Reddit reviews The Art and Craft of Problem Solving

First, consistently solving A1 and B1 is a great start! Puts you well above the typical. Be sure to pay attention to how you write it up: Putnam graders are very strict and solutions most often get 0, 1, 9, or 10 points. Be also aware of what your goals are and don’t get anxious, you’re not looking to solve everything, so it's good to fully solve one problem before moving on. Putnam problems in particular often have short clean solutions that are really satisfying to find.

You also can't beat just working through problems. Putnam 1985-2000 by Vakil, Kedlaya, Poonen is fantastic as it gives many ways of solving or approaching each of the problems. It also gives just the right level of hints. This way you can learn both by working through the problem and by seeing the different perspectives. For example, with a single problem there may be a long brute-force solution, a quick but hard to discover solution, and a quick solution based on advanced math (you can use most things that come up in an undergrad math curriculum, even elliptic curves).

The Art and Craft of Problem Solving is a great read for general strategies and practice, and will remain relevant throughout any later work.

Mathematical Olympiad Challenges by Andreescu and Gelca shows off a few major problem solving styles and has a great selection of problems. I studied it in high school and it ended up being very important for me getting Putnam Fellow.

Earlier I had also studied Problem-Solving Strategies but that may be too big and not as focused on Putnam type of problems

Check out math competition questions, and hang out on the AoPS forums. The Art and Craft of Problem Solving by Paul Zeitz is a good start, though it's far from sufficient for getting proficient at problem solving.

It's difficult to make recommendations without being certain of what you actually know and what you imagine mathematics to be like. A lot of university-level mathematics is technical and requires familiarity of high-level concepts. This is in contrast to softer popular mathematics, which is more related to solving problems and contest questions. One of the things I've noted about pre-university students passionate about mathematics is that they assume that the subject is only about problem-solving and fail to take into mind the level of technical knowledge that must be learnt and memorized to be a mathematician.

If you're simply looking for problems to solve, try The Art and Craft of Problem Solving by Zeitz or Problem Solving Strategies by Engel. Generally any book geared to the Olympiad or regional competitions will be alright. Here, you're not looking for a specific body of knowledge, but rather an approach to thinking and persevering when handling tough problems.

But if you're looking to learn more about 'technical' mathematics, you'll need to know the basics of numbers and sets. Numbers & Proofs by Allenby is a good introduction, using an approach that gets you to actively solve problems. Once you get past that, then you can try your hand on analysis or group theory or linear algebra or even basic graph theory. But keep in mind that with 'technical' mathematics, all knowledge is built on understanding of previous fields, so don't rush through it or you'll get discouraged by any difficulty or unfamiliarity you'll encounter.

Hi, here I will post some great books, some free (by Santos), some not (others).

Junior problem seminar: Santos

Number Theory for Mathematical contests: Santos

The Art and Craft of Problem solving: Zeitz

Problem-Solving Strategies: Engel

Mathematical Olympiad Treasures: Andreescu, Enescu

Mathematical Olympiad challenges: Andreescu, Gelca

Problems from the book: Andreescu

Those are more or less the "general" books, they always contain the main topics of mathematical olympiads, they usually aren't focused on just one topic, for one-topic books see here: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=319&t=405377

This passage appears to come from page 72, problem section 3.1, of The Art and Craft of Problem Solving by Paul Zeitz, which does not claim to be a rigorous book; most descriptions of elementary symmetric polynomials do specify that the terms are products of distinct factors, while this book just lists the three-variable and four-variable elementary symmetric polynomials and then describes some of their characteristics.

It's a good question that's hard to answer exhaustively. Here are some pointers. Maybe they can help to start a discussion.

1. Paul Zeitz writes the following advice in The Art and Craft of Problem Solving (emphasis added):
   
   > It isn't hard to acquire a modest amount of mental toughness. As a beginner, you most likely lack some confidence and powers of concentration, but you can increase both simultaneously. You may think that building up confidence is a difficult and subtle thing, but we are not talking here about self-esteem or sexuality or anything very deep in your psyche. Math problems are easier to deal with. You are already pretty confident about your math ability or you would not be reading this. You build upon your preexisting confidence by working at first on "easy" problems, where "easy" means that you can solve it after expending a modest effort. As long as you work on problems rather than exercises, your brain gets a workout, and your subconscious gets used to success. Your confidence automatically rises.
   
   
2. A useful term to know is self-efficacy, which means "one's belief in one's ability to succeed in specific situations or accomplish a task". The Wikipedia article mentions four factors affecting self-efficacy, which are worth looking at. "The experience of mastery is the most important factor determining a person's self-efficacy. Success raises self-efficacy, while failure lowers it." This is consistent with Zeitz' advice to start with easy problems and then gradually increase the level of difficulty.
   
   
3. Another factor affecting self-efficacy is "vicarious experience". This is "most effectual when we see ourselves as similar to the model", but I still think it helps to know that many eminent mathematicians have experienced feelings of self-doubt at some points in their careers. Here are some quotes that illustrate this; they're from Advice to a Young Mathematician, which is well worth reading.
   
   > One struggles unsuccessfully with small problems and one has serious doubts about one's ability to prove anything interesting. I went through such a period in my second year of research, and Jean-Pierre Serre, perhaps the outstanding mathematician of my generation, told me that he too had contemplated giving up at one stage. Only the mediocre are supremely confident of their ability. The better you are, the higher the standards you set yourself — you can see beyond your immediate reach. — Michael Atiyah
   
   > When I arrived in Moscow in my last year of graduate study, Gel’fand gave me a paper to read on the cohomology of the Lie algebra of vector fields on a manifold, and I did not know what cohomology was, what a manifold was, what a vector field was, or what a Lie algebra was. — Dusa McDuff
   
   
4. Finally, in my opinion there's nothing wrong with getting nervous in office hours and fumbling with easy questions. I've done this myself several times. It's just the nerves talking. As one gets more comfortable and relaxed with the situation (the fourth factor affecting self-efficacy), one gets better at keeping calm and tackling the questions one piece at a time.

This is good, with respect to learning tips and tricks for competitions, I think you're best off getting a book.

http://www.amazon.co.uk/Art-Craft-Problem-Solving/dp/0471789011

Is good

http://www.amazon.com/Art-Craft-Problem-Solving/dp/0471789011

Doing math is all about problem solving and this is a really terrific book for building your intuition and confidence in problem solving.

Well the first book details much of what you would need to know as a first year grad student in math and has recommendations for other books as well.

This book might also be something to try or perhaps Knuth's Concrete Math.

You could also try what this guy did, only with the MIT math curriculum on MIT OCW and without the time constraints.

Here are a couple of suggestions:

• Take up competitive programming. Go to http://uva.onlinejudge.org/ and do the problems from there (they have a few book suggestions as well). Aim to participate one day in ACM-ICPC or Google CodeJam.
  
  
• Go deeper into Mathematical areas relevant to programming such as graph theory, number theory, combinatorics, etc. Rudiments of most of these can be picked up at your level. Go through a book like The Art and Craft of Problem Solving
  
  
• Explore functional programming languages (read this: Advanced Programming Languages ) to improve your programming range.
  
  I am not an especially good problem solver, but I have done fairly OK financially. These are suggestions that I wish someone had given me when I was your age - it would have made my career slightly more fun!

I can't answer your question directly since I haven't fully read CTCI, but I'm going to throw another recommendation out there. I started doing Codility problems but wasn't happy with my ability solve them.

I could arrive at the correct solutions, but it took me a long time and always felt messy, like I was going in circles or down more dead ends than I should be. Maybe you don't have that problem. Anyway, finally I found this book, The Art and Craft of Problem Solving. It's aimed a little more at mathematics, but, damn was it helpful for me. The difference in my solutions before and after reading this book is huge.