Kirkman's Schoolgirls Wearing Hats and Walking through Fields of Numbers

Abstract

Imagine fifteen young ladies at the Emmy Noether Boarding School—Anita, Barb, Carol, Doris, Ellen, Fran, Gail, Helen, Ivy, Julia, Kali, Lori, Mary, Noel, and Olive. Every day, they walk to school in the Official ENBS Formation, namely, in five rows of three each. One of the ENBS rules is that during the walk, a student may only talk with the other students in her row of three. These fifteen are all good friends and like to talk with each other—and they are all mathematically inclined. One day Julia says, “I wonder if it’s possible for us to walk to school in the Official Formation in such a way that we all have a chance to talk with each other at least once a week?” “But that means nobody walks with anybody else in a line more than once a week,” observes Anita. “I’ll bet we can do that,” concludes Lori. “Let’s get to work.” And what they came up with is the schedule in TABLE 1.