Math 380 A, The Art of Problem Solving

Autumn 2011

• Meeting time and location     Tu-Th  3:30-4:50pm, location TBA
  
• Office hours                              
• M, 5-6pm, PDL C-342;
  
• M, 6-8pm, PDL C-36;
• by appointment (with either instructor)
  

• Instructors    
  

• Textbook:                             "The Art and Craft of Problem Solving", by Paul Zeitz
                                            Second Edition (Wiley & Sons),  available at the University Bookstore

• Class discussion board on Catalyst  (will be opened on the first day of the quarter)
  

• Syllabus and schedule  (in .pdf format)
• Homeworks and solutions and handouts    
  
• Final Exam                                                        (New!)
• Presentations
  
• Grading Policy
  
• Tips for writing up solutions
  
• The Putnam Competition
• Other sources of material

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Homeworks and Solutions and Handouts:


• To be posted at a later time
  

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Final Exam:


• TBA
  

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Grading Policy:



Homeworks (9)	40%
Participation / Presentations (4)	35%
In-class Final	25%
Total	100%



  In special circumstances, we will allow you to turn in one homework late, at the cost of 50% of the grade. No make-up final will be offered.

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Presentations:

Along with the 3-5 homework problems, each week, you will get 3-5 "presentation" problems. These will not be turned in, but you must solve them nonetheless. You must be ready to come to the board and explain them to the other people in the class. We will let you volunteer for each problem; however, keep in mind that each student will have to present 4 times during the quarter. If the same people keep volunteering each time, we will have to "designate a volunteer" from among the people with less than 4 presentations.

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Tips for writing up solutions


Say you're doing homework or taking an exam. You're working hard, focused, and all of a sudden a light bulb gets lit above your head: you have figured out how to solve one of the problems. Good for you! But here's the hard, unforgiving truth: that warm tingle you're feeling is no guarantee that

1) you have actually solved the problem (are you sure you have all the cases covered?...) and
2) you will get any or all credit for it.


Solving a problem is only part of the work: now you have to convince the grader that you have done so.



Life is tough. Graders may be even tougher. But here's a list of things which, when done properly, will yield a nice writeup which will appease any reasonable grader.

1. Before you start writing, organize your thoughts. Make a list of steps to the solution. Figure out what intermediary results you will need to prove. For example, if the problem involves induction, always start with the base case, and continue with proving that "true for n means true for n+1". Make sure that the steps follow from each other logically, with no gaps.


2. After tracing a "road to proof" either in your head or (preferably) on scratch paper, start writing up the solution. The best way to start is with an outline of what you will be doing!


3. Complete each step of the "road" before you continue to the next one.


4. When making statements like "it follows trivially", listen for quiet, nagging doubts. If you yourself aren't 100% convinced, how will you convince another? Even if it seems to follow trivially, check again. Small exceptions may not be obvious. The strategy "I am sure it's true, even if I don't see it; if I state it's obvious, maybe the grader will believe I know how to prove it" has occasionally lead its user to a score of 0 out ot 10.


5. Organize your solution on the page; avoid writing in corners or perpendicular to the normal orientation. Avoid, if possible, post-factum insertion (if you discover you've missed something, rather than making a mess of the paper by trying to write it over, start anew!)
   


6. Before writing each phrase, formulate it completely in your mind. Make sure it expresses an idea. Starting to write one thing, then changing course in mid-sentence and saying another thing is a sure way to create confusion.


7. Be as clear as possible.  Avoid, if possible, long-winded phrases. Use as many words as you need -- just do it clearly.
   


8. If necessary, state intermediary results as "claims" or "lemmas" which you can prove right after stating them. If you cannot prove one of these results, but can prove the problem's statement from it, state that you will assume it, then show the path from it to the solution. You may get partial credit for it.


9. When you're done writing up the solution, go back and re-read it. Put yourself in the grader's shoes: can someone else read your writeup and understand the solution? Must one look for things in the corners? Are there "miraculous" moments? etc.


As always, ease comes with practice, and you'll get plenty of that in this class. Even though it may be annoying at first to go through all of these steps when writing up a solution, do it! Eventually, it will become second nature, and at that point you will have learned how to write mathematical solutions.


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The William Lowell Putnam Mathematical Competition

This course can serve as a preparatory class for undergraduate mathematical contests, in particular the Putnam Competition. You can find more information about the Putnam at UW here.

IMPORTANT:

Participation in the Putnam or any other competition is COMPLETELY VOLUNTARY and NOT a required part of the course. 


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Other sources of material

• IMO problems (problems only)
  
• The Putnam Competition (slide down the page for a table with problems and solutions)
  
• The Bay Area Mathematical Olympiad (some of these may be too easy, but some will be hard)
  
• 19 Proofs of the Euler Theorem (just for fun)
  
• Catalan numbers (and the objects they count) (courtesy of Richard Stanley)
  

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