Below I am giving you a handful of problems. You don't need to do all of them,
but you must do the first problem. Please explain your answers and include your
explanation as comments in your file.

Question #1:
Let st be your student id number. Enter the formula on
the blackboard. The answer to the question determines which two other questions
you should answer.

Question #2:
Pick a small value $0 \leq \epsilon \leq 0.01$ and a large value of
$d>60$ (make it so you can easily change these values).
Let $z := (1-\epsilon) e^{2 \pi i/d}$. Look at the powers of $z$ and,
using the command pointplot, plot
the points [x, y] where $x$ and $y$ are the real and imaginary
parts of $z^r$ for $0 \leq r \leq 5d$. Describe in words what you observe,
in particular, give a description of how the list of points change
through this sequence.
Explain what happens when you change
the values of $\epsilon$ and $d$.

Question #3:
Pick a small value $0 \leq \epsilon \leq 0.1$
(make it so you can easily change this value).
Let $z := 1 + \epsilon i$. Look at the powers of $z$ and,
using the command pointplot, plot
the points [x, y] where $x$ and $y$ are the real and imaginary
parts of $z^r$ for $0 \leq r \leq 100$. Describe in words what you observe,
in particular, give a description of how the list of points change
through this sequence.
Explain what happens when you change
the values of $\epsilon$ and why.

Question #4:
Pick a small value $0 \leq \epsilon \leq 0.1$
(make it so you can easily change this value).
Let $z := (1+\epsilon) i$. Look at the powers of $z$ and,
using the command pointplot, plot
the points [x, y] where $x$ and $y$ are the real and imaginary
parts of $z^r$ for $0 \leq r \leq 100$. Describe in words what you observe,
in particular, give a description of how the list of points change
through this sequence. Explain what happens when you change
the values of $\epsilon$ and why.

Question #5:
Pick a small value $0 \leq \epsilon \leq 0.1$
(make it so you can easily change this value).
Let $z := (1+\epsilon)/\sqrt{2} + i(1+\epsilon)/\sqrt{2}$.
Look at the powers of $z$ and,
using the command pointplot, plot
the points [x, y] where $x$ and $y$ are the real and imaginary
parts of $z^r$ for $0 \leq r \leq 100$. Describe in words what you observe,
in particular, give a description of how the list of points change
through this sequence. Explain what happens when you change
the values of $\epsilon$ and why.

Question #6:
Find the number of solutions to the equation
\[ x_1 + x_2 + x_3 + x_4 + x_5 = n \]
where the $x_i$ are non-negative integers and
$x_1 \leq x_2 \leq x_3 \leq x_4 \leq x_5$.
Find the answer for $0 \leq n \leq 10$. Use this
data to look up your answer in the Online Encyclopedia
of Integer Sequences. Explain how you know your answer is
right.

Question #7:
Find the number of solutions to the equation
\[ x_1 + x_2 + x_3 + x_4 + x_5 = n \]
where the $x_i$ are non-negative integers
and of the variables $x_1$, $x_2$ and $x_3$, exactly two of them
are odd.
Find the answer for $0 \leq n \leq 10$. Use this
data to look up your answer in the Online Encyclopedia
of Integer Sequences. Explain how you know your answer is
right.

Question #8:
Plot the solutions to the equations
\[ y^3~~(mod~n) = x^2 + x -2~~(mod~n) \]
for $0 \leq x,y < n$ and for $n = 88,89,90,91$ (each $n$ should
be on a separate graph).
What do you observe that is different about the graphs where
$n$ is prime and $n$ is not prime.

NOTE THAT THE PENALTY FOR HANDING THIS IS LATE IS QUITE STRICT. I WANT YOU TO
DO THIS IN THE ALOTTED TIME.
You should be able to complete this quiz within the class time.
If you finish after the class time your overall grade for this assignment
will be reduced by 20% per hour (or part thereof). Make sure that your file is uploaded
by 12:30pm.
Upload your worksheet to the course
moodle.

You are expected to work alone on this assignment. Any indication of academic dishonesty
will result in a $0$ for the assignment and possible higher penalties.