.202 The Rev. T. P. Kirkman on the Puzzle of 



134 6i3 746 237 5s 



245 723 11735.1 66 



356 135 224 4ei 7? 



467 256 331 572 14 



571 367 452 643 2i0 



612 474 563 715 32 



723 54i 675 126 43 

 If we operate with the substitution 



6543217 1237456 _ 



1234567 lasher -*' 

 it becomes J = ^H^ = 



643 112 775 536 24 



532 723 6ie447l5 

 421 634 527 3si 7«0 



317 545 431 262 67 



216 456 342l73 5l0 



175 367 253 714 42 



764 27i I6462533O 



which has the set of triplets made with 1234567 which Mr. 

 Woolhouse employs. 



This J is by our definition the same solution as H, ; and it is 

 a derangement of the model group formed with the two circular 

 factors 1765432 and 1234567, or, what is the same thing, with 

 the circular factors 1234567 and 1765432. 



If we operate on J with the substitution 



1234567 2176543 

 1234567 4567123' 



it becomes the third of Mr. Woolhouse's forms. 



A rigorous mathematical discussion of the problem of arran- 

 ging 12?i + 3 young ladies, to walk out daily till every pair have 

 once, and once only walked abreast, 6n + 1 being a prime number, 

 such as to supply a direct method of finding and exhausting all 

 the different solutions, has not hitherto been given. 



The following method will be found rigorous and satisfactory. 

 Mr. Anstice showed, in his elegant memoir in vol. vii. of the 

 Cambridge and Dublin Journal, that n primary triplets A 1 B 1 C 1 , 

 A 2 B 2 C 2 . . A 4 B 4 C 4 can always be found such that, by continual 

 additions of unity to every element, all the duads possible with 

 6n + 1 capitals shall be once, and once only exhausted. Thus 

 for thirteen capitals, 1, 2, 3, 4, 5 . . . 12, 13, we may take the 

 triplets 1 . 2 • 5, 4 • 6 • 12, and the first day's arrangements of 27 



