CH. x] kirkman's sohool-girls problem 211 



d> e >/> g, h,j, and 42 points denoted by 1, 2, ... 42, placed at 

 equidistant intervals on the circumference of a circle. Then if 

 the arrangement on the first day is (a. 5. 6), (6. 26, 27), (c. 3. 10), 

 (d. 24. 31), (e. 19. 34), (/. 40. 13), (g. 39. 16), (h. 18. 37), (j. 21. 42), 

 (9.11.22), (30.32.1), (35.41.2), (14.20.23), (17.29.12), 

 (38. 8. 33), (7. 15. 25), (28. 36. 4), a two-step cyclical per- 

 mutation of the numbers gives arrangements for 21 days. 

 Next, arrange the 9 foci in triplets by any of the methods 

 already given so as to obtain arrangements for 4 days. From 

 the numbers 1 to 42 we can obtain four typical triplets 

 not already used, namely (1. 5. 21), (2. 6. 22), (3. 7. 23), 

 (14. 28. 42). From each of these triplets we can, by a three- 

 step cyclical permutation, obtain an arrangement of the 42 girls 

 for one day, thus getting arrangements for 4 days in all. 

 Combining these results of letters and numbers we obtain 

 arrangements for the 4 days. Thus an arrangement for the 

 first day would be (a. c.j), (b. d. g), (e.f. h), (1. 5. 21), (4. 8. 24), 

 (7. 11. 27), (10. 14. 30), (13. 17. 33), (16. 20. 36), (19. 23. 39), 

 (22. 26. 42), (25. 29. 3), (28. 32. 6), (31. 35. 9), (34. 38. 12), 

 (37. 41. 15), (40. 2. 18). For the second day the corresponding 

 arrangement would be (d. f. c), (e. g. a), (h. j. b), (2. 6. 22), 

 (5. 9. 25), &c. 



Analytical Methods. The methods described above, under 

 the headings One-Step, Two-Step and Three-Step Cycles, involve 

 some empirical work. It is true that with a little practice 

 it is not difficult to obtain solutions by them when n is a low 

 number, but the higher the value of n the more troublesome 

 is the process and the more uncertain its success. A general 

 arithmetical process has, however, been given by which it is 

 claimed that some solutions for any value of n can be always 

 obtained. Most of the solutions given earlier in this chapter 

 can be obtained in this way. 



The essential feature of the method is the arrangement of 

 the numbers by which the girls are represented in an order 

 such that definite rules can be laid down for grouping them in 

 pairs and triplets so that the differences of the numbers in each 

 pair or triplet either are all different or are repeated as often as 



14—2 



