Revised report due Tuesday, October 3.

Your **main goal** is to address Question 5 (you may want
to, but do not have to, organize your answer along the lines suggested
by Question 6).

Also address **one** of the following three:

- Questions 7
**and**8;**or** - Question 9 (be sure to read "Visualizing Iteration" after Question 9);
**or** - Question 10 [note that "Question 10" on page 11 has a typographical error, and is printed as "Question 1: 0"].

Revised report due Thursday, October 12.

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

- The distribution of the
**number of steps**the Euclidean algorithm takes to compute the GCD of two numbers, as a function of the size of the numbers; and - the distribution of the
**GCD itself**of two numbers, as a function of the size of the numbers.

In particular, address Questions 2 and 3, **and**
either of the following:

- Questions 4
**and**5;**or** - Question 6.

Questions 4 and 5 guide you in further analyzing one interesting feature of the distribution of the GCD's, while Question 6 guides you in looking at how the Euclidean algorithm works on Fibonacci numbers.

Note that there is not much to say, theoretically, in Questions 2 and 3; here, you will simply need to analyze data very carefully. You can still make somewhat precise statements about what you can expect. Questions 4-6, however, guide you to proving some interesting things about the Euclidean algorithm.

Revised report due Thursday, November 2.

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 9.

Your **main goal** is to answer the question "For which m
does the set of non-zero squares (mod m) form a cyclic difference set
with (m-1)/2 elements?". Questions 1-3 help you discover this
experimentally. Theoretically, Questions 4-8 help you answer for
which m are there (m-1)/2 distinct non-zero squares, and Questions
9-10 help you answer for which of **those** do we get
cyclic difference sets.

Revised report Thursday, November 30.

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.

Revised report due Friday, December 8 (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.

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.)