Suggested Projects for Math 640: EXPERIMENTAL MATHEMATICS, Spring 2015

First posted: April 5, 2015

This version: May 6, 2015.

Because this semester there was so much homework, the projects are optional, and there is no due date. I'll be glad to post them, if you wish, whenever you finish it.

Feel free to propose your own project! As long as it uses Maple (or even Mathematica, or SAGE) in a non-trivial way it is fine. I prefer that it would involve Probability and/or Finance, but you are welcome to do it on anything mathematical.

NOTE: Any project can be done by n people, with 1 ≤ n < infinity

Untaken Projects

• Study at depth the Shepp Urn, following this beautiful article, with different criteria besided positive expected gain. Also generalize it for several kinds of balls (not just balls labeled -1 and 1). Implement everything in Maple

• Write a program that inputs an arbitrary backgammon position, with some exposed pieces, and computes the probability of at least one piece being captured by the person who is about to roll the pair of fair standard dice. (and more generally, for general loaded dice)

• More ambitiously, Write a program that tells you how to play Backgammon optimally, for a general board with b points and house-size h, and any arrangement of pieces. (For the usual one, b=24, h=6).

• Study probabilistic generalizations of the centepede game

• [This contains MANY possible projects] Use the brilliant method of this article to investigate the limiting distribution (if it exists) of your favorite sequence of combinatorial random variables. [Ask me for suggestions]

• Use the vocabularies in this directory to write a program that automatically designes crossword puzzles (with and without black squares)

• Use the vocabularies in this directory to study English as a Stochastic process.