the Fifteen Young Ladies. 199 



it has, so far as I know, been published in print except this guess 

 of Prof. Sylvester's in 1861. 



It may be worth while to show the connexion between this 

 tactical problem and the Theory of Groups, which has lately 

 acquired new interest by the competition of last year for the 

 prize of the Imperial Institute of France on that subject, and which 

 presents to the investigator a most valuable region of truth yet un- 

 explored. 



The three solutions of Mr. "Woolhouse (Lady's Diary, 1862) 

 are all derangements of the group G, 



7 1234567 



1 2345671 



2 3456712 



3 4567123 



4 5671234 



5 6712345 



7 12 3 4 5 



6 



of the seventh order made with fifteen elements, 1234567 12.34567 0, 

 considered as consecutive, 1 counting for 8, 2 for 9, &c, on the 

 partition 



15 = 7'2 + M=A« + B&, 



which is determined by the two vertical circular factors 1234567 

 and 1234567, the fifteenth element being undisturbed. 



nis 



There are -— different derangements of G x , made by altering 



the horizontal order of the vertical rows. 



124 325 5i6 634 7t0 

 235 436 627 745 liO 



346 547 731 156 22O 



457 651 I42 267 3s0 

 561 762 2.53 3 7 i 440 

 672 I73 364 412 5 5 

 713 2u 475 523 660 



G . 



Three of them are 



G : 



124 346 523 615 7:0 



124 Sis 5i6 623 770 



235 4.57 634 726 liO 



235 426 657 734 HO 



346 56i 745 I37 22O 



346 537 76i I45 220 



457 672 I06 2n 3s0 



457 641 I72 256 3*6 



561 7 13 267 352 440 



561 752 2i3 3 67 4*0 



672 I24 37i 463 550 



672 163 324 4 7 1 5s0 



713 235 4l2 574 660 



713 274 435 5l2 660 



The derangements G^ G 2 , G 3 are the three solutions of the 



