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



semi-circumference we may confine ourselves to the relation 

 p+q = r. In the geometrical methods described below, we 

 usually first determine the dimensions of the triangles to be 

 used in the solution, and then find how they are to be arranged 

 in the circle. 



If (y — l)/3 scalene triangles, whose sides are p, q, r, can be 

 inscribed in the circle so that to each triaDgle corresponds an 

 equal complementary triangle having its equal sides parallel to 

 those of the first and with its vertices at free points, then the 

 system of 2 (y — l)/3 triangles with the corresponding diameter 

 will give an arrangement for one day. If the system be per- 

 muted cyclically y — 1 times we get arrangements for the other 

 y — 1 days. No two girls will walk together twice, for each 

 chord occupies a different position after each permutation, and 

 as all the chords forming the (y — 1)/3 triangles are unequal the 

 same combination cannot occur twice. Since the triangles are 

 placed in complementary pairs, one being y points in front of 

 the other, it follows that after y — 1 permutations we shall 

 come to a position like the initial one, and the cycle will be 

 completed. If the circle be drawn and the triangles cut out 

 to scale, the arrangement of the triangles is facilitated. The 

 method will be better understood if I apply it to one or two of 

 the simpler cases. 



The first case is that of three girls, a, b, c, walking out for 

 one day, that is, n = 3, m = 0, y = 1. This involves no discussion, 

 the solution being (a. b. c). 



The next case is that of nine girls walking out for four days, 

 that is, n = 9, m — 0, y = 4. The first triplet on the first day 

 is (1. 1c. 5). There are six other girls represented by the points 

 2, 3, 4, 6, 7, 8. These points can be joined so as to form 

 triangles, and each triangle will represent a triplet. We want 

 to find one such triangle, with unequal sides, with its vertices 

 at three of these points, and such that the triangle formed by 

 the other three points will have its sides equal and parallel to 

 the sides of the first triangle. 



The sides of a triangle are p, q, r. The only possible values 

 are 1, 2, 3, and they satisfy the condition p + q = r. If a 



