On the Puzzle of the Fifteen Young Ladies. 527 



indices. Now the number of triplets possible with n things is 

 less than that of those which must be employed in the columns 

 to be added to the given one. We have a right to expect that 

 the pair ah will be associated equally with the remaining letters ; 



that is, the number of triplets to be added, which is^ Sn(Bn— 1)— n, 



will be divisible by that of those possible with n symbols, which 



is-gK . n— 1 . n— 2 j in other words, 9 is divisible by n— 2, which 



confines n to the values 3, 5, and 11. The force of this reason- 

 ing lies in the position, tbat there cannot be less than n families, 

 all being symmetrical. 



The problem can be solved for the two first values of n ; but 

 I doubt greatly its solvability fom=ll. 



There are 11 x 15 triplets possible with 11 things, and 15 

 columns of 11 triads are required to be added in the solution; 

 thus we may safely predict that every triplet of the 11 x 15 will 

 be once employed. And it is reasonable to anticipate, on account 

 of the symmetry to be expected, that the 15 columns will fall 

 into groups of one or more columns, which can all be formed 

 from the first added group by cyclical permutation either of n, 

 or of n— 1, or of n— 2 letters; for to suppose such permutation 

 to be made with less than n — 2 letters, would involve the ad- 

 mission that some one triplet of the 11 x 15 would be unaffected 

 by it, which is next door to absurd. The only groups into which 

 the 15 columns of letters, considered apart from subindices, can 

 fall, are groups of 1, or of 3, or of 5, if all is symmetrical ; and 

 these cannot be produced from each other by cyclical permuta- 

 tion of 11, nor of 10, nor of 9 letters. I venture to affirm, 

 though I do not pretend to have demonstrated, that the problem 

 cannot be solved for n a prime number greater than 5. 



The following is a better arrangement than that which I have 

 before given : — 



a l b 1 c l a i d l e l afi^d, 



aj> a c a a a d a e a a a b^d, 



a x a^a 3 

 b Ah 



CjC 2 C 3 



d t d s d 3 



0| >',l' t 



"a'-a 

 a 3 d 3 e 3 



c 3 a 2 e l 



2 «. 2 tr 2 



a 3 b 3 c 3 

 d 3 b x c 2 



e :A c l 



/ a u 3"3 



a 2, e \ c \ 



b l e 9 c 3 

 «j<? a e 2 



^i^a^a 

 ff 2 e 3 c 3 



e,b a d.. 



!/ 2 



cbvd, 



a t b 3 e 3 

 a 2 b 1 e 1 

 a 3 c 2 d 2 



<? 2 c,</ 3 



a x c 3 d 3 

 a 2 c i d 1 

 a 3 b s e 2 

 c 2 b 3 e 1 

 <h b \ e a 



The second and third groups of added columns, looking at 

 letters ajjiirt from subindices, are made by cyclical permutation 

 of cde in the first. The subindices axe made by cyclical permu- 

 tation of 123, under all the letters bede, the second and third 

 groups from the first. 



