# Variations of the Counterfeit Coin Problem

Summer 2010

Student: Kayla Pope
Adviser: John Frohliger

The original counterfeit coin problem involves using a balance scale to determine a single counterfeit from among a collection of coins. This faculty/student team generalized this problem to the case in which the number ofcounterfeit coins is unknown. They expressed possible solutions and weighings as vectors and used dot products to characterize the results of weighings. A weighing scheme is a tree diagram in which the nodes correspond to weighing vectors and the edges leaving a node are determined by the results of that weighing (and the preceding weighings).

Their most unexpected result involved the size of a minimal weighing tree. This team speculated that, with the right choice of vectors for the nodes, a successful tree could be generated as long as the number of leaves exceeded the number of possible solution vectors. Surprisingly, Kayla Pope and John Frohliger were able to prove that this is not always the case.

Kayla Pope, a native from Oshkosh, Wis., began this project the summer after her first year attending St. Norbert. She is majored in mathematics while minoring in computer science and plans to finish her degree requirements in three years. Kayla is a member of the SNC Women's Soccer team, and in Fall 2010, she was among the team's leading scorers.

Balanced Sequences and Egyptian Fractions

Student: Haoqi Chen
Adviser: Professor Teena Carroll

Balanced sequences, sequences whose first k terms have the same sum and product, were built in this research project. The first balanced sequence found was related to the well known Sylvester sequence, whose reciprocalssum to one.

Investigating this connection leads to the ancient idea of an Egyptian fraction representation of a rational number a/b. Enumeration problems were a primary focus of this work, including finding how many Egyptian fraction representations of one there are which use only even denominators.

Using the denominators present in these representations, this student/faculty team found a connection to the so called Pythagorean spiral sequence, which encodes an infinite number of Pythagorean triples. Through this exploration they discovered a way of generating sequences of this type, which partitions the integers into equivalence classes of Pythagorean spiral sequences using the recursion relationship which defines Sylvester's sequence.

Haoqi Chen attended St. Norbert as a mathematics major with interests in physics and computer science. He was an international student from China who graduated in 2012 and had goals to then pursue a graduate degree, possibly in Industrial and Systems Engineering.

This research began as a multiplication error with interesting consequences in Dr. Poss's Calculus II class, which Haoqi enrolled in Spring 2009, his first semester at St. Norbert. He continued to think about the problem for all of winter break and the collaborative research with Dr. Carroll began while Haoqi was a student in her Advanced Foundation of Math course that following spring. Haoqi has given two different talks on the results.