194 kirkman's school-giels problem [ch. X 



The question was propounded in 1850, and in the same year 

 solutions were given for the cases when n = 9, 15, and 27 ; but 

 the methods used were largely empirical. 



The first writer to subject it to mathematical analysis was 

 R. R. Anstice who, in 1852 and 1853, described a method for 

 solving all cases of the form 12m + 3 when 6m +1 is prime. 

 He gave solutions for the cases when re =15, 27, 39. Sub- 

 stantially, his process, in a somewhat simplified form, is covered 

 by that given below under the heading Analytical Methods. 



The next important advance in the theory was due to 

 B. Peirce who, in 1860, gave cyclical methods for solving all 

 cases of the form 12m + 3 and 24m +9. But the processes 

 used were complicated and partly empirical. 



In 1871 A. H. Frost published a simple method applicable 

 to the original problem when n = 15 and to all cases when n is 

 of the form 2 2m — 1. lb has been applied to find solutions 

 when n = 15 and n = 63. 



In 1883 A. Bray (a name assumed by G. D. L. Harison) and 

 E. Marsden gave three-step cyclical solutions for 21 girls. These 

 were interesting because Kirkman had expressed the opinion 

 that this case was insoluble. 



Another solution when n = 21, by T. H. Gill, was given in 

 the fourth edition of this book in 1905. His method though 

 empirical appears to be applicable to all cases, but for high 

 values of m it involves so much preliminary work by trial and 

 error as to be of little value. 



A question on the subject which I propounded in the 

 Educational Times in 1906, attracted the attention of L. A. 

 Legros, H. E. Dudeney and 0. Eckenstein, and I received from 

 them a series of interesting and novel solutions. As illustra- 

 tions of the processes used, Dudeney published new solutions 

 for w=27, 33, 51, 57, 69, 75, 87, 93, 111 ; and Eckenstein for 

 w=27, 33, 39, 45, 51, 57, 69, 75, 93, 99, 111, 123, 135. 



I now proceed to describe some of the methods applicable 

 to the problem. We can use cycles and combinations of them. 

 I confine my discussion to processes where the steps of the 

 cycles do not exceed three symbols at a time. It will be con- 

 venient to begin with the easier methods, where however a 



