In a boarding school there are fifteen schoolgirls who always take their daily walks in rows of threes. How can it be arranged so that each schoolgirl walks in the same row with every other schoolgirl exactly once a week? Solution of this problem is equivalent to constructing a Kirkman Triple System of order . The following table gives one of the 7 distinct (up to permutations of letters) solutions to the problem.

Sun | Mon | Tue | Wed | Thu | Fri | Sat |

ABC | ADE | AFG | AHI | AJK | ALM | ANO |

DHL | BIK | BHJ | BEG | BDF | BEF | BDG |

EJN | CMO | CLN | CMN | CLO | CIJ | CHK |

FIO | FHN | DIM | DJO | EHM | DKN | EIL |

GKM | GJL | EKO | FKL | GIN | GHO | FJM |

(The table of Dörrie 1965 contains a misprint in which the and entries for Wednesday and Thursday are written simply as .)

**References**

Abel, R. J. R. and Furino, S. C. ``Kirkman Triple Systems.''
§I.6.3 in *The CRC Handbook of Combinatorial Designs*
(Ed. C. J. Colbourn and J. H. Dinitz). Boca Raton, FL: CRC Press, pp. 88-89, 1996.

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations and Essays, 13th ed.* New York: Dover, pp. 287-289, 1987.

Dörrie, H. §5 in *100 Great Problems of Elementary Mathematics: Their History and Solutions.*
New York: Dover, pp. 14-18, 1965.

Frost, A. ``General Solution and Extension of the Problem of the 15 School Girls.'' *Quart. J. Pure Appl. Math.* **11**, 26-37, 1871.

Kirkman, T. P. ``On a Problem in Combinatorics.'' *Cambridge and Dublin Math. J.* **2**, 191-204, 1847.

Kirkman, T. P. *Lady's and Gentleman's Diary*. 1850.

Kraitchik, M. §9.3.1 in *Mathematical Recreations.* New York: W. W. Norton, pp. 226-227, 1942.

Peirce, B. ``Cyclic Solutions of the School-Girl Puzzle.'' *Astron. J.* **6**, 169-174, 1859-1861.

Ryser, H. J. *Combinatorial Mathematics.* Buffalo, NY: Math. Assoc. Amer., pp. 101-102, 1963.

© 1996-9

1999-05-26