Is the shorter solution the smarter solution?

anemone

MHB POTW Director
Staff member
Hi MHB,

I have a question that has been bothering me for years. I wish to ask for your opinion regarding this issue.

Consider the following scenario where there are two people, A and B. A is quite smart and capable of solving many challenging math problems using succinct methods, but though B isn't that bad, B solves them using longer and more tedious methods than A.

I am wondering how do mathematicians perceive B? Does this imply that B is is not as smart as A and is less capable as well?

Thank you in advance for any response you provide and sorry if this has been discussed before.

MarkFL

Staff member
I think no implication should be drawn about two people based on their solution, or the length of it or method used. It may very well be the case that the shorter solution is the result of experience, rather than pure intelligence.

Bacterius

Well-known member
MHB Math Helper
[JUSTIFY]It may also just be that A is better than B at exploiting the structure of a problem to solve it faster than through the mechanical application of an algorithm. But depending on your definition of intelligence, that doesn't make him "smarter" than B, in some absolute sense. Perhaps B is better at developing those algorithms to start with, whereas A isn't very good at that. And so on. Intelligence is not a one-dimensional metric, and I think trying to reduce it to a scalar is both misguided and misleading. After all, solving challenging math problems is just one skill-set, and equating intelligence to just this one skill-set is clearly biased and not particularly useful, in my opinion.

But then again, I am not really a mathematician. I just play one on the internet [/JUSTIFY]

anemone

MHB POTW Director
Staff member
I think no implication should be drawn about two people based on their solution, or the length of it or method used. It may very well be the case that the shorter solution is the result of experience, rather than pure intelligence.
Thank you Mark for your opinion. I always think if someone is able to solve a problem in a more elegant manner than someone's else, then that person is the winner of the game. And the loser (I am referring to myself) has no excuse but to admit that they are the loser, the weaker.

But I agree with what you have mentioned, our life is an accumulation of experiences from which we can then use to tackle challenging problems.

[JUSTIFY]It may also just be that A is better than B at exploiting the structure of a problem to solve it faster than through the mechanical application of an algorithm. But depending on your definition of intelligence, that doesn't make him "smarter" than B, in some absolute sense. Perhaps B is better at developing those algorithms to start with, whereas A isn't very good at that. And so on. Intelligence is not a one-dimensional metric, and I think trying to reduce it to a scalar is both misguided and misleading. After all, solving challenging math problems is just one skill-set, and equating intelligence to just this one skill-set is clearly biased and not particularly useful, in my opinion.

But then again, I am not really a mathematician. I just play one on the internet [/JUSTIFY]
Hey, what you just said is like an awakening for me. You're right, Bacterius, one cannot equate intelligence with how well-versed one is at being able to solve difficult problems. Intelligence cannot be measured using only one yardstick, the yardstick which measures how capable someone is at solving hard problems. It encompasses a wide range abilities, such as doing well in formal and informal circumstances, the ability to deal with harsh conditions and survive and the ability to process information and hence produce useful material from it.

I feel better after reading the replies, thanks.

chisigma

Well-known member
Hi MHB,

I have a question that has been bothering me for years. I wish to ask for your opinion regarding this issue.

Consider the following scenario where there are two people, A and B. A is quite smart and capable of solving many challenging math problems using succinct methods, but though B isn't that bad, B solves them using longer and more tedious methods than A.

I am wondering how do mathematicians perceive B? Does this imply that B is is not as smart as A and is less capable as well?

Thank you in advance for any response you provide and sorry if this has been discussed before.
The question posed by anemone is very suggestive and my reponse risks to betray Her. I normally post a solution if this solution is as possible 'timesaving' and 'immediatly understandable' , otherwise I prefer to renunce. That is due to a combination of two bad qualities I have: pride and indolence. Just to do an example, when I see the step by step solution of an indefinite integral reporting kilometers of trivial steps, just to use the words of Stephen Howking about the 'Scroedinger's Cat'... I unsecure my pistol! ...

Kind regards

$\chi$ $\sigma$

Last edited:

anemone

MHB POTW Director
Staff member
The question posed by anemone is very suggestive and my reponse risks to betray Her. I normally post a solution if this solution is as possible 'timesaving' and 'immediatly understandable' , otherwise I prefer to renunce. That is due to a combination of two bad qualities I have: pride and indolence. Just to do an example, when I see the step by step solution of an indefinite integral reporting kilometers of trivial steps, just to use the words of Stephen Hopkins about the 'Scroedinger's Cat'... I unsecure my pistol! ...

Kind regards

$\chi$ $\sigma$
Hi chisigma, I appreciate you taking the time to reply to this thread.

Looks like some of the time we feel exactly the same regarding this matter...

Prove It

Well-known member
MHB Math Helper
I think it comes down to what is the general purpose of solving these problems.

As an educator, I find that while there is a great beauty in being able to solve a problem as succinctly as possible, I tend to always make sure that the solution can be understood by those who are not at the level I am at. This often means having to put in the trivial steps, as the student's understanding and fluency might not be enough to follow the more succinct solution.

Also as an educator of mathematics, one needs to remember that there are MANY ways to solve any problem. Unless you are testing the student's use of a particular method, any valid method must be considered to be correct and in fact be encouraged. This is vital, as too much criticism of a person's solution will make them feel like their hard work isn't paying off, when it's actually the exact opposite. It is their hard work that will ensure their success and help them gain the fluency and finesse required to know what is the quickest or most succinct method for future problems. Also, sometimes a student won't know what to do to start with, so might try one method which might take a while, but actually get to a solution, but in having tried something, other ideas may come to them which enable them to refine their solution.

So I personally do not see any difference in intelligence, as intelligence is built on hard work, not some entity that exists and has a limit. The only possible difference is that the more succinct solutions have probably come from a person with more experience. Since the hard work must be encouraged, all valid solutions need to be considered correct and equally valid and beautiful.

Ackbach

Indicium Physicus
Staff member
I think it comes down to what is the general purpose of solving these problems.

As an educator, I find that while there is a great beauty in being able to solve a problem as succinctly as possible, I tend to always make sure that the solution can be understood by those who are not at the level I am at. This often means having to put in the trivial steps, as the student's understanding and fluency might not be enough to follow the more succinct solution.

Also as an educator of mathematics, one needs to remember that there are MANY ways to solve any problem. Unless you are testing the student's use of a particular method, any valid method must be considered to be correct and in fact be encouraged. This is vital, as too much criticism of a person's solution will make them feel like their hard work isn't paying off, when it's actually the exact opposite. It is their hard work that will ensure their success and help them gain the fluency and finesse required to know what is the quickest or most succinct method for future problems. Also, sometimes a student won't know what to do to start with, so might try one method which might take a while, but actually get to a solution, but in having tried something, other ideas may come to them which enable them to refine their solution.

So I personally do not see any difference in intelligence, as intelligence is built on hard work, not some entity that exists and has a limit. The only possible difference is that the more succinct solutions have probably come from a person with more experience. Since the hard work must be encouraged, all valid solutions need to be considered correct and equally valid and beautiful.
Well said! These were some of the thoughts running around in my head as well. Sometimes, it is the one who communicates clearly who is perceived as the clearer thinker, and not necessarily the person who solves the harder problem. To communicate clearly is to evidence the deepest level of thinking about a particular problem.

anemone

MHB POTW Director
Staff member
I think it comes down to what is the general purpose of solving these problems.

As an educator, I find that while there is a great beauty in being able to solve a problem as succinctly as possible, I tend to always make sure that the solution can be understood by those who are not at the level I am at. This often means having to put in the trivial steps, as the student's understanding and fluency might not be enough to follow the more succinct solution.

Also as an educator of mathematics, one needs to remember that there are MANY ways to solve any problem. Unless you are testing the student's use of a particular method, any valid method must be considered to be correct and in fact be encouraged. This is vital, as too much criticism of a person's solution will make them feel like their hard work isn't paying off, when it's actually the exact opposite. It is their hard work that will ensure their success and help them gain the fluency and finesse required to know what is the quickest or most succinct method for future problems. Also, sometimes a student won't know what to do to start with, so might try one method which might take a while, but actually get to a solution, but in having tried something, other ideas may come to them which enable them to refine their solution.

So I personally do not see any difference in intelligence, as intelligence is built on hard work, not some entity that exists and has a limit. The only possible difference is that the more succinct solutions have probably come from a person with more experience. Since the hard work must be encouraged, all valid solutions need to be considered correct and equally valid and beautiful.
Hi Prove It,

I salute your desire to ensure everything you write is easy to understand and follow for those who read your posts, and so even those who are equipped with only an understanding of basic elementary math can follow your posts without struggling.

I think you're absolutely right when you say we have to keep in mind there are always many ways to solve any mathematical problem and as long as each method results in the correct answer, then they are all valid and good methods.

I like what you've written and I appreciate your taking the time to reply to my thread.

Now, I've gotten the feeling that the sky is the limit, depending on how hard a person is willing to work towards achieving something or how much one wants to get it done, and intelligence is something that is very abstract. While it's true that some people are just more naturally intelligent than others, one can work harder to learn how to go beyond what comes naturally.

Thanks again!

Well said! These were some of the thoughts running around in my head as well. Sometimes, it is the one who communicates clearly who is perceived as the clearer thinker, and not necessarily the person who solves the harder problem. To communicate clearly is to evidence the deepest level of thinking about a particular problem.
First, thanks for chiming in with your thoughts, Ackbach.

I must say I agree with you as well! One who thinks more clearly has a better ability to communicate their ideas and demonstrate what has been done which can be one indicator of one's "intelligence" to the public. If what he/she said is unintelligible to others, then even the most brilliant piece of work may only ended up being discarded.

Thank you for the thought!

ModusPonens

Well-known member
I both agree and disagree with the respective points made in this thread. Let me make my own points.

One first thing one has to realize, when wanting to do mathematics, is the hard truth that there are people who we will never surpass, no matter how much work we put in. Can you honestly say that, if you work hard enough, you will prove the Riemann hypothesis? I think it was Michael Atyah who mentioned in an interview that he knew a guy who dedicated his entire life of hard work to proving the Riemann hypothesis. He didn't do it.

On the other hand, intelligence without hard work is worthless. This is an even more important thing to realize. Our western culture admires geniuses and, at the same time, think that hard work is not admirable. In school the kids who work hard are made fun of. The cool thing is to be lazy. This has an effect on people, namely me. I thought I was so bright and that hard work was so lame that I didn't succeed in college. Now I'm back, many years later, with a more experience.

Third and foremost point: the important thing is not how one is perceived by others; it's the importance of the work you do that counts and the pleasure you get doing it. If you are in mathematics to be apreciated _ which is what 99% of people are doing _ you will have a great deal of stress and suffering with school. I had this attitude for many, many years and college was torture to me because I hated being graded _ as if my value as an individual could be defined by a set of grades. I still have this attitude in me, but I'm trying to actively change it. I try to be there to learn the wonderful theorems, the beautiful patterns, shapes and structures. The goal of life is to be happy and to avoid suffering. If you are doing this because of fame you will suffer. If you're doing this because of the pleasure of learning and discovering, you will be happy. In the end, what's important to a mathematician is the quality of his work due to the pleasure of discovery. Erdös had it right when he said that those who were in mathematics for the fame would never make it.

Sooooo, to answer your question: mathematician A will be more praised. But mathematician B possibly could do more important work than mathematician A, if he works hard. Most importantly, mathematician B could be happier.

mathbalarka

Well-known member
MHB Math Helper
I completely agree with the points made by ModusPonens and believe it to be the best answer among all the others.

It's completely correct that the hard work you do to prove RH is worthless, if you are not a born genius (I will come to this point later).

I especially am very impressed by ModusPonens' conclusion, that A will be praised, but B can realize he is happy with himself. This is one of the beautiful points made in this thread, and I realized this piece of fact from the very day I beginning doing mathematics, that it certainly is very satisfactory to prove something very hard, which in turn has been done by A before, regardless of the fact that the proof is larger and less elegant than A. As one can see, I am much like B who, unlike A, has less-elegant solutions. But I am happy with that, as I have mentioned before, as proving something is much more satisfactory than conceiving others' proofs.

About A, you can do nothing about the fact that A is a genius and will always be superior than B at something, but also bear in mind that the peoples who discovers things, like A, is much less praised than peoples who draw conclusion from discoveries done by people like A. There are lots such example, which I am omitting here on purpose.

I think my points are made through this and the previous posts, and I can safely end this post at this point by making a quote from Poincare :

Facts do not speak.

And adding once again that one who makes all the facts speak are much more praised than those who discovers the facts.

Balarka
.

zzephod

Well-known member
Hi MHB,

I have a question that has been bothering me for years. I wish to ask for your opinion regarding this issue.

Consider the following scenario where there are two people, A and B. A is quite smart and capable of solving many challenging math problems using succinct methods, but though B isn't that bad, B solves them using longer and more tedious methods than A.

I am wondering how do mathematicians perceive B? Does this imply that B is is not as smart as A and is less capable as well?

Thank you in advance for any response you provide and sorry if this has been discussed before.
If you do maths for beauty then the elegant solution is very much more satisfying that the ugly brute force solution. The problem is that often we are not smart enough to find an elegant solution to a result we needed yesterday, and have to resort to brute force whether we like it or not. Later we may be able to refine our work and finally get close to the ideal of timeless beauty that we strive for...

.

Well-known member
some tiimes elegant solution comes because of understanding of the formula and usage. This may come with understanding in depth and experience and some times solving a similar problem.
Basically usage of proper tools(formula or trick) makes it elegant

Klaas van Aarsen

MHB Seeker
Staff member
It takes extra time and effort to reduce a brute force solution to a short and elegant solution. For a real world problem that is usually not necessary.
But in my opinion, the "better" mathematician takes that extra time to dot the i's and cross the t's.

And when there is more work to be done, or more of the same work to be done, it's easier and better to build on a short and efficient solution, than on a lazy and sloppy solution.
In other words, it is not necessarily the smarter mathematician that comes up with these elegant solutions. It's the mathematician that does not stop when he already has an answer, the one that keeps going, has endurance, and enjoys making the solution "perfect". The one who can also enjoy looking at the perfection in another's solution.

After a while and with some experience, it becomes more natural to do so.
Then it seems easy to conjure up these perfect solutions.
Never mind the time and effort spent to reach that point.
It's sleight of mind to make it seem magical!

alane1994

Active member
The elegant methods employed by the person A is likely through years of experience of struggling through the methods that person B uses. Elegance usually follows from learning from past experiences and recognizing when you can do certain things. That is more often than not why math builds upon itself. Differential Equations for example. You can have ODE IVP's that you can solve in a method that is very much a brute force way. Then later you learn Laplace Transforms which, strictly speaking, are a more elegant way. Neither person is wrong in how they do it, just different. Person B isn't to be looked down upon, but rather admired for persevering through the struggles that mathematics provide.

These are just my rambling and in-eloquent thoughts on this.

anemone

MHB POTW Director
Staff member
Hi ModusPonens,

Yes, sometimes, no matter how meticulous we are, how many resources we have used in order to solve a hard problem, we still may never be able to solve it, ant I agree there are always smarter people around the world and we just have to accept who we are, without whining about our own lack of ability or knowledge. Sometimes, some things are just not meant to be.

We need to tell ourselves no matter what we do, it should be because we love it, we are not doing it to get some flattering compliments from others, we are not doing it to impress others, it's all about us. We do it because we gain much satisfaction from doing what we like and for that, we should be very grateful.

Hi mathbalarka, thanks for the thought.

Yes, the most important thing is we have to enjoy what we are doing and time takes passes very quickly if we like what we're doing and we should not pay attention to the perceptions of others regarding our work. Now, I see the point that we have not only to be satisfied by our own finding of the proofs, we need also to draw as many conclusions from it as we can and see how to manipulate it to draw benefits from it. And we have to express our thoughts clearly to the public regarding our "new" findings, if any. Then and only then, is our mission is accomplished!

Hi zzephod, thank you for taking the time to reply in my topic.

Yes, more often than not, in the middle of the process of finding an elegant way to solve a hard math problem beautifully, we can get to a point where we are fretting and tired. If we are not determined enough, we will say to ourselves, "Fine. I give up." It's so much easier to give up rather than cudgeling our brain hard to think of something promising to solve the problem. But, what you just told me reminds me of a very wise saying, "When you think you have reached the end of your rope, tie a knot and hang on." I think now if we truly have the heart to learn, we can take days or even months to try to think of all possible alternatives to solve it. But if it is a problem that needs to be attended to urgently, then we may have to resort to brute force.

I realize that you have a deep and thorough understanding of the fundamentals of mathematics and hence the tricks that you use to solve some of my challenge problems seem very pretty and smart. That is, I agree with you that a complete understanding of any of the fundamental theories in math can be a great help for us to draw many useful short-cuts and tricks to tackle any hard math problem!

Hey, I can't agree more with your point where you state the learning process is never a never ending process...while we enjoy providing "perfect/smart" solutions and feel proud and happy for ourselves, we can also enjoy reading the perfection in the solutions of others and learn from them. There is nothing shameful about jotting down "tricks" we have observed from others for our future reference, there is no disgraceful being less competitive at times and most of all, we should accept the level which we are at and don't cry like a big baby when we are not as smart or or as capable as somebody else.

Hi alane1994, thanks or your thought.

I value each and every one's opinion in the thread. And I think one (the experienced one) has to also have the great memory because when he is solving a current difficult and challenging math problem, though he has the lay of the land a little better, if he can't recall that he has already solved a problem of this type before, then he might solve the problem again and maybe this time using a different approach, with the equally likely chance that the new approach is either better than the first approach, or the same or it even can be a more tedious method compared to the first method that he previously thought of.

And yes, no one should be looked upon with any degree of disdain for not being able to solve a hard math problem using the simple route, except if he/she is after perfection and he/she is the one looking down on him/herself.

Deveno

Well-known member
MHB Math Scholar
There are a lot of facets to this beautiful question, and I would like to address some of them separately:

First of all, the "long and tedious route" often has the advantage that each step on the ladder is "transparent", and may be conceptually easier (if more time-consuming) to validate. Think of it like the difference between jumping 4 feet all at once, or climbing step-by-step. Mathematical proofs and problems are not unlike file-compression formats, if one uses "higher-level algorithms", more "decoding/encoding" apparatus is required.

That said, mathematics is not a mundane manufacturing process like assembling furniture, but more of an art-form: painting with the colors of reason. As such, the element of style enters into it, with a clear community preference for elegance and minimality. A proof that is 4 scant lines long is admired over one that does an equally good job in 10 pages. Rest assured, though, the 4 line proof will almost certainly make greater demands of the reader!

In truth, however, the vast majority of us lay in the no-man's land between the two extremes. Our brilliant pithy insights are littered with some sections where we trudge along in a dull and workman-like fashion. We are all, at our own unique rates, travelling the path that leads from ignorance through knowledge, to understanding. It's a long and treacherous road.

As regard our dear colleagues A and B, it may depend on the venue they are involved in. If all that matters is "getting some answer" (perhaps they are applied mathematians doing calculations for a new super-collider) then a sound result is all that matters (although A may perhaps find more time to sleep at night). If A and B are teachers, A may do very well with an "honors class" and fail to get his/her point across with a "remedial one". B may have the OPPOSITE problem: the slower students appreciate the patient step-by-step guidance, while the brighter students get bored, and lose interest. Different approaches may make more sense in different contexts.

In practice, what often happens is this: mathematician B discovers a result, and mathematician A later revises it in a more polished form. One often sees this in texts where a comment is given parenthetically (this is based on a a presentation by....), meaning the orginal ideas are borrowed, but "cleaned up" (perhaps due to newer methods being available).

Which brings me to another facet of your question: the historical aspect. We do not learn mathematics in a vacuum, we are unfortunately all bound to the era we happen to be born in. A and B may well have learned their trade at different times, and different places, affecting the methods available to them to learn in the first place. These things cannot be helped, what matters is not your past, but where you aim for your future. That moment of decision, of turning the will of "someday" to the decision of "now" is the only moment you ever have. And so, it may be that A might become B, or vice-versa, according to the winds of fate.

I think, more or less, we are ALL "B" (even the prodigies, just on a different level) aspiring to be "A". This is as it should be. And if "A" and "B" cease to be able to dialogue amongst themselves, I think we're all in big trouble.

mathbalarka

Well-known member
MHB Math Helper
Deveno said:
I think, more or less, we are ALL "B" (even the prodigies, just on a different level) aspiring to be "A".
Exactly, point well taken. +1.

anemone

MHB POTW Director
Staff member
Well said, Deveno!

Grand or not, everyone has talent. When you express your talents you create joy and fulfillment in yourself and for others and that's all that counts.

ZaidAlyafey

Well-known member
MHB Math Helper
Most of the time it is not easy to realize that a problem has a shorter solution. This is because we don't have that much experience and broad view of the problem. It is possible that the person with the shorter solution has already solved a similar problem or seen it somewhere else. It is what is called the art of problem solving. Sometimes we appreciate a solution because it is using a method that we haven't thought of before. We usually think of the most straight forward way to solve a problem but as the time goes by we are recruited with many instruments and approaches to tackle that problem differently or elegantly. Generally knowledge and experience are the main differences but it might be possible that that person is an out of box thinker ,he thinks different that ordinary people or sees a certain problem from another point of view.

Turgul

Member
This has somewhat been touched on, but is worth saying explicitly: shorter solutions need not be better than longer ones. Certainly a solution that gets right to the heart of the problem is very nice, but there are many terribly clever solutions to problems which are completely unenlightening.

There is a reason multiple solutions to a problem are valuable. Different solutions speak to different parts of the problem. Some solutions make it clear how one might solve similar problems, while different solutions might generalize in a different direction. There is certainly something aesthetically pleasing about a short solution, but they often depend upon clever tricks and happenstance regarding the exact problem at hand. Personally, I tend to prefer solutions which speak to the "why" of the answer and people who can make that clear, in my experience, have much better command of the mathematics.