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A thousand million leagues

  • Tony Barnard (a1)


Some time ago a friend, who was organising a chess league, rang me up and asked “How many different ways are there of arranging a roundrobin league tournament?” Thinking of his phone bill I told him I’d ring him back. Which was just as well, because the problem was much more difficult dian I had realised. Indeed it is unsolved in general. In the language of graph theory it is the problem of finding the number of onefactorisations of a complete graph. These and related objects have been the focus of considerable study both in combinatorics and recreational mathematics and, although the enumeration question remains unanswered in general, it is, even for small numbers of players, remarkably full of interest.



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10. Wallis, W.D., Street, A.P., and Wallis, J.S., Combinatorics: Room squares, Sum-free sets, Hadamard matrices, Lecture Notes in Mathematics 292, Springer, Berlin (1972). Chapter VIII: Room squares of side 7.
11. Wallis, W.D., “One-factorizations of graphs: tournament applications”, College Math. Journal 18 (2) (1987) 116123.
12. Wallis, W.D., “A tournament problem”, J. Austral. Math. Soc. Ser. B 24 (1983) 289291.

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A thousand million leagues

  • Tony Barnard (a1)


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