Revised report due Wednesday, March 7.

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 Wednesday, March 21.

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 Wednesday, April 11.

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.

Final note: Some of your answers could depend on how you index the Fibonacci sequence. The standard numbering is that F_1=1, F_2=1, but the notebook we are using defines F_1=1, F_2=2. You may use either one, but some things may be easier to state and/or prove using one indexing or the other.

Revised report due Wednesday, April 18.

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, May 11.

There are **three main goals** of this lab:

- Consider the well-known
*Euler relation*on vertices, edges, and faces of "most" polyhedra: E = V + F - 2. Show how constructions, such as stellation, affect this formula. In other words, show the inductive step of an inductive proof of Euler's formula. In other words, show that if you start with a polyhedron that satisfies Euler's relation, then stellate it (or do other similar operations), the resulting new polyhedron will still satisfy Euler's relation.[Caveats: Euler's formula is only true for polyhedra that are "spherical", that is, those that can be thought of as a sphere with all the roundness smoothed out into flat spots corresponding to the faces of the polyhedron. For instance, it is not true if you make a polyhedron that resembles a donut! So, you cannot expect a "proof" to show it is always true. But your inductive step can still work, because you can't make a donut-polyhedron out of a sphere-polyhedron by stellating!]

- Find all regular polyhedra. Prove your list is complete.
- Find formulas (beyond Euler) for polyhedra with just triangles
and squares. In other words, find relations among the numbers of
vertices, edges, triangles, and squares.
To attack this problem, it is a good idea to first consider polyhedra with just triangles. Then try polyhedra with just squares. Then, finally, try mixtures of triangles and squares.

There are not so many formal questions in the text this time that are as directly related to the main goals. But the general discussion of polyhedra may be helpful to you.

Revised report Friday, May 11.

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.)