Books for Problem-Solving Skills (i.e. Putnam)

[bacte2013]

Dear Physics Forum advisers,

My name is Phoenix Kim, a rising junior with major in mathematics and an aspiring applied mathematician & algebraist. I wrote this email to seek your recommendation on great books for problem-solving skills (techniques, strategies, etc.) in the mathematical competition, specifically the Putnam Competition; I firmly decided to prepare myself in order to compete in the Putnam Competition. However, Putnam Competition will be my first participation in the mathematical competition as I was never involved in any form of mathematical competition...So I am a quie rookie. I would like to pick up a book or two on the problem-solving in mathematics and use it (or them) in conjunction with the Putnam problems. I see a lot of people recommend following books: "How to Prove It" by G. Polya, "Problem-Solving Strategies" by Engel, "Problem-Solving Through Problems" by Larson, "Putnam and Beyond" by Andreescu, and "Art and Craft of Problem Solving" by Zeitz. However, I do not know which book should I start with since all of those books seem to cover similar materials.

Also what approach should I take in order to prepare for Putnam? Is my strategy of reading the problem-solving books and doing Putnam problems good approach? The thing is that I do not know quite a lot of mathematical topics since I just started to pursue a track in mathematics on the last semester. However, I have been studying the following books by my own: M. Artin's Algebra and Hoffman/Kunze's Linear Algebra. I hope those books are great for theoretical contents of Putnam.

PK

 

[verty]

How good do you think you are at solving problems now?

Here is an example question. You are in a room with no metal objects except for two iron rods. Only one of them is a magnet. How can you identify which one is a magnet?

Can you solve that in 1 minute? (This question is from Martin Gardner, or was at least included in one of his books.)

 

[bacte2013]

^
I am actually a newbie in the problem-solving. I never did any mathematics competition...Regarding to your interesting problem, my conclusion is that I cannot identify the magnet without knowing those iron rods' movements. Is that question from Martin Gardner's Mathematical Games column? My research mentor gave the series of books to me.

 

[verty]

Keep pondering that question, I think you'll figure it out before too long and it'll suddenly seem obvious.

My point in asking it was to give a little demonstration of what a competition is going to be like. My guess was that 1-minute would be a typical time for such a question. So that's the level you need to reach. I don't know which book to suggest though.

 

[bacte2013]

^
Do I need to read all problem-solving books I suggested? I am really not sure which book should I start for the Putnam. I am feeling overwhelmed since this is my first time preparing for the mathematical competition.

 

[verty]

I'll give the answer to that magnet question now.

One way to tackle it is to simplify the problem. If you can simplify the problem so that solving the simpler problem solves the original problem, that can work well. In this case, a simpler problem is to consider that you have only one bar. If you have one iron bar and want to determine if it is a magnet, how could you do that? One would have to use the Earth's magnetic field, it's the only thing the magnet would interact with. So the solution to that problem is to suspend the bar by its center and see if it points toward the north (or any particular direction). This solves the original problem because if you test one bar, you will know if it is or isn't the magnet, so that is enough to answer it.

Another way is to look for a weak spot. The weak spot in this question is the word "iron": the bars are made of iron and one can easily magnetize an iron bar by running the pole of a magnet along its length multiple times. So take bar B and attempt to magnetize bar A. If bar A is magnetized, you have two magnets and can test for repulsion. But if bar B was not a magnet, bar B is still not a magnet and there will be no repulsion.

The third solution is the hardest one to see, I think. Attraction or repulsion happens at the poles of a magnet, not at the center. So touch the end of bar A to the center of bar B. If there is attraction, bar A is a magnet, otherwise not.

I chose this question because most visitors here will have had enough knowledge to be able to solve it, but it is still highly difficult to do in a quick time. So that is the whole "skill" part of it, finding an answer when you have the knowledge.

The Andreescu book looks great but I would not choose it yet because I don't think you are ready for it. You should choose an easier book, I don't have time to pick one out but find one for high-schoolers that is more easy and tackle that first.

 

[verty]

Right, I think you see what I was trying to do, I was showing you that it's not just about knowledge. One can have knowledge and still do poorly. And when you turn that paper over and look at those problems that look so scary, it is inevitable for one's heart to beat more quickly and for one to be flustered, etc. And it only gets worse as one starts to work on the problems because one may make mistakes or find that one's strategy isn't working, etc.

So I absolutely don't blame you for struggling with the magnet question. I wanted to show you that there is more than just knowledge at play and it isn't simply a matter of learning the subject very well. It *is* difficult to answer a question the first time one sees it, in a short time. And that is hard to appreciate until one has been put in a situation where it counts. But I think you'll find that problems don't scare you as much now and you will already be better able to handle them.

To finish what I was saying about books, Zeitz and Andreescu both look nice to me actually. Zeitz looks to be a good book to start with and I think they would compliment each other quite nicely, so that would not be a bad way to go. Then also, Polya's book is very famous and very cheap as well, so I think it would be worth getting as well.

If you have questions, please ask and I will endeavor to answer them.