Exploring Discrete Mathematics Summary

Monday - Friday, July 7 - 11, 2008

Day 1 - Monday, 07 July 2008
We started by generating a table to represent the sample space for finding the sum when tossing two normal 6-sided dice and then organizing the results by using a frequency table. We were then challenged to find another set of two 6-sided dice that would produce the same results as the sum and frequencies when tossing the "normal" dice - under the conditions that each face of a die must be a positive integer. After much discussion and generation of ideas, we were able to find at least one other pair of 6-sided dice that would accomplish the desired outcome.

Day 2 - Tuesday, 08 July 2008
Our leader, Brian, introduced the idea of generating functions to the group and led a discussion on their relevance to finding the solution to the problem that was set on Day 1. We used a generating function and polynomial algebra (expanding, factoring, and evaluating) to produce the same result that we had obtained on Day 1. Further, we were able to use this same approach to prove the uniqueness of our solution. We then considered some follow-up questions such as using dice with different number of sides and using more than two dice. We ended today's session with a neat application of generating functions to the Fibonacci sequence. Our homework was to read through previous years' Discrete Math Group projects as well as some articles on alternate dice.

Day 4 - Thursday, July 10, 2008
We solved the problem of coming up with an alternate pair of dice, using tetrahedral dice, by using its associated generating function. We then investigated further methods for obtaining the same results as tossing tetrahedral dice (e.g. using a coin and an octahedral die). It was left as an exercise to show that a particular 2-sided die and 18-sided die produce an equivalent sample space to the normal two 6-sided dice. Further follow-up questions were posed. We then discussed cyclotomic polynomials as possible math topic that we can include in our project. The session ended with some exposure to Catalan numbers found in three examples: polygon dissections, sub-diagonal restricted lattice paths, and rooted binary trees.

Day 5 - Friday, July 11, 2008
Brian presented a beautiful tie from Catalan numbers to binomial coefficients. We then started working in two different groups.
Bree, Allen, Andrew, and Ted started looking at the combinatronics project from 2005 on Catalan numbers. They delegated different parts of the project to each other and spent time revising it by addressing editor notes provided from the refereed process.
Vicki, Brian, and Armando started developing the combinatronics project on alternate dice using generating functions from this year's topic.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.