CH. x] kirkman's school-girls problem 217 



(a9. al5. 612), (a8. a7. 617), (a2. al6. 6 9), (a6. alO. 68), 

 (al8. all. 65), (al4. al7. 66), (a4. al3. 618), (al2. ol. 616), 

 (63. 610. 61), (62. 613. 67), (614. 615. 611), (jfc. al9. 619). 



In the case of 39 girls we may also extend the method 

 used above by which for 27 girls we obtained the solution 

 (1. 25. 5), (14. 12. 18), .... "We thus get a solution for 39 girls 

 as follows : (1. 25. 18), (14. 38. 31), (27. 12. 5) ; (15. 16. 10), 

 (28. 29. 23), (2. 3. 36); (17. 19. 35), (30. 32. 9), (4. 6. 22); 

 (21. 24. 33), (34. 37. 7), (8. 11. 20); (13. 26. 39). From this 

 the arrangements for the first 13 days are obtained either by 

 a one-step or a three-step cyclical permutation of the numbers. 

 The single triplets from each of which an arrangement for one 

 of the other six days is obtainable are (1. 5. 15), (2. 6. 16), 

 (3. 7. 17); (1. 9. 20), (2. 10. 21), (3. 11. 22). From each of 

 these the arrangement for one day is obtainable by a three-step 

 cyclical permutation of the numbers. 



These examples of the use of the Focal and Analytical 

 Methods are given only by way of illustration, but they will 

 serve to suggest the applications to other cases. When the 

 number taken as base is composite, the formations of the series 

 used in the Analytical Method may be troublesome, but the 

 principle of the method is not affected, though want of space 

 forbids my going into further details. Eckenstein, to whom 

 the development of this method is mainly due, can, with the 

 aid of a table of primitive roots and sets of numbers written on 

 cards, within half an hour obtain a solution for any case in 

 which n is less than 500, and can within one hour obtain a 

 solution for any case in which n lies between 500 and 900. 



Number of Solutions. The problem of 9 girls has been 

 subjected to an exhaustive examination. The number of solu- 

 tions is 840, if an arrangement on Monday, Tuesday, Wednesday 

 and Thursday, and the same arrangement on (say) Monday, 

 Tuesday, Thursday and Wednesday are regarded as identical. 

 [If they are regarded as different the number of possible 

 independent solutions is 20,160.] Any of these 840 solutions 

 can however be deduced from any other of them by inter- 

 changes, and thus there is only one fundamental solution. The 



