Ma 17 How to Solve It
 

Ma 17 How to Solve It

Instructor:
Abhishek Saha
Term:
Fall 2007/2008
Meeting:
Tuesday 19:30–21:30, Sloan 159

Course description

This class is intended for those who wish to develop and practise their skills at solving mathematical problems, particularly those which do not involve very much in the way of advanced mathematical concepts. As such, the course has no formal prerequisites, although students who are familiar with some concepts from calculus, linear algebra, combinatorics, and elementary number theory may find that to their advantage. Those with interest in math competitions will likely find the course interesting and relevant. In particular, those who wish to take the Putnam, are strongly encouraged to take the course. Freshmen are welcome to take the course, and some have done so in the past with spectacular success. Those who wish to attend the classes without registering are welcome to do so.

Tentative plan

  • October 2: Basic problem solving techniques, induction, recursion, descent.
  • October 9: Combinatorics, with emphasis towards pigeon-hole principle and counting arguments.
  • October 16: Basic calculus and some combinatorial geometry
  • October 23: Trigonometry, complex numbers, algebraic techniques, polynomials
  • October 30: More algebraic techniques, linear algebra, generating functions
  • November 6: Analysis
  • November 13: Number theory
  • November 20: Invariants, and many applications
  • November 27: Probability, Putnam primer.
  • December 4: Miscellaneous problems

    Grading and homework

    Grading is based on weekly homework assignments, attendance, and class participation. Each homework set will consist of six problems. Of these, the first three will be homework problems in the traditional sense of the word, in so far as they will be based on material taught in previous classes. The remaining three will be problems that you are supposed to 'think about' but not necessarily solve. These are designed to motivate the theory that will be taught in the next class. In order to pass, a student will have to attend at least nine of the ten classes and write up a correct solution of at least one of the problems each week. Any of the six problems may be solved for credit; however the solution must be mathematically correct, complete, well written, and on time.

    Collaboration policy

    You may discuss a homework problem with another Caltech student only after you spend at least one hour of time thinking about it yourself. However all solutions must be written independently.

    Written sources

    Many homework problems will be from actual mathematical competitions (Putnam, IMO, national olympiads, etc.). Reading homework problem solutions from books, the Internet or anywhere else is not allowed. Apart from that, you are welcome to consult any books whatsoever.

    Additional materials

    Problem sets

    1. Think about these [pdf] problems before the first class.
    2. Homework [pdf] due October 9. Solutions[pdf]
    3. Combinatorics handout [pdf]. Homework [pdf] due October 16. Solutions[pdf]
    4. Homework [pdf] due October 23. Solutions [pdf]
    5. Homework [pdf] due October 30. solutions [pdf]
    6. Homework [pdf] due November 6. Solutions [pdf]
    7. Analysis handout [pdf]. Homework [pdf] due November 13. solutions [pdf]
    8. Homework [pdf] due November 20. [pdf]
    9. Homework [pdf] due November 27. solutions [pdf]
    10. Homework [pdf] due December 4. Solutions [pdf]
    • Abhishek Saha
    Sloan 260
    saha@its