't look back. Are these steps from How To Solve It in 1945 so different from Eric Raymond's "how to ask smart questions" from the 1990s(?) or from Eric Lippert's "How to debug small programs" from 2014 ( https://ericlippert.com/2014/03/05/how-to-debug-small-programs/ https://ericlippert.com/2014/03/05/how-to-debug-small-progra... )? His example "I wrote this program for my assignment and it doesn’t work." is still a common approach to "problem solving" nearly a decade later, not just for student coders either. People seem to be missing: - the belief that problems are solvable. (Instead apparently believing that some people know the answer or that the problem cannot be solved; not understanding that someone can start unknowing, work to solve, and then know). - that solving something involves a mental model of how a thing works, in some context. (Instead apparently believing that solving is random or guesswork or luck or flash of inspiration, but not solved by any kind of knowable process). - that solving something involves a feedback loop of gathering information, testing hypotheses, learning from the result, looking back (Instead trying one thing and stopping, not seeing if anything could be learned from the result, not trying to expand a mental model from what was known). Even that this is a pet frustration of mine with some of my coworkers and internet programming askers, I still resist writing down the problem or thinking real hard (as far as I can do such a thing).
ergocoder 3y ago
I read it and it doesn't help much. What helps with solving problems like math and algorithmic problems is to go through a lot of problems to see different patterns and strategies of solving problems. I'm talking about going through thousands of problems. That is very effective.
- hgsgm 3y ago
  Maybe if you followed Polya you wouldn't need thousands of problems.
- rahimnathwani 3y ago
  This is consistent with a sentence on page 1 of the book: 'The student should acquire as much experience of independent work as possible.'
  - jvanderbot 3y ago
    Not only that, my take on this book is it's meant to help you classify those patterns and better recall them by going through consistent triage and process.
- mathisfun123 3y ago
  > I'm talking about going through thousands of problems. you don't need thousands of problems. you don't even need hundreds, unless, no offense, your medium-term memory is very poor. personal anecdote 1: in between undergrad and grad school i decided i was gonna try this "solve all of the problems" approach, as opposed to my usual "sit there and ponder approach", in order to prepare for eventual quals in grad school. i started with calculus, using apostol's calculus (famous for its rigor and difficulty right?). some sections have double digits (maybe even 100? i don't remember) problems and invariably (no pun intended) by the time i got about a quarter of the way through they got trivially easy. i did finish and do all the problems in both volumes. i didn't feel i learned any of it better than the first time i took calc (wherein i didn't solve many at all beyond assignments). i did not keep on with this kind of slavish dedication and just skimmed the rest of the books. i didn't end up doing a phd in math but i did take math and cs theory classes and i did well. personal anecdote 2: after my MS i did hundreds of leetcode problems. it was roughly the same phenomenon: in every category it only took about a dozen to be able to solve the remainder trivially (yes even hard DP problems). and i'm willing to bet (if you're on this board) your memory is better than mine (i smoked incredible amounts of pot in high school...).
modeless 3y ago
It's prompt engineering for humans.
bee_rider 3y ago
> What is the unknown? What are the data? What is the condition? … > Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem? What sort of problems are they solving, that they can somehow identify the relevant data to the point that they know once they’ve chomped their way through the data, the problem is done? It seems oddly constructed (I can only imagine that I know I’ve only been given relevant data if I’m working a textbook problem or playing a video game; somebody has set the problem up for me, but clearly this was written by somebody prestigious, so that isn’t it).
- UltimateEdge 3y ago
  I think the author is talking about maths problems, or proofs of theorems/propositions. A problem might give you one or more mathematical objects, and ask you to show that some further condition holds true. To get started, you might consider how the given properties of those objects will help you to achieve your goal (and typically you would need to use every given property).
- bonoboTP 3y ago
  It's about advanced school exercises or math contest questions that are designed by someone, not real-world/research problems, where you don't even know how to best frame the issue or whether it's even solvable or the appropriate thing to tackle at the time.
  - bee_rider 3y ago
    Oh! That’s a huge difference, haha.
- jxramos 3y ago
  > Did you use all the data... It's a good articulation that informs one while working on complex stuff. Here's a recent example of this where the above advice came to mind while reading over this advice the other day (I believe someone linked it on HN) > A good way to stress-test this sort of false argument is to try to run the same argument without the initial assumption that X is false. If one can easily modify the argument to again lead to a contradiction, it shows the problem wasn’t with X – it was with the argument. https://terrytao.wordpress.com/career-advice/be-sceptical-of-your-own-work/#:~:text=A%20good%20way%20to%20stress%2Dtest%20this%20sort%20of%20false%20argument%20is%20to%20try%20to%20run%20the%20same%20argument%20without%20the%20initial%20assumption%20that%20X%20is%20false.%20%C2%A0If%20one%20can%20easily%20modify%20the%20argument%20to%20again%20lead%20to%20a%20contradiction%2C%20it%20shows%20the%20problem%20wasn%E2%80%99t%20with%20X%20%E2%80%93%20it%20was%20with%20the%20argument https://terrytao.wordpress.com/career-advice/be-sceptical-of....
RcouF1uZ4gsC 3y ago
Or you can just be like von Neumann From: https://www.tennessean.com/story/opinion/columnists/teachable-moments/2013/12/28/john-von-neumann-had-mad-passion-for-mathematics/4228849/ https://www.tennessean.com/story/opinion/columnists/teachabl... >George Pólya, one of his university teachers, said, “I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him, he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.”
boppo1 3y ago
This summary really undersells the book IMO. It's one of the more interesting books I've read in that it is not structured linearly. He introduces a few ideas which have very particular language he defines in the remainder of the book which is essentially a dictionary. If you don't come from a math-y background and you are trying to get into serious mathematics, this definitely helps to 'lift the veil' on how you might think about math. Most of the criticism I've heard about the book comes from Paraphrased from memory: "I'm not claiming to have discovered anything special here. What I think I'm doing is describing the mental operations that lots of people perform, consciously or unconsciously, to solve different sorts of problems. You probably already do some or all of what I'm describing." I found that he was describing some thought patterns that I am extremely familiar with because I use them so often. Making them explicit was pretty cool. I haven't gotten around to his follow-up works yet, but I look forward to them.
- sfpotter 3y ago
  If you don't come from a math-y background and are trying to "get into serious mathematics", the only way to do this is to get a PhD. I realize this might come across as gatekeeping, but the reality is that each subfield of math (or any scientific discipline) has its own set of tactics for approaching problems, which have been developed over the years by the people actually in the trenches. These problem solving strategies and habits of mind aren't written up anywhere, and even if they were, they wouldn't be useful to read. The way you learn them is by getting the training that comes with a PhD: they are passed down as part of the mentoring process. It's not clear to me that it could happen any other way. I think the right way to look at this is that someone getting a PhD is basically a "journeyman researcher", much like a journeyman in one of the trades. Unfortunately, this often goes sideways (particularly if an advisor is bad or if there aren't enough support systems). But much like the only way to become an electrician is to apprentice yourself, the same goes for becoming a researcher. I think this is for good reason.
  - veqq 3y ago
    There are a few books which admirably explain the assumptions etc. - A Programmers Introduction to Mathematics (amazing for building intuition, explaining how to read mathematics papers, understanding beauty of proofs etc.) - Elements of Mathematics for Economics and Finance (a quick jog through typical the high school and college math a non-mathematician would study) - Papula's Mathematik für Ingenieure und Naturwissenschaftler (rigorous (for practitioners) and comprehensive. Starts from nothing, with sets, then teaches you fractions, then quadratic equations until you're doing vector analysis and Laplace transformations) - Foundations and Fundamental Concepts of Mathematics by Howard Eves (wide overview of the man developments in mathematics, up to its publication)
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paulpauper 3y ago
this may work for simple problems. i have found it's more like: 1. study extensive background/prior art 2. have major epiphany to solve problem 3. try to get around inevitable roadblocks and complications during the process of solving it
- hgsgm 3y ago
  You should read the book. It fills in about 100 more steps/options around those 3 general steps. Getting from 1 to 2 and from 2 to 3 is the aim of the book
greymalik 3y ago
What is meant by “the condition”? I am imagining that in x + 3 = 7 that x is the unknown and 3 and 7 are the data? Is the condition equality and, if so, how do you use that information? Some more complicated examples would probably help me understand.
- hgsgm 3y ago
  Yes. It's more relevant in a word problem, where you need to extract the condition from text and translate it into syntactical math notation so you can compute. "John wants to have 7 pizzas. He has already made 3 pizzas. How many more pizzas does he need to make?" Data: 7 pizzas needed. 3 pizzas made. Unknown: X pizzas more to make Condition: X+3=7 Or for geometry diagram problems (diagram geometry is word problems where the words are pictures), where you have to extract the relationships from the diagram. The book has more examples.
steno132 3y ago
The best people I work with don't use this stuff. They solve problems based on pattern matching. Having read hundreds of books and papers over the years, no problem seems new to them. They can rapidly find a solution, especially compared to someone trying to derive a answer from first principles like Polya suggests.
- felipefar 3y ago
  I believe pattern matching works best when solving problems with little variation from previous ones. Novel problems requires a lot more maturing and thinking, which is when slower, longer term thinking like Polya suggests goes a long way.
- hgsgm 3y ago
  If what you say is true, why did it take so long to prove Fermat's Last Theorem? Polya's book is distilling hard fought PhD wisdom into a book children can understand. It's for solving novel hard problems, not trivial ones.
  - raincole 3y ago
    The best people the GP worked with are probably not capable to solve Fermat's Last Theorem at the first place...
dang 3y ago
Related. Surely there have been others? How to solve it - https://news.ycombinator.com/item?id=17726102 https://news.ycombinator.com/item?id=17726102 - Aug 2018 (1 comment) George Pólya: How to Solve It (1945) - https://news.ycombinator.com/item?id=13255739 https://news.ycombinator.com/item?id=13255739 - Dec 2016 (1 comment) George Pólya – How to Solve It (1945) - https://news.ycombinator.com/item?id=12823608 https://news.ycombinator.com/item?id=12823608 - Oct 2016 (1 comment) G. Polya: How to Solve It - https://news.ycombinator.com/item?id=153425 https://news.ycombinator.com/item?id=153425 - April 2008 (1 comment)
ryandv 3y ago
How well does this book work for solving problems in Walter Rudin's "Principles of Mathematical Analysis?"
- sverona 3y ago
  Baby Rudin, despite the somewhat patronizing name, is a second analysis text. There are many things that should have explanations that he expects you to already know. Swallow your pride and start with Abbott. You'll thank me later. Now here's what I really think: I don't know your situation, but the odds are better than even that you should forget about it entirely. Doubly so if you're out of school. The bizarre math pride held by college kids (I did Math 55! I took a class with Stephen Wolfram's kid! I did all the exercises in Hartshorne! I proved X at age Y!) is real fucking stupid, coming from someone who badly wanted it. Your academic career or lack thereof will be determined about 90% by the people around you. It may piss you off, but you'll find it to be the same bitter truth I did.
  - ryandv 3y ago
    My question was more rhetorical and for _synchronicity's_ sake than anything; I wanted to see what sort of replies would pop out. As a recreational student, I've already committed to this text, difficulty be damned. I am no stranger to intellectual masochism. Nor do I care one iota for an "academic career," only for an understanding of results that were taken for granted in my prior CS mathematics undergraduate education at UWaterloo, which involved a few years of formal proof. Would that I could have finished said undergraduate program, but tuition is expensive, and being rendered homeless partway through a CS program certainly complicates the financing, especially absent the monetary privileges afforded to one of a more affluent upbringing. I appreciate the elitist and somewhat exclusionary language, though, despite admitted lack of knowledge regarding my situation.
    - sverona 3y ago
      You're right, I'm sorry. I took my frustrations out on you. Funny, I was also homeless in the middle of undergrad. I don't know what real analysis has to do with CS, except as a pre-pre-prerequisite to boring old ML math, but it's not my place. Good luck.
lcuff 3y ago
On a totally orthogonal axis: My biggest problem with tackling math problems early in life was psychological. My inner critic would say nasty things to me when I didn't get the answer right away, making it impossible to stick with exploring the problem. Once I overcame that, I got lots better at math. And working on yet another axis, the most useful tip I've gotten on how to think about problems is to "think in extremes". Perhaps more applicable to physics than math, but a classic example is the puzzle where a ship in a lock (closed system/giant bathtub) has an engine break down. As the engine is being hauled out for repairs, the chain breaks and the engine drops into the lock beside the ship and falls to the bottom of the lock. Does the water in the lock rise, fall or stay the same? The only person I ever told the problem to that solved it described his technique: Imagine the engine is the size of a pea and weighs 20 tons. Shazaam: The water in the lock falls because it is now displacing its volume, instead of its weight in water.
- recursivecaveat 3y ago
  I like to use extremes and the intermediate value theorem, ie if you have two extreme points, and some continuous curve connecting them, it passes through all the points in the middle. It's a good shortcut for "is it possible to balance X and Y" questions. Like is there a hexagon with 'equal' area and perimeter? It's easy to conceptualize ones on each side of the ratio, and you can smoothly deform between them so the ratio is continuous, hence there is such a hexagon.
- hibernator149 3y ago
  Thinking in extremes helped me to understand the Monty Hall problem. Imagine there are a hundred doors instead of three and 98 wrong ones are eliminated. Should you switch?
- apocadam 3y ago
  If you don't mind, could you share how you overcame that? I think for me at least a lot of it is/was not being able to be comfortable struggling with a hard problem for an extended amount of time, probably due to being used to only solving fairly trivial exercises and thus feeling dumb/the problem feeling impossible if it wasn't clearly solvable under a short timeframe.
markc 3y ago
Rich Hickey (creator of Clojure) references Polya several times in his classic talk "Hammock Driven Development". Here's a transcript: https://github.com/matthiasn/talk-transcripts/blob/master/Hickey_Rich/HammockDrivenDev.md https://github.com/matthiasn/talk-transcripts/blob/master/Hi... I've long been impressed by Hickey's problem solving skills, so I took much of this talk to heart, and even bought a copy of HTSI. Can't say it really helped me any more than Rich's talk (as a programmer) but I'm thinking I'll give it another look.
wayeq 3y ago
How to Solve it (202?) Ask GPT7 to think for you
jzer0cool 3y ago
Anyone with an interesting example to share using these steps?
ethanwillis 3y ago
I find it interesting how the link at the bottom of the page is a clearly hacked link.
- OmarShehata 3y ago
  What do you mean by "hacked"? It's just an old domain that's expired and now redirects to the hosting website. Nothing nefarious going on. Here's the original, which does indeed have a translation as one would expect: https://web.archive.org/web/20111005170154/http://webhostinggeeks.com/science/g-polya-be https://web.archive.org/web/20111005170154/http://webhosting...
R41 3y ago
Loved the way polia's explained the basics of solving challenges by asking questions - - Can you find a problem analogous to your problem and solve that? - Can you add some new element to your problem to get closer to a solution? etc we always asks new onboard people to read this .It's like a bible for problem solving .
im_lince 3y ago
In our team we recommend this book to new joiners. The principles and methods shared by polya is valid for solving the not only the mathematical problems also engineering problems in effective way.
OldGuyInTheClub 3y ago
It's easy when you have John von Neumann as a student.

203 points by GamerUncle 3y ago

I remember this book as being written for teachers and was about about prompting students as you help them solve problems. I read it as a student, and felt it was somewhat formative even though I don't think I ever explicitly applied anything it said.

sverona 3y ago

Mathematics and Plausible Reasoning, in two volumes also by Polya, is the closest thing I've ever found to an explanation of how a working mathematician goes about her business. It has exercises, too. Great fun when I was in undergrad.

3abiton 3y ago
Any recommendations of similar books?
- i_am_a_peasant 3y ago
  How to Prove it is pretty nice
  - vkazanov 3y ago
    I have both the book discussed here, and How to prove it, and back in the day the Proof book was a revelation, unlike Polya's one, which was interesting but that's about it How to prove it just made proofs click, and I wish I understood the material earlier, by high school the latest. I am convinced that all the half intuitive half ("kind of like") mechanical ("just remember this") explanations we get in school and undergrad years are a huge waste of everyone's time.
    - i_am_a_peasant 3y ago
      There are other books in the literature that mainly focus on how to go about proving things rather than proving specific things. Some titles that come up are: Mathematical Proofs: A transition to Advanced Mathematics by Chartrand, Polimeni and Zhang. Book of Proof: Richard Hammack Jay Cumming’s Proofs I remember I once had a book that advertised itself as a book on Discrete Mathematics but it turned out to be more in line with the titles above, was a gem, if I find it I'll let you know. But it's nice to see I'm not the only guy who gets a kick out of this stuff. I wouldn't bother with things from the foundations of mathematics area, like Russel or Frege and whoever else came from that era, back when people still wrote an 800 page treatise on what a "number" even is. A lot of the material coming from that period seems to have been thrown down the drain.
      - sverona 3y ago
        > I remember I once had a book that advertised itself as a book on Discrete Mathematics but it turned out to be more in line with the titles above, was a gem, if I find it I'll let you know. Every discrete math book I've ever seen has a strong proof component; it's supposed to be an undergraduate's introduction to rigor. The classes taught out of those books may not actually be that rigorous, depending on whether the students are going to be expected to rigorously prove things later, or if it's just Leetcode combinatorics. But for a guess, maybe Kenneth Rosen?
        i_am_a_peasant 3y ago
        Ah yes, I think this was it, I see a note that I have the 7th edition and I wanted to get a hold of the 8th.
  - retrocryptid 3y ago
    +1

dingosity 3y ago

Weird. I was just talking about this book to my offspring. You're reading my mind, @GamerUncle.

js2 3y ago

Ah yes, just four steps: 1. Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Look back. --- Compare with the Fenyman Algorithm: 1. Write down the problem. 2. Think real hard. 3. Write down the solution. https://wiki.c2.com/?FeynmanAlgorithm https://wiki.c2.com/?FeynmanAlgorithm (The discussion on FeynmanAlgorithm links back to Polya's book since not everyone is Feynman.)

bonoboTP 3y ago
Or his steps to do science: 1. Guess 2. Compute consequences 3. Compare with experiment - if they disagree, the guess was wrong. https://www.youtube.com/watch?v=OL6-x0modwY https://www.youtube.com/watch?v=OL6-x0modwY
morkalork 3y ago
>Write down the problem Sometimes this step is enough to get you 80% of the way to solving a problem.
jiggawatts 3y ago
Reminds me of the official Microsoft guidance for big projects like migrating Exchange to the cloud, merging an Active Directory, or whatever. They’re all verbatim the same, except for the product name. 1. Gather requirements. 2. Devise plan. 3. Run a proof of concept 4. Run production migration …etc. It’s about as helpful as someone telling you to “do your job”.
- keyle 3y ago
  Classic: How to draw an owl: 1. carefully darw a circle as the head of the owl 2. draw the rest of the owl
jodrellblank 3y ago
You say 'ah yes' as if there's no value in these steps. Compare with more common steps: 1. Don't try to understand the problem, just stare at the given information blankly. 2. Instead of devising a plan or thinking real hard, try to "guess the teacher's password" and parrot any relevant sounding words for this ... or any related thing - https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-the-teacher-s-password https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-t... and the linked https://www.lesswrong.com/posts/48WeP7oTec3kBEada/two-more-things-to-unlearn-from-school https://www.lesswrong.com/posts/48WeP7oTec3kBEada/two-more-t... - as if the universe will accept the words (map) as the solution (territory). 3. Carry out incantations handed down from above, regardless of whether they could be applicable to this problem; they are not understood which leaves room for them being potentially powerful and relevant. 4. Give up. If the incantations of power didn't help, what else is there to do? Don't look back. Are these steps from How To Solve It in 1945 so different from Eric Raymond's "how to ask smart questions" from the 1990s(?) or from Eric Lippert's "How to debug small programs" from 2014 ( https://ericlippert.com/2014/03/05/how-to-debug-small-programs/ https://ericlippert.com/2014/03/05/how-to-debug-small-progra... )? His example "I wrote this program for my assignment and it doesn’t work." is still a common approach to "problem solving" nearly a decade later, not just for student coders either. People seem to be missing: - the belief that problems are solvable. (Instead apparently believing that some people know the answer or that the problem cannot be solved; not understanding that someone can start unknowing, work to solve, and then know). - that solving something involves a mental model of how a thing works, in some context. (Instead apparently believing that solving is random or guesswork or luck or flash of inspiration, but not solved by any kind of knowable process). - that solving something involves a feedback loop of gathering information, testing hypotheses, learning from the result, looking back (Instead trying one thing and stopping, not seeing if anything could be learned from the result, not trying to expand a mental model from what was known). Even that this is a pet frustration of mine with some of my coworkers and internet programming askers, I still resist writing down the problem or thinking real hard (as far as I can do such a thing).

ergocoder 3y ago

I read it and it doesn't help much. What helps with solving problems like math and algorithmic problems is to go through a lot of problems to see different patterns and strategies of solving problems. I'm talking about going through thousands of problems. That is very effective.

hgsgm 3y ago
Maybe if you followed Polya you wouldn't need thousands of problems.
rahimnathwani 3y ago
This is consistent with a sentence on page 1 of the book: 'The student should acquire as much experience of independent work as possible.'
- jvanderbot 3y ago
  Not only that, my take on this book is it's meant to help you classify those patterns and better recall them by going through consistent triage and process.
mathisfun123 3y ago
> I'm talking about going through thousands of problems. you don't need thousands of problems. you don't even need hundreds, unless, no offense, your medium-term memory is very poor. personal anecdote 1: in between undergrad and grad school i decided i was gonna try this "solve all of the problems" approach, as opposed to my usual "sit there and ponder approach", in order to prepare for eventual quals in grad school. i started with calculus, using apostol's calculus (famous for its rigor and difficulty right?). some sections have double digits (maybe even 100? i don't remember) problems and invariably (no pun intended) by the time i got about a quarter of the way through they got trivially easy. i did finish and do all the problems in both volumes. i didn't feel i learned any of it better than the first time i took calc (wherein i didn't solve many at all beyond assignments). i did not keep on with this kind of slavish dedication and just skimmed the rest of the books. i didn't end up doing a phd in math but i did take math and cs theory classes and i did well. personal anecdote 2: after my MS i did hundreds of leetcode problems. it was roughly the same phenomenon: in every category it only took about a dozen to be able to solve the remainder trivially (yes even hard DP problems). and i'm willing to bet (if you're on this board) your memory is better than mine (i smoked incredible amounts of pot in high school...).

modeless 3y ago

It's prompt engineering for humans.

bee_rider 3y ago

> What is the unknown? What are the data? What is the condition? … > Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem? What sort of problems are they solving, that they can somehow identify the relevant data to the point that they know once they’ve chomped their way through the data, the problem is done? It seems oddly constructed (I can only imagine that I know I’ve only been given relevant data if I’m working a textbook problem or playing a video game; somebody has set the problem up for me, but clearly this was written by somebody prestigious, so that isn’t it).

UltimateEdge 3y ago
I think the author is talking about maths problems, or proofs of theorems/propositions. A problem might give you one or more mathematical objects, and ask you to show that some further condition holds true. To get started, you might consider how the given properties of those objects will help you to achieve your goal (and typically you would need to use every given property).
bonoboTP 3y ago
It's about advanced school exercises or math contest questions that are designed by someone, not real-world/research problems, where you don't even know how to best frame the issue or whether it's even solvable or the appropriate thing to tackle at the time.
- bee_rider 3y ago
  Oh! That’s a huge difference, haha.
jxramos 3y ago
> Did you use all the data... It's a good articulation that informs one while working on complex stuff. Here's a recent example of this where the above advice came to mind while reading over this advice the other day (I believe someone linked it on HN) > A good way to stress-test this sort of false argument is to try to run the same argument without the initial assumption that X is false. If one can easily modify the argument to again lead to a contradiction, it shows the problem wasn’t with X – it was with the argument. https://terrytao.wordpress.com/career-advice/be-sceptical-of-your-own-work/#:~:text=A%20good%20way%20to%20stress%2Dtest%20this%20sort%20of%20false%20argument%20is%20to%20try%20to%20run%20the%20same%20argument%20without%20the%20initial%20assumption%20that%20X%20is%20false.%20%C2%A0If%20one%20can%20easily%20modify%20the%20argument%20to%20again%20lead%20to%20a%20contradiction%2C%20it%20shows%20the%20problem%20wasn%E2%80%99t%20with%20X%20%E2%80%93%20it%20was%20with%20the%20argument https://terrytao.wordpress.com/career-advice/be-sceptical-of....

RcouF1uZ4gsC 3y ago

Or you can just be like von Neumann From: https://www.tennessean.com/story/opinion/columnists/teachable-moments/2013/12/28/john-von-neumann-had-mad-passion-for-mathematics/4228849/ https://www.tennessean.com/story/opinion/columnists/teachabl... >George Pólya, one of his university teachers, said, “I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him, he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.”

boppo1 3y ago

This summary really undersells the book IMO. It's one of the more interesting books I've read in that it is not structured linearly. He introduces a few ideas which have very particular language he defines in the remainder of the book which is essentially a dictionary. If you don't come from a math-y background and you are trying to get into serious mathematics, this definitely helps to 'lift the veil' on how you might think about math. Most of the criticism I've heard about the book comes from Paraphrased from memory: "I'm not claiming to have discovered anything special here. What I think I'm doing is describing the mental operations that lots of people perform, consciously or unconsciously, to solve different sorts of problems. You probably already do some or all of what I'm describing." I found that he was describing some thought patterns that I am extremely familiar with because I use them so often. Making them explicit was pretty cool. I haven't gotten around to his follow-up works yet, but I look forward to them.

sfpotter 3y ago
If you don't come from a math-y background and are trying to "get into serious mathematics", the only way to do this is to get a PhD. I realize this might come across as gatekeeping, but the reality is that each subfield of math (or any scientific discipline) has its own set of tactics for approaching problems, which have been developed over the years by the people actually in the trenches. These problem solving strategies and habits of mind aren't written up anywhere, and even if they were, they wouldn't be useful to read. The way you learn them is by getting the training that comes with a PhD: they are passed down as part of the mentoring process. It's not clear to me that it could happen any other way. I think the right way to look at this is that someone getting a PhD is basically a "journeyman researcher", much like a journeyman in one of the trades. Unfortunately, this often goes sideways (particularly if an advisor is bad or if there aren't enough support systems). But much like the only way to become an electrician is to apprentice yourself, the same goes for becoming a researcher. I think this is for good reason.
- veqq 3y ago
  There are a few books which admirably explain the assumptions etc. - A Programmers Introduction to Mathematics (amazing for building intuition, explaining how to read mathematics papers, understanding beauty of proofs etc.) - Elements of Mathematics for Economics and Finance (a quick jog through typical the high school and college math a non-mathematician would study) - Papula's Mathematik für Ingenieure und Naturwissenschaftler (rigorous (for practitioners) and comprehensive. Starts from nothing, with sets, then teaches you fractions, then quadratic equations until you're doing vector analysis and Laplace transformations) - Foundations and Fundamental Concepts of Mathematics by Howard Eves (wide overview of the man developments in mathematics, up to its publication)

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paulpauper 3y ago

this may work for simple problems. i have found it's more like: 1. study extensive background/prior art 2. have major epiphany to solve problem 3. try to get around inevitable roadblocks and complications during the process of solving it

hgsgm 3y ago
You should read the book. It fills in about 100 more steps/options around those 3 general steps. Getting from 1 to 2 and from 2 to 3 is the aim of the book

greymalik 3y ago

What is meant by “the condition”? I am imagining that in x + 3 = 7 that x is the unknown and 3 and 7 are the data? Is the condition equality and, if so, how do you use that information? Some more complicated examples would probably help me understand.

hgsgm 3y ago
Yes. It's more relevant in a word problem, where you need to extract the condition from text and translate it into syntactical math notation so you can compute. "John wants to have 7 pizzas. He has already made 3 pizzas. How many more pizzas does he need to make?" Data: 7 pizzas needed. 3 pizzas made. Unknown: X pizzas more to make Condition: X+3=7 Or for geometry diagram problems (diagram geometry is word problems where the words are pictures), where you have to extract the relationships from the diagram. The book has more examples.

steno132 3y ago

The best people I work with don't use this stuff. They solve problems based on pattern matching. Having read hundreds of books and papers over the years, no problem seems new to them. They can rapidly find a solution, especially compared to someone trying to derive a answer from first principles like Polya suggests.

felipefar 3y ago
I believe pattern matching works best when solving problems with little variation from previous ones. Novel problems requires a lot more maturing and thinking, which is when slower, longer term thinking like Polya suggests goes a long way.
hgsgm 3y ago
If what you say is true, why did it take so long to prove Fermat's Last Theorem? Polya's book is distilling hard fought PhD wisdom into a book children can understand. It's for solving novel hard problems, not trivial ones.
- raincole 3y ago
  The best people the GP worked with are probably not capable to solve Fermat's Last Theorem at the first place...

dang 3y ago

Related. Surely there have been others? How to solve it - https://news.ycombinator.com/item?id=17726102 https://news.ycombinator.com/item?id=17726102 - Aug 2018 (1 comment) George Pólya: How to Solve It (1945) - https://news.ycombinator.com/item?id=13255739 https://news.ycombinator.com/item?id=13255739 - Dec 2016 (1 comment) George Pólya – How to Solve It (1945) - https://news.ycombinator.com/item?id=12823608 https://news.ycombinator.com/item?id=12823608 - Oct 2016 (1 comment) G. Polya: How to Solve It - https://news.ycombinator.com/item?id=153425 https://news.ycombinator.com/item?id=153425 - April 2008 (1 comment)

ryandv 3y ago

How well does this book work for solving problems in Walter Rudin's "Principles of Mathematical Analysis?"

sverona 3y ago
Baby Rudin, despite the somewhat patronizing name, is a second analysis text. There are many things that should have explanations that he expects you to already know. Swallow your pride and start with Abbott. You'll thank me later. Now here's what I really think: I don't know your situation, but the odds are better than even that you should forget about it entirely. Doubly so if you're out of school. The bizarre math pride held by college kids (I did Math 55! I took a class with Stephen Wolfram's kid! I did all the exercises in Hartshorne! I proved X at age Y!) is real fucking stupid, coming from someone who badly wanted it. Your academic career or lack thereof will be determined about 90% by the people around you. It may piss you off, but you'll find it to be the same bitter truth I did.
- ryandv 3y ago
  My question was more rhetorical and for _synchronicity's_ sake than anything; I wanted to see what sort of replies would pop out. As a recreational student, I've already committed to this text, difficulty be damned. I am no stranger to intellectual masochism. Nor do I care one iota for an "academic career," only for an understanding of results that were taken for granted in my prior CS mathematics undergraduate education at UWaterloo, which involved a few years of formal proof. Would that I could have finished said undergraduate program, but tuition is expensive, and being rendered homeless partway through a CS program certainly complicates the financing, especially absent the monetary privileges afforded to one of a more affluent upbringing. I appreciate the elitist and somewhat exclusionary language, though, despite admitted lack of knowledge regarding my situation.
  - sverona 3y ago
    You're right, I'm sorry. I took my frustrations out on you. Funny, I was also homeless in the middle of undergrad. I don't know what real analysis has to do with CS, except as a pre-pre-prerequisite to boring old ML math, but it's not my place. Good luck.

lcuff 3y ago

On a totally orthogonal axis: My biggest problem with tackling math problems early in life was psychological. My inner critic would say nasty things to me when I didn't get the answer right away, making it impossible to stick with exploring the problem. Once I overcame that, I got lots better at math. And working on yet another axis, the most useful tip I've gotten on how to think about problems is to "think in extremes". Perhaps more applicable to physics than math, but a classic example is the puzzle where a ship in a lock (closed system/giant bathtub) has an engine break down. As the engine is being hauled out for repairs, the chain breaks and the engine drops into the lock beside the ship and falls to the bottom of the lock. Does the water in the lock rise, fall or stay the same? The only person I ever told the problem to that solved it described his technique: Imagine the engine is the size of a pea and weighs 20 tons. Shazaam: The water in the lock falls because it is now displacing its volume, instead of its weight in water.

recursivecaveat 3y ago
I like to use extremes and the intermediate value theorem, ie if you have two extreme points, and some continuous curve connecting them, it passes through all the points in the middle. It's a good shortcut for "is it possible to balance X and Y" questions. Like is there a hexagon with 'equal' area and perimeter? It's easy to conceptualize ones on each side of the ratio, and you can smoothly deform between them so the ratio is continuous, hence there is such a hexagon.
hibernator149 3y ago
Thinking in extremes helped me to understand the Monty Hall problem. Imagine there are a hundred doors instead of three and 98 wrong ones are eliminated. Should you switch?
apocadam 3y ago
If you don't mind, could you share how you overcame that? I think for me at least a lot of it is/was not being able to be comfortable struggling with a hard problem for an extended amount of time, probably due to being used to only solving fairly trivial exercises and thus feeling dumb/the problem feeling impossible if it wasn't clearly solvable under a short timeframe.

markc 3y ago

Rich Hickey (creator of Clojure) references Polya several times in his classic talk "Hammock Driven Development". Here's a transcript: https://github.com/matthiasn/talk-transcripts/blob/master/Hickey_Rich/HammockDrivenDev.md https://github.com/matthiasn/talk-transcripts/blob/master/Hi... I've long been impressed by Hickey's problem solving skills, so I took much of this talk to heart, and even bought a copy of HTSI. Can't say it really helped me any more than Rich's talk (as a programmer) but I'm thinking I'll give it another look.

wayeq 3y ago

How to Solve it (202?) Ask GPT7 to think for you

jzer0cool 3y ago

Anyone with an interesting example to share using these steps?

ethanwillis 3y ago

I find it interesting how the link at the bottom of the page is a clearly hacked link.

OmarShehata 3y ago
What do you mean by "hacked"? It's just an old domain that's expired and now redirects to the hosting website. Nothing nefarious going on. Here's the original, which does indeed have a translation as one would expect: https://web.archive.org/web/20111005170154/http://webhostinggeeks.com/science/g-polya-be https://web.archive.org/web/20111005170154/http://webhosting...

R41 3y ago

Loved the way polia's explained the basics of solving challenges by asking questions - - Can you find a problem analogous to your problem and solve that? - Can you add some new element to your problem to get closer to a solution? etc we always asks new onboard people to read this .It's like a bible for problem solving .

im_lince 3y ago

In our team we recommend this book to new joiners. The principles and methods shared by polya is valid for solving the not only the mathematical problems also engineering problems in effective way.

OldGuyInTheClub 3y ago

It's easy when you have John von Neumann as a student.