218 kirkman's school-girls problem [ch. X 



total number of possible arrangements of the girls in triplets 

 for four days is (280)7(4!); hence the probability of obtaining 

 a solution by a chance arrangement is about 1 in 300,000. 



In the case of 15 girls the number of solutions is said to be 

 65 x (13!), but I do not vouch for the correctness of this result. 

 The number of fundamental solutions has not yet been definitely 

 ascertained, but it has been shown that it is not less than seven 

 and not more than eleven. The total number of ways in which 

 the girls can walk out for a week in triplets is (455) 7 ; so the 

 probability that any chance way satisfies the condition of the 

 problem is very small. 



Harisoris Theorem. If we know solutions for Kirkman's 

 Problem for 31 girls and for 3m girls we can find a solution 

 for 3lm girls. The particular case of this when 1 = 1 was 

 established by Walecki and given in the earlier editions of 

 this work. Harison's proof, given in 191 C, of the more general 

 theorem is as follows : — 



If the school-girls be denoted by the consecutive numbers 

 from 1 to 3lm and the numbers be divided into 31 sets, each of 

 m consecutive numbers, each of these sets can, by the method 

 for the 31 problem, be divided in (31 — 1)/2 collections of groups 

 of three sets, so that every set shall be included once in the 

 same group with every other set. 



In one of these collections, each group of three sets (in- 

 volving 3m numbers) can, by the method for the 3m problem, 

 be arranged in triplets for (3m — 1)/2 days, so as to have each 

 number in each of the sets composing the group included once 

 in the same triplet with every other number in the set to which 

 it belongs and with every number in the other two sets in the 

 group. This will give arrangements of all the numbers for that 

 number of days. 



In the remaining (31 — 3)/2 collections, each group can be 

 arranged in triplets for m days, so as to have each number in 

 each of the sets composing the group included once in the 

 same triplet with each number of the other two sets. In the 

 first arrangement in each collection, the first triplet in each 

 group is composed of the first number of each set. In the 

 second arrangement, the first triplet is composed of the first 



