Homeworks and Solutions and Handouts:
- Check your UW Canvas page (for registered students only)
- TBA
- 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.
- 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!
- Complete each step of the "road" before you continue to the next one.
- 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.
- 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!)
- 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.
- Be as clear as
possible. Avoid, if possible, long-winded phrases.
Use as many words as you need -- just do it clearly.
- 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.
- 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.
- 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)
- 20
Proofs
of the Euler Theorem (just for fun)
- Catalan
numbers (and the objects they count) (courtesy of
Richard Stanley)
Final Exam:
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Grading Policy:
| Homeworks (9) |
40%
|
| Participation / Presentations (4) | 30% |
| In-class Final |
30% |
| Total |
100% |
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Presentations:
Along with the 4-5 homework problems, each week, you will get 4-5 "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
Solving a problem is only part of the work: now you have to convince the grader that you have done so.
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.
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.
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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
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