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



The geometrical representation is sufficiently obvious. In 

 the methods used by Legros and Eckenstein, previously de- 

 scribed, the girls were represented by 2y equidistant points 

 on the circumference of a circle and a point at its centre. 

 It is evident that we may with equal propriety represent all 

 the girls by symbols placed at equidistant intervals round 

 the circumference of a circle : such solutions are termed non- 

 central. The symbols may be 1, 2, 3, ... n, or letters Oj, 6 1( c u 

 a*, b q , c 2 , .... Any triplet will be represented by a triangle 

 whose sides are chords of the circle. The arrangement on any 

 day is to include all the girls, and therefore the triangles re- 

 presenting the triplets on that day are w/3 in number, and as 

 each girl appears in only one triplet no two triangles can have 

 a common vertex. 



The complete three-step solution will require the determina- 

 tion of a system of (a — l)/2 inscribed triangles. In the first 

 part of the solution n/3 of these triangles must be selected to 

 form an arrangement for the first day, so that by rotating this 

 arrangement three steps at a time we obtain triplets for n/3 days 

 in all. In the second part of the solution we must assure ourselves 

 that the remaining {n — 3)/6 triangles are such that from each of 

 them, by a cyclical permutation of three steps at a time, an 

 arrangement for one of the remaining (n — 3)/6 days is obtainable. 



As before we begin by tabulating the possible differences 

 [1], [2], [3], ... [(n — l)/2], whose values denote the lengths of 

 the sides p, q, r of the possible triangles, also, we have either 

 p + q = r or p + q + r = n. From these values of p, q, r are 

 formed triads, and in these triads each difference must be 

 used three times and only three times. Triangles of these 

 types must be then formed and placed in the circle so that 

 the side denoting any assigned difference p must start once 

 from a number of the form 3m, once from a number of the 

 form 3m + 1, and once from a number of the form 3m + 2. 

 Also an isosceles triangle, one of whose sides is a multiple of 

 three, cannot be used : thus in any particular triad a 3, 6, 9, ... 

 cannot appear more than once. Save in some exceptional 

 cases of high values of n, every triangle, one of whose 



