I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where ½ proof = 0, and it is demanded for proof that every doubt becomes impossible.
--Carl Friedrich Gauss
Weekly problem sets will be posted every Wednesday, and due the following Wednesday (unless otherwise noted) before the start of class. Your write-ups should be neat, legible, and stapled. Proofs should be written in complete sentences. In general, late homework will not be accepted for a grade, and never without prior arrangement. Illness and family emergency are the two main reasons why an extension on homework may be granted.
Challenge problems will posted regularly throughout the term. The challenge problems will be more difficult than those on your problem sets. In general, challenge problems require proofs. You will solve ten challenge problems by the end of the quarter. To encourage you to be thinking about these challenge problems, two solutions are due by October 1st, four additional solutions are due by October 29th, and the final four problems are due by November 12. Turn in your challenge problem solutions on a separate sheet of paper from your problem set solutions.
I strongly encourage you to collaborate with others on your problems sets and challenge problems. When learning difficult mathematics, it is helpful to discuss ideas and confusions with your classmates. If you choose to collaborate (please do!), keep the following guidelines in mind:
Unfortunately, I must assign you a number to measure your comprehension of course material. In mathematics, a proof is either correct, or not. On the other hand, a proof can be almost right, but there is some small detail that keeps it from being an airtight argument. Each problem will be graded on a scale of 0 to 5, depending the extent to which the proof deviates from being correct. In general, a solution will loose 1 point for a minor defect and 2 points for a major defect. You will never loose points for being creative.
Here are is an example of a proof in which a few details are omitted, and how it could be modified to become correct.