1U3 



CHAPTER X. 

 kirkman's school-girls problem. 



The Fifteen School-Girls Problem — first enunciated by 

 T. P. Kirkman, and commonly known as Kirkmaris Problem 

 — consists in arranging fifteen things in different sets of 

 triplets. It is usually presented in the form that a school- 

 mistress was in the habit of taking her girls for a daily walk. 

 The girls were fifteen in number, and were arranged in five 

 rows of three each so that each girl might have two companions. 

 The problem is to dispose them so that for seven consecutive 

 days no girl will walk with any of her school-fellows in any 

 triplet more than once. 



In the general problem, here discussed, we require to 

 arrange n girls, where n is an odd multiple of 3, in triplets to 

 walk out for y days, where y = (n — l)/2, so that no girl will 

 walk with any of her school-fellows in any triplet more than once. 



The theory of the formation of all such possible triplets in 

 the case of nine girls is comparatively easy, but the general 

 theory involves considerable difficulties. Before describing any 

 methods of solution, I will give briefly the leading facts in the 

 history of the problem. For this and much of the material of 

 this chapter I am indebted to 0. Eckenstein. Detailed refer- 

 ences to the authorities mentioned are given in the bibliography 

 mentioned in the footnote * 



* The problem was first published in the Lady's and Gentleman's Diary for 

 1850, p. 48, and has been the subject of numerous memoirs. A bibliography 

 of the problem by O. Eckenstein appeared in the Messenger of Mathematics, 

 Cambridge, July, 1911, vol. xli, pp. 33—36. 



B. R. 13 



