the kirkman schoolgirls problem Friday Afternoon Mathematics Undergraduate Seminar Dana C. Ernst Northern Arizona University November 13, 2015

the kirkman schoolgirls problem The Problem In 1850, the Reverend Thomas Kirkman, posed an innocent-looking puzzle in the Lady’s and Gentleman’s Diary, a recreational mathematics journal: Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast. Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast. Here “abreast” means “in a group,” so the girls are walking out in groups of 3, and each pair of girls should only be in the same group once. It turns out that this problem is harder than it looks. Is it even possible? Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast. 1

the kirkman schoolgirls problem Let’s try to see what happens if we try to move beyond 7 days. Simpler Problem 1 15 young ladies in a school walk out in groups of 3 for 8 days in succession. Can you arrange the girls in walking groups so that no pair of girls ever walks in the same group of three more than once? What’s special about 7 days with 15 schoolgirls? 2

the kirkman schoolgirls problem Let’s try to reduce the number of schoolgirls. Simpler Problem 2 6 young ladies in a school walk out in groups of 3 for 2 days in succession. Can you arrange the girls in walking groups so that no pair of girls ever walks in the same group of three more than once? Any observations? 3

the kirkman schoolgirls problem OK, let’s try an odd multiple of 3 that is smaller than 15. Simpler Problem 3 9 young ladies in a school walk out in groups of 3 for 4 days in succession. Can you arrange the girls in walking groups so that no pair of girls ever walks in the same group of three more than once? Why did I pick 4 days? [applet] 4

the kirkman schoolgirls problem Let’s return to Kirkman’s original problem. Do you think it has a solution? Solution to Kirkman’s Schoolgirl Problem It turns out that there are 7 distinct solutions (up to relabeling of the schoolgirls) of the problem. Here is one possible solution. Sun Mon Tues Wed Thurs Fri Sat 1, 6, 11 1, 2, 5 2, 3, 6 5, 6, 9 3, 5, 11 5, 7, 13 11, 13, 4 2, 7, 12 3, 4, 7 4, 5, 8 7, 8, 11 4, 6, 12 6, 8, 14 12, 14, 5 3, 8, 13 8, 9, 12 9, 10, 13 12, 13, 1 7, 9, 15 9, 11, 2 15, 2, 8 4, 9, 14 10, 11, 14 11, 12, 15 14, 15, 3 8, 10, 1 10, 12, 3 1, 3, 9 5, 10, 15 13, 15, 6 14, 1, 7 2, 4, 10 13, 14, 2 15, 1, 4 6, 7, 10 5

the kirkman schoolgirls problem Combinatorial Design Theory Kirkman’s puzzle is a prototype for a more general problem: If you have n schoolgirls, can you create groups of size k such that each smaller set of size t appears in just one of the larger groups? If you have n schoolgirls, can you create groups of size k such that each smaller set of size t appears in just one of the larger groups? Such an arrangement is said to be of type S(t, k, n), which is called a Steiner system (or combinatorial design). For example, solutions to the original Kirkman problem are of type S(2, 3, 15). If you have n schoolgirls, can you create groups of size k such that each smaller set of size t appears in just one of the larger groups? 6

the kirkman schoolgirls problem Combinatorial Design Theory (continued) Applications of CDT are plentiful: Finite geometry, tournament scheduling, lotteries, mathematical biology, algorithm design and analysis, networking, group testing, cryptography, etc. One of the fundamental problems in CDT is determining whether a given S(t, k, n) exists and if one exists, how many are there? 7

the kirkman schoolgirls problem Combinatorial Design Theory (continued) For example, is S(2, 3, 7) possible? The answer is yes. Here is the Fano plane, which is the finite projective plane of order 2. 8

the kirkman schoolgirls problem Combinatorial Design Theory (continued) Many combinations of t, k, and n are quickly ruled out by divisibility obstacles. For example, we already discovered that S(2, 3, 6) is not possible. For combinations that aren’t immediately tossed out, there’s no easy way to discover whether a given combination is possible. For example, it turns out that S(2, 7, 43) is impossible, but it is for complicated reasons. However: In January 2014, Peter Keevash (Oxford) established that, apart from a few exceptions, S(t, k, n) will always exist if a few divisibility requirements are satisfied. This is a big deal in the world of CDT! 9

the kirkman schoolgirls problem References ∙ There is a great write up of Kirkman’s original problem, its connection to CDT, and a brief summary of Keevash’s new result in Quanta Magazine. ∙ There is a nice applet for playing with the nine schoolgirl problem in Quanta Magazine. ∙ The Wikipedia article on The Kirkman Schoolgirl Problem has a lot of cool information. ∙ Also, check out the Wikipedia articles on Steiner systems, Combinatorial Design Theory, and the Fano plane. ∙ You can find Peter Keevash’s article about the existence of Steiner systems over on the arXiv. 10

the kirkman schoolgirls problem Here’s a variation on the problem involving 9 schoolgirls. The Nine Prisoners Problem 9 prisoners are taken outdoors for exercise in rows of 3 such that each adjacent pair of prisoners is linked by handcuffs for 6 days in succession. Can the prisoners be arranged over the course of the 6 days so that each pair of prisoners shares handcuffs exactly once? 11

the kirkman schoolgirls problem Here’s another variation on Kirkman’s problem. The Social Golfer Problem 20 golfers wish to play in groups of 4 for 5 days. Is it possible for each golfer to play no more than once with any other golfer? 12