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



a quadrilateral (1. 2. 4. 8) ; and the two similar quadrilaterals 

 (6. 7. 9. 13) and (11. 12. 14. 3) obtained by five-step cyclical 

 permutations. These four quartets give an arrangement for 

 the first day. From them the solutions for the other days are 

 obtained by one-step cyclical permutations. 



When n = 5, we may arrange initially the 25 points in five 

 quintets, namely, k and four equidistant points, (k. 6. 12. 18. 24); 

 a pentagon (2. 3. 5. 13. 22); and the three similar pentagons 

 (8. 9. 11. 19. 4), (14. 15. 17. 1. 10), (20. 21. 23. 7. 16) obtained 

 by six-step cyclical permutations. These five quintets give 

 an arrangement for the first day. From them the solutions for 

 the other days are obtained by one-step cyclical permutations. 

 There is a second solution (k 6. 12. 18. 24), (1. 2. 15. 17. 22), &c. 



Hitherto the case when n = 6 has baffled all attempts to 

 find a solution. 



When n = 7, we may initially arrange the 49 points in 

 seven groups, namely (k. 8. 16. 24. 32. 40. 48), a group 

 (1. 2. 5. 11. 31. 36. 38) and five similar groups obtained 

 by successive additions of 8 to these numbers. There are 

 three other solutions: in these the second group is either 

 (2. 3. 17. 28. 38. 45. 47), or (3. 4. 6. 18. 23. 41. 45), or 

 (3. 4. 14. 17. 26. 45. 47). 



When n = 8 there are three solutions. If the first group 

 is (k. 9. 18. 27. 36. 45. 54. 63), the second group is either 

 (1. 2. 4. 8. 16. 21. 32. 42), or (2. 3. 16. 22. 24. 50. 55. 62), or 

 (3. 4. 7. 19. 24. 26. 32. 56) : the other groups in each solution 

 being obtained by successive addition of 9 to the numbers in 

 the second group. 



When n = 9, one solution is as follows. Divide the girls into 

 nine classes, each containing nine girls, denoted by ol, a2, . . . a9 ; 

 61, 62, ... 69 ; and similarly for the letters c, d, e,f, g, h, k. The 

 arrangement on the first day is made up of the a group, the 

 6 group, etc. The arrangement on the second day is of nine 

 groups as follows: first group, kl, a5, 63, c8, dl, e9, /4, gQ, h2; 

 second group, k2, a3, 66, c4, dl, e8, f9, go, h*7 ; third group, 

 kZ, a8, 64, c7, d5, el, fl, g9, A6 ; fourth group, M, o7, 61, c5, d8, 

 e6, /3, g% h9 ; fifth group, k5, a9, 68, c2, d6, el, fl, g4>, hS ; 



