Project 1: The dirty work problem
In real life, bigger is not always better. For example, if we are dividing unpleasant house chores, then we prefer a small portion. Can you formulate a
fair division problem where each player wants the smallest fair and try to solve it? What fairness criteria do you want to use?
Project 2: Amicable pairs
Two integers n and m are called an amicable pair if the sum of the proper divisors of n equals m and vice versa. For example, 220 and 284 form an amicable pair.
The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110 and
284 = 1+ 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110
Also, the proper divisors of 284 are 1, 2, 4, 71, and 142 and
220 = 1 + 2+ 4 + 71 + 142.
What sorts of properties are known about amicable pairs? How many amicable pairs are there? Are there existing connections to cryptography?
Your project should culminate in the creation or explanation of a cipher incorporating amicable numbers.
Project 3: Negotiating to settlement in divorce
Cake-cutting algorithms are not generally applicable to real-world situations involving indivisible goods, whose value is destroyed if they are divided.
People see this conflict arise very often in splitting up property in divorce. Can you formulate a fair division problem where players want to fairly divide
a set of property that involve indivisible goods?
Project 4: Creating a cipher
Suppose you and your best friend have your own alphabet that you use to exchange messages securely. Furthermore, suppose you have a long message to send to your friend
that is time sensitive! In this project, you will design your own alphabet and cipher, and determine how to use a computer program to help you translate between your
original message and the encrypted message written in your alphabet. Note that this is not only an artistic and computational task: you must design your cipher so that
it is difficult for unintended recipients to crack. How many layers of encryption can you incorporate? How strong is your cipher? In what ways is it vulnerable to attack?
Project 5: Exact Division
We have seen in the Cut and Choose Method that it's better to be the chooser than the cutter. Can you come up with a fairness criteria that balance the asymmetry?
Can you formulate a fair division problem that uses the criteria you proposed, and try to solve it?
Project 6: Mersenne numbers
A Mersenne number is a positive integer of the form Mn=2n-1. When Mn is prime, we call Mn a
Mersenne Prime. While we can certainly construct infinitely many
Mersenne numbers (how?), does this mean there are infinitely Mersenne primes? How might one go about finding Mersenne primes? What is GIMPS? Are there existing
connections to cryptography? Your project should culminate in the creation or explanation of a cipher which incorporates Mersenne primes.
Project 7: Online division
In traditional cake-cutting setting, all players are available at the time of the division. However, in real life, people may show up late or leave early.
Can you simulate a fair division problem where players arrive and depart during the process of dividing a resource?
Project 8: Fibonacci Numbers
The famous Fibonacci series is a series of positive integers in which each number (called a Fibonacci number) is determined by summing the two preceding numbers.
The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, … and so on. The Fibonacci numbers are ubiquitous in mathematics, and they are still studied in
modern times. For example, a theorem of Zeckendorf (1970s) states that every positive integer can be written uniquely as the sum of distinct, non-consecutive
Fibonacci numbers. For example, 10 = 2 + 5 and 17 = 13 + 3 + 1. What other properties (new and old) are known about Fibonacci numbers? Are there existing
connections to cryptography? Your project should culminate in the creation or explanation of a cipher incorporating the Fibonacci numbers.