1 70 On the Triads made moith Fifteen Things, 



The last three pairs of columns exhibit three systems of 

 subindices, which systems may be cyclically permuted, the 

 letters remaining undisturbed: this gives us two sets more 

 each of six final columns, making three sets of six columns, 

 any set of which being added to the primary column will solve 

 the problem. If we now permute the letters abode cyclically 

 in these 6x3 columns, we shall obtain four times 6x3 co- 

 lumns more, making 6x 14- columns in all, additional to the 

 system of seven above written ; so that we can complete the 

 primary column into a solution in fifteen ways, and this with- 

 out repeating any triad : by this process we shall exhaust the 

 455 triads that are possible with fifteen things. All this I 

 have long known ; but it never occurred to me to observe that 

 the additional 6x14 were exactly 12 x 7, and thus to make 

 the pleasing variation, by which Mr. Sylvester has extended the 

 puzzle over the remaining twelve weeks of the quarter. We 

 see that this is, as Mr. Cayley expected, a matter of cyclical 

 permutation ; not of thirteen (a negative which he has proved), 

 but of five letters. 



I obtained this property of the triads made with fifteen 

 things four years ago, by observing that, if you substitute in 

 Q ]5 , at page 195 of the second volume, N. S. of the Cambridge 

 and Dublin Mathematical Journal, in the place of the small 

 letters, the second instead of the first arrangement of D 8 given 

 on the preceding page, Q 15 can be broken up into seven co- 

 lumns of five triads each, so as to solve this problem of the 

 fifteen young ladies ; but Q 15 , as it stands at page 195, cannot 

 be so broken up. The solution of Mr. Cayley at page 52 

 above is obtained by a like tentative process; and, in fact, 

 all the solutions are of the same form, and can be made 

 identical with the one above written, by disturbing the alpha- 

 betical order, or that of the subindices in certain triads, or 

 both, in the first column. The question has yet to be mathe- 

 matically treated: I do not feel satisfied with knowing how 

 to form thirty-five triads, which are found on trials but not 

 certainly proved before trial, to be capable of the required 

 arrangement. We want to know, before trial, toby a school 

 of fifteen can thus be marched out till every pair have walked 

 together, and why a school of twenty-one can also, or cannot. 

 I have a strong opinion, but will not undertake to prove the 

 negative, that twenty-one cannot be thus arranged. 



Permit me, in conclusion, to enunciate the following pro- 

 positions: — 



Theor. I. 5 x S m+1 symbols can be arranged in |(5-3 m+1 — l) 

 columns of triads, each column containing all the symbols, 

 and so that every duad shall be once, and once only, employed. 



Theor. II. If r be any prime number, and otherwise not, 



