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Math is a vast, rich field of opportunities for anyone with the curiosity to explore it, says one student immersed in an EXCEL research project this summer.
“I had always wondered how you could know where to look initially to seek undiscovered knowledge,” says Kerem Ok ’03, an electrical and computer engineering major. “This program introduced me to the process.”
EXCEL Scholars assist faculty members with research while earning a stipend. Working with Chawne Kimber, assistant professor of mathematics, Ok is examining the modified Josephus Problem.
According to Kimber, the mathematical problem was posed in the writings of biblical scholar Flavius Josephus circa 80 A.D. The ancient combinatorial puzzle involves the generation of ordered lists of numbers, called permutations, by successively removing every k-th element from the ordered list (0, 1, 2 n).
Ok explains as follows: Imagine that an even number of people of two “types” (called here “A” and “B”) are sitting in a circle. There is an equal number of A’s and B’s. Find a number “k” such that when every k-th person in the circle is eliminated, all the A’s are removed first, leaving all the B’s. He says “k” values that are greater than the number of people in the circle are possible because one can continue going around and around it.
“The project involves coming up with an efficient recursive algorithm in the Java computer program to calculate the ‘k’ values for different seating configurations,” says Ok. “The software creates all permutations of the given input string, then eliminates them using repeated rotation and comparison.”
Ok says the computer program comes in very handy for this version of the Josephus Problem since as the counting continues, the solutions for “k” also grow in value, so finding a solution by hand becomes impractical. For example, he says, the solution for a circle containing AAABABBB turned out to be k = 162.
“The problem is open to further research in many aspects,” the student comments. “Seating arrangements containing an unequal number of A’s and B’s could be considered. A correlation between the k values for reversed seating arrangements could be looked into. For example, if the AAABABBB solution is 162, what would be the solution for BBBABAAA?”
Ok says Kimber “has been extremely supportive of everything I did during this project.” “She has a way of making any subject interesting and knows how to direct the student for the maximum enrichment the EXCEL program is meant to provide,” says Ok, who describes Kimber as “so approachable yet professional” and someone he views as a good friend. Likewise, the professor says that Ok is “a very bright, self-motivated student who makes insightful observations.”
A graduate of Bilken University Preparatory School in Ankara, Turkey, Ok is a member of Lafayette’s German Club, the International Students Association, and the Muslim Students Association. He is a writing associate for English as a Second Language and participates regularly in intramural sports.