How to Write Up a Challenging Problem as a Paper

My usual structure:

You may do this project at almost any point during the semester. Here's the procedure...

Pick a problem/topic and clear it with me. The final deadline is 6 weeks before the end of the term. (Please do it earlier. Everyone will be happier.)
Within two weeks of picking your problem, write a rough draft and give it to me to critique.
Within one week of getting your rough draft back, revise your rough draft and hand in a final copy.

Here are some guidelines for picking the problem...

You may choose your problem from any of the chapters covered in the course.
No two people/groups may do the same problem (which is why you have to clear your problem with me).
Problems assigned for handed-in homework are ineligible.
Odd-numbered problems are ineligible when they have hints in the back of the book.
Good sources for appropriate problems are the review sections for the various chapters.
If you have an idea for a problem or topic that is not in the book, suggest it.

Writing Guidelines:

There are some common problems which arise when learning mathematical writing. I recognize that you have almost certainly not done this before, and that this is probably all new to you.

(1) Technology -- Both the rough draft and the final copy must be computer-processed, including any pictures or graphs needed. I should see nothing handwritten on your paper. Here are fixes for some standard difficulties...

Want to square something? Use the superscript command to make it look as it should. There's also a subscript command. These are usually kept with modifiers like "bold" and "italic."
Need an equation? In MSWord (my nemesis), there's an Insert Object -> Equation command.
MSWord also has a picture-drawing utility, which you can use to make graphs and such.
You can create equations, graphs, pictures, etc. in Mathematica, and probably also in Maple, and copy/paste them into your paper. (This works differently on the Mac and PC platforms.)
Special symbols can often be found in a "symbol" font.
Trauma on a Mac? I can probably help.

(2) Format -- One way to think about the assignment is to pretend you're writing a short paper, and the subject is the problem at hand. There should be an introduction, which often consists of a statement of the problem, a body, which often consists of a discussion and solution of the problem, and a concluding statement, which usually places the solution in context. (Examples of common conclusions: "This amount is realistic because..." or "...which is what we expected because...") There should not be sections labelled as corrresponding to parts (a), (b), etc. of a problem, if indeed the problem has such parts.

(3) Writing style -- While the style of your writeup should be similar to any regular old group homework problem, this assignment requires a bit more formality. (In my dreams, writing this will help you in writing up your regular homework...) Your reader will not have a textbook handy when perusing your paper, so don't refer to the text in such a way that you confuse the reader. For example, restate the problem, but do not quote it directly. ("Find the area under the curve" becomes "Suppose we wanted to find the area under the curve...") Similarly, if your chosen problem has several parts, don't list them at the beginning of the paper. Bring them up as they occur. And, because you're controlling the flow of the writing, you can discuss the parts in a different order than they're presented in the problem if that's more convenient!

Another example, taken from an email sent on this topic:
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Instead of writing
"(a) Find the miniflarg of the flozzlenuck."
write something like
"Now let's find the miniflarg of the flozzlenuck."
or
"Suppose we wanted to find the miniflarg of the flozzlenuck..."
or
"What about the miniflarg of the flozzlenuck? To find..."

That way you're introducing what you're trying to do without actually posing it as a command.

You don't need to have intro and conclusion paragraphs, but you should have a couple of sentences on each end that serve as a frame for the problem. (You don't want people to be confused at the beginning or wonder what happened to the punch line of the problem.) One good way to end is to make a statement that ties together a few parts of the problem.
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When I critique your rough draft, I will not point out every flaw. In an English-style writing assignment, a faculty member generally doesn't make detailed grammatical comments and such. Because you don't have much experience with mathematical writing, I usually do some editing---I'll correct grammar errors and usage errors as I see them... after all, mathematical grammar is slightly different from English grammar. So, when you're revising your paper, don't forget to read it over a few times to catch flaws that I didn't mark. And just as should happen with any other paper, the exposition should improve somewhat between the rough draft and the final copy.