Math 2.0

Combinatorics Designs and configurations Problem

Kirkman's schoolgirl problem

Builds on Steiner system

Problem

Fifteen schoolgirls go for a walk on each of seven days, in five rows of three. Arrange the rows for the seven days so that no two girls walk in the same row on more than one day.

Remarks

Each girl has 1414 companions and meets 22 of them per day, so in a solution every two girls walk together exactly once. So a solution is a Steiner triple system S(2,3,15)S(2, 3, 15) whose 3535 triples split into 77 days of 55 disjoint triples each. A Steiner triple system with such a split into parallel classes is called a Kirkman triple system. The problem has solutions; up to renaming the girls, there are seven different ones. Thomas Kirkman posed the problem in 1850. The general question, for vv girls with v≡3(mod6)v \equiv 3 \pmod 6, was solved by Ray-Chaudhuri and Wilson in 1971: a Kirkman triple system of order vv exists for every such vv.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionFinite set, definition, no evidence levelFinite setdefinitionBlock design, definition, no evidence levelBlock designdefinitionSteiner system, definition, no evidence levelSteiner systemdefinitionKirkman's schoolgirl problem, problem, no evidence levelKirkman's schoolgirl …problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Family of sets definition
  7. Ordered pair definition
  8. Existential quantifier definition
  9. Universal quantifier definition
  10. Cartesian product definition
  11. Subset definition
  12. Uniqueness quantifier definition
  13. Binary relation definition
  14. Function definition
  15. Codomain definition
  16. Injective function definition
  17. Surjective function definition
  18. Bijective function definition
  19. Equinumerous sets definition
  20. Finite set definition
  21. Block design definition
  22. Steiner system definition
  23. Kirkman's schoolgirl problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Kirkman%27s_schoolgirl_problem
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Changed2 November 2026

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