Math 2.0

Combinatorics Designs and configurations Problem

Euler's thirty-six officers problem

Builds on Mutually orthogonal Latin squares

Problem

Thirty-six officers come from six regiments, with one officer of each of six ranks in each regiment. Arrange them in a 6×66 \times 6 square with two conditions. Each row and each column contains exactly one officer of each regiment and exactly one officer of each rank.

Remarks

Writing the regiment and the rank of the officer in each cell gives two Latin squares of order 66. The arrangement asked for exists if and only if these two squares are orthogonal. So the problem asks for a pair of orthogonal Latin squares of order 66. Euler posed the problem in 1782 and believed it had no solution. Gaston Tarry proved in 1901, by a case analysis, that there is no solution. Euler also conjectured that no orthogonal pair exists for any order n≡2(mod4)n \equiv 2 \pmod 4. Bose, Shrikhande and Parker disproved this around 1959 to 1960: an orthogonal pair of order nn exists for every nn except 22 and 66.

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 levelElementdefinitionMatrix, definition, no evidence levelMatrixdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered 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 setdefinitionLatin square, definition, no evidence levelLatin squaredefinitionMutually orthogonal Latin squares, definition, no evidence levelMutually orthogonal L…definitionEuler's thirty-six officers problem, problem, no evidence levelEuler's thirty-six of…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. Matrix definition
  3. Proposition concept
  4. Set definition
  5. Implication definition
  6. Predicate 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. Latin square definition
  22. Mutually orthogonal Latin squares definition
  23. Euler's thirty-six officers problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Mutually_orthogonal_Latin_squares
About this page
TagWXJAVQKFR6
URIm2://WXJAVQKFR6
Kindproblem
WritingW0 stub
MSC05B15
StatusThis copy of the site has no build data. Run m2 status WXJAVQKFR6.
AboutMutually orthogonal Latin squares
UsesMutually orthogonal Latin squares
Changed2 November 2026

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