The Kirkman Schoolgirls 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.”

Here “abreast” means “in a group,” so the girls are walking out in groups of three, 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? We will begin by tinkering with a simpler problem and then spend some time playing with Kirkman's original problem. Time permitting, we will also discuss generalizations of the problem that form the backbone of a branch of mathematics called combinatorial design theory.

This talk was given at the Northern Arizona University Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) on Friday, November 13, 2015.

Transcript

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

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2. 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
   

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3. 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
   

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4. 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
   

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5. 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
   

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6. 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
   

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7. 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
   

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8. 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?
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9. 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.
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10. 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
    

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11. 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.
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12. 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
    

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13. 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
    

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