214 



KIRKMANS SCHOOL-GIRLS PROBLEM 



[CH. X 



thus. If any term in the residue series is greater than 3m re- 

 place it by its complementary number y — e. In this way, from 

 the residue series, we get a derivative series dl, d2, d3, ... such 

 that any 3m consecutive terms comprise all the integers from 

 1 to 3m. The first half of this series may be divided into three 

 equal divisions thus: (1) dl, d4, dl, ... ; (2) d2, d5, d8, ...; 

 (3) d3, d6, d9, .... If the displacement is such that the first 

 numbers in the second, third, and fourth columns are contained 

 in different divisions, each difference must occur twice, and it 

 will give a possible solution. Other possible regular arrange- 

 ments give other solutions. Applying this to our case we have 

 -the residue series, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1. The 

 derivative series is 2, 4, 5, 3, 6, 1, 2, 4, 5, 3, 6, 1. The three 

 divisions are (i) 2, 3 ; (ii) 4, 6 ; (iii) 5, 1. The cyclical arrange- 

 ment we started with and the consequent differences are shown 

 in the left half of the accompanying table. A cyclical change 



as described above of the vertical lines of the symbols in the 

 first column gives the arrangement in the right half of the 

 table. Here each difference occurs twice and accordingly this 

 gives a possible arrangement of the triplets, namely, 2, 6, 5 ; 

 3, 9, 1 ; 11, 7, 8 ; 10, 4, 12. These are the suffixes of the '6's. 

 We have now to use six of these 'b's in connection with the 

 basic ' a 's already determined, keeping the other six ' b '& for 

 the remaining two triangles. 



For instance we may obtain a scalene solution by taking as 

 a suffix of the 'b' associated with any pair of 'a's, a number 

 equal to the sum of the suffixes of the ' a 's. We thus get as a 

 solution (o,2. a8. 610), (a4. a3. 67), (a6. all. 64), (al2. a9. 68), 

 (a5. a7. 612), (alO. ol. 611), (62. 66. 6 5), (63. 69. 61), 

 (Jc. al3. 613). Or we might take as a suffix of the '6 ' associated 

 with any pair of ' a's, the number midway between the suffixes 



