Improving Problem Solving Skills: Tips from Polya's How to Solve It Book

[StatOnTheSide]

I am trying to improve my problem solving skills. I am an engineer and I have a PhD in EE. All through, the only thing I have done is to solve problems. I am familiar with most of the things mentioned in Polya's book.

I am interested in being able to solve the very hard problems that usually appear in the international Olympiads and Putnams. Consider a problem like

"In the plane, n lines are given (n ≥ 3), no two of them parallel. Through every
intersection of two lines there passes at least an additional line. Prove that all lines
pass through one point."

If you know that the heuristic to be used here is that of the solution to Sylverster's problem, then the answer to this is trivial. There is another interesting problem which is

"We start with the state (a, b) where a, b are positive integers. To this initial state we
apply the following algorithm:

while a > 0, do if a < b then (a, b) ← (2a, b − a) else (a, b) ← (a − b, 2b).

For which starting positions does the algorithm stop? In how many steps does it stop,
if it stops? What can you tell about periods and tails?

The same questions, when a, b are positive reals."

Here, the heuristic is to look at the x in (x,y) mod(a+b) and to make deductions based on the fact that it is constant (= 2a mod(a+b)).

My question is the following:

Are there books training you to come up with heuristics to arrive at useful conjectures? In other words, is there a book which gives useful suggestions to enable students to come up with useful conjectures?

Polya describes ways to do that but are truly elementary for me. TO solve problems at the level I have mentioned above, it must be possible to come up with heuristics or tricks using which, useful conjectures can be made.

Is there a book for training students to come up with truly smart conjectures? I am familiar with Engel's book and Andreescu's book. They teach the tricks. THey do not talk about "conjecture making" and ways to train yourself to do that which is what I am mostly interested in. THat to not elementary conjectures but truly advanced ones.

Conjecturing is the heart of solving problems. Please advise if you can think of other ways to train myself to be able to come up with smart conjectures. Also, please feel free to correct me if my question does not make sense.

 

[Petek]

You might be interested in Polya's two books Mathematics and Plausible Reasoning. https://www.amazon.com/dp/0691025096/?tag=pfamazon01-20 is subtitled Induction and Analogy in Mathematics and https://www.amazon.com/dp/4871878341/?tag=pfamazon01-20 is subtitled Patterns of Plausible Inference. They are more advanced than How to Solve it. You can view excerpts on Amazon with the provided links.

 

[StatOnTheSide]

Hi Petek. Those are good books but I still feel that he talks mostly about standard techniques like induction and 'looking at patterns'. I am more interested in the non-standard ways which seem really enigmatic. Like the problems I have posted. I am curious to know as to what thought process prompted the guy to think of looking at mod(a+b).

In other words, I am trying to figure out if there is a book which can help me become a good "Conjecturer". The non-standard problems in Olympiads require a different thinking and I am very curious to know as to how the kids there manage to come up with deductions based on such seemingly weird observations. Some of the tricks involve
looking at mod 2 or mod 3 of some quantity, looking at a maximum or minimum of some quantity and making deductions etc.

I feel that conjecturing is an art in itself and problem solution is the extension of it in the sense that if the conjecture is right, then formal expression of the "rightness" of a conjecture is the solution to a problem but the crux of the matter is to come up with conjectures. That being the case, it makes sense to get intense training in "Conjecturing" to arrive at useful deductions.

Are there any books which take this approach and provide useful training?

 

[homeomorphic]

You could try The Art and Craft of Problem Solving by Paul Zeitz. I haven't read it, but he was very successful in math competitions with his students.

Another thing about conjecturing is just having a really good understanding of math and what's going on behind the scenes. For that, the best example would have to be Visual Complex Analysis by Tristan Needham--incidentally, he is friends with Paul Zeitz and teaches at the same university.

You could also try to look into the literature on creativity.

Keep in mind that math-competition-style problems are like 100 m races, whereas what real mathematicians do is more like run marathons.

 

[StatOnTheSide]

Thanks for the feedback H. I really appreciate it.

You could try The Art and Craft of Problem Solving by Paul Zeitz. I haven't read it, but he was very successful in math competitions with his students.

I have his book. He has some suggestions but he resorts to the same approach as Engel does. He teaches certain tricks which can be used to arrive at useful deductions. Pigeonhole principle, extreme principle, invariance etc. It looks as though most Olympiad trainers have a common set of tricks based on which they formulate problems whose solution falls into one of the standard trick categories.

Another thing about conjecturing is just having a really good understanding of math and what's going on behind the scenes.

This is exactly what I am trying to figure out. Is there a book which takes a mathematical subject and explains, with the help of a very tough question, the mental process involved in arriving at useful conjectures?

Mason, Burton and Stacey's Thinking Mathematically is a really good book. It has good material on these lines. I read an online article by Schoenfeld about thinking mathematically and even that helped a lot. He mentions one of the things that happens to me all the time. Whenever I look at a problem, I think of an approach and I get fixated on it. On the other hand, when I see an experienced mathematician work on the same problem, it is amazing to see how many approaches he takes in an attempt to arrive at a useful conjecture.

This handicap of not being able to creatively think of different connections to a given problem is what bothers me the most. I could solve almost all the problems from Mason, Burton and Stacey's book "Thinking Mathematically" which is not too easy but not super hard either. The main problem is with problems in the Olympiad where I feel that there is a big need for the student to be very creative to be able to deduce from acute observations.

In the invariance approach, sum of squares of a sequence is constant and so you make some deductions based on that. In another sequence, modulo n = 2 times a constant so deductions are made based on that etc.

In the so called coloring approach, parity of number of black squares and white squares is important. So if you cut off the four corners of the chess board, there is no way to cover the rest of it using dominoes.

The above are basically some examples of tricks. If I come across a trick, I can solve another problem which can be solved using the same trick; however, I get struck in cases where I am not familiar with a trick that needs to be used.

The reason I posted this thread is to request the members of this forum to kindly suggest me sources where they talk about how various people have managed to come up with tricks to solve some tough problems.

I tried to do a lot of research myself and came across this article
http://www.ams.org/notices/199701/comm-rota.pdf
where the author says that every mathematician has a limited number of tricks. If that really is true, then I am not sure if my question has an answer because there does not seem to be mathematicians with a lot of tricks to tackle problems with.

Two other books I have recently ordered are

Karl Popper: Conjectures and refutations : the growth of scientific knowledge
Imre Lakatos : Proofs and Refutations

I am yet to read them but I am hoping that they will have useful information for me.

If you have any suggestions, I would greatly appreciate it if you can kindly post it here.

 

[homeomorphic]

Maybe "The Psychology of Invention in the Mathematical Field" by Hadamard. Again, just a book I would like to read, not one I have read. There are a lot of books that I haven't read that are on my radar screen, since I have looked for them or noticed them.

 

[StatOnTheSide]

I just ordered it :). Once again, thanks very much for the feedback H!

 

[fourier jr]

how to solve problems by wickelgren is good too. one thing that's different about it is that wickelgren isn't a mathematician, but a psychologist so he has a slightly different perspective. he writes in the intro that a lot of the content was influenced by artificial intelligence & computer simulation of problem solving, which might be interesting to an electrical engineer.

 

[StatOnTheSide]

Thanks Fourier_jr. The reviews of this book said that the math content is not impressive at all but I am going to get the book because the psychological aspects might be helpful. I read Schoenfeld's "Learning to think mathematically" where he explains his experience with training students with problem solving. The one thing that was really impressive was his observation regarding how an experienced mathematician thinks and how a novice thinks. A novice tries one line of thinking and keeps attacking the problem based only on that line of attack while the experienced one looks at a variety of approaches and gets the answer.
More interesting is the fact that at the end of the training, the kids who were novice at the beginning started to move towards trying more approaches. He says that at the end, the kids still tried just one approach till the end and then discarded it and the moment they tried the next approach, they got the answer. There is a big element of self regulation involved when attempting to solve a tough problem.
I have an intuitive feeling that "brain muscles" have to be developed to try to come up with different tricks and as many possible useful conjectures as possible. It is contextual and I am sure that majority of exericise problems involve straightforward application of principles and definitions or maybe the use of induction or the standard approach of looking at patterns if there are any obvious ones. But like I said, the main problem arises when there are problems which require a nonstandard approach. For that, the critical thing seems to be the ability to be able to come up with tricks which can possibly help make useful deductions. In his article, Ho Jun Wei says "Do not spend too much time on any single approach or conjecture, especially if you’re not sure it’s the right one" and he also says "the well-prepared problem solver never runs out of things to try". The link to his webpage is
http://hcmop.wordpress.com/2012/03/23/how-to-approach-an-olympiad-problem-by-ho-jun-wei/
From this, I feel that the name of the game is to develop "brain muscles" that can systematically and tactfully fish out different observations based on which, useful conjectures can be formulated.

I wish there was a book which would throw more light on how to develop the "useful conjecturing brain muscles". As another wise man told me in his PM, practice is probably the only way to develop those brain muscles.:)