Revised report due Thursday, October 16.

Your **main goal** is to determine **when**
the sequence converges (that is, **for which values** of
*a, b*, and *x_0* does it converge), and, when it does
converge, to **which limit**? These questions are
closely related, and working on either one will help you with the other.

The first part of this goal (when does it converge?) involves addressing Question 5 (you may want to, but do not have to, organize your answer along the lines suggested by Question 6). The second part of this goal (about finding the limit) involves addressing Questions 7 and 8.

Revised report due Thursday, October 23.

Your **main goal** is to answer the question "Which
properties of the parameters *p, q, r, s* determine the symmetry
of the parametric curves

Revised report due Thursday, November 13.

Your **main goal** is to investigate the following:

- The
**number of steps**the Euclidean algorithm takes to compute the GCD of two Fibonacci numbers; and - the
**GCD itself**of two Fibonacci numbers.

This is largely a restatement of Question 6, which provides a good way
of attacking the problem: sorting the Fibonacci numbers by how far
apart they are in the Fibonacci sequence, starting with those that are
close to each other in the sequence. In particular, it would be more
valuable to **prove** some of the results you get by
answering Question 6 than to completely answer
**without** proof the main goal listed above.

Also compare the answers you get to the corresponding answers for non-Fibonacci numbers in Questions 2 and 3.

Revised report due Thursday, November 20.

Your **main goal** is to determine **when**
the sequence has:

- attracting fixed points;
- repelling fixed points; and
- 2-cycles.

In other words, for **which values of** *a* do you
get attracting fixed points, repelling fixed points, and 2-cycles, and
what are the values of these fixed points and 2-cycles?

This goal can be approached by answering Questions 1, 2, 3, and 7.
Question 4 may also be helpful in finding values of *a* to
exclude.

Question 6 is also very interesting, but is more of a lead-in to the rest of the lab, which you might consider doing for your optional 7th lab, if you like the idea of chaos, or want to learn more about it.

Revised report due Friday, December 12 (last day of finals).

Your **main goal** is to find the chromatic polynomial
of the following two kinds of graphs:

- The path on
*n*vertices; and - The cycle on
*n*vertices.

This is Questions 7 and 9 (and just a very few parts of Question 5 as warmups). Exercises 4-10 are also good practice with some of the tools you'll need, though these exercises do not directly answer our main questions.

Revised report Friday, December 12 (last day of finals).

Your **main goal** is to answer the question: "What
should the survey-taker do with the results?" In other words, what is
your estimate of the proportion of True Yesses as a function of the
proportion of reported yesses? Answer this in the most general
setting, where the probabilities of answering the real question (dime
lands heads) and the answer to the decoy question being yes (penny
lands heads) are variables.

This is some mixture of Questions 1-3, 8, and 11.

No official revisions, but I encourage you to consult with me as you write your report.

Talk to me in advance so that we can set up a reasonable main goal for you to pursue.

Also note that, **instead** of an optional 7th lab, you
may turn in a re-revision of any of the first six reports, or you may
turn in nothing at all. (See syllabus for how all this
affects your grades.)