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LXXII. On the Puzzle of the Fifteen Young Ladies. BytheUev. 

 Thomas P. Kirkman, M.A., Rector of Croft with Southworth, 

 Lancashire. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



WHILE I fully appreciate the analytic value of Mr. 

 Spottiswoode's observations on my problem of the 

 fifteen young ladies in your May Number, I shall hope for his 

 pardon if I say, that, so far as I can discover from what he has 

 written, his solution, like my own and all that I have yet heard 

 of, is accomplished simply by the rule of thumb. When he has 

 supplied the demonstration, that from his seven groups, each of 

 eight terms, not one must, but one can, be selected " in such a 

 manner that no combinations recur," I will confess that all the 

 tentative process is avoided. 



I do not believe, although I am far from denying, that 63 

 young ladies can be handled day after day like the 15. That 

 this can be done with 5 x3 m+1 young ladies, I have proved at 

 p. 259, vol. v. of the Cambridge and Dublin Mathematical 

 Journal. 



The following arguments in support of the opinion that the 

 problem cannot be generalized for the case of Sn young ladies, 

 n being a prime number greater than 5, may be deserving of 

 attention, although I do not offer them as a demonstration of 

 the negative. 



Let it be required to march out day by day in threes, until 

 every pair have walked together, all the 3n ladies, 



«, a 2 a 3 

 b l b 2 b 3 



C-\ Ccy Cq 



N» N 2 N 3 , 



consisting of three sisters a, three sisters b, three sisters c, &c, 

 n being a prime number. 



As the data are symmetrical in a, b, c . . . N, and there is 

 nothing in the restriction, that each pair shall walk once and 

 once only together, which is unsymmetrical, and as the whole 

 column is to walk out every day, it is to be expected that the 

 sum of the columns will be also symmetrical in these n letters. 

 The families will exhibit no special preferences or dislikes towards 

 each other, when we consider the letters apart from the sub- 



