the Fifteen Young Ladies. 203 



young ladies will be 



1 . 2 . 5 4 . 6 . 12 3ab led Sef §gh lOij 11H 13isV. 



It is required to determine ab, cd, efgh... M, which are in 

 some order the numbers 1 2 3 ... 12 (to be counted as 14, 15 

 . . . 25), in every possible way, so as to solve the problem. 



I shall content myself here with enunciating the two arithme- 

 tical conditions, which are necessary and sufficient. 



First, it is required that, supposing a > b, od, &c, 



(a-b), (c-d), (e-f), [g-k), (i-j), (k-l) 



shall be six (or for the general case of 12?i + 3 shall be Sn) dif- 

 ferent numbers > 0. 



Secondly, it is required that the twelve numbers (or for the 

 general case the 6n numbers denned by the capitals and small 

 letters) 



fl-3, b-Z, c-7, d-1 . . . £-11, /-ll 



shall be in some order the twelve numbers 12 3 4... 12, or for 

 the general case the 6n numbers 12 3... 6n, when estimated as 

 residues to modulus 13 (or 6/z+l). 



To find the numbers «, b, c, &c, we have only to write out 

 the congruences 



(«-3r+(6-3r+(c-7r+(^-7) w +...+!(A:-iir+ 



{l-ll)=r + 2 m + 3 m + . . . +12™ (mod. 13,) 



for as many values of m as we require. We can thus obtain by 

 a solution of these congruences every possible system of the 2>n 

 duads which can satisfy the second condition. The number of 

 these systems which satisfy also the first, is that of the different 

 solutions of the problem. 



It is perfectly certain that for the case of 12/z + 3 = 15, the 

 only systems possible will thus turn out to be, for addition to the 

 capitals 3, 5, 6, those read in the first lines of G^ G%, G 3 . 



Mr. Anstice's method of constructing the n primary triplets of 

 the 6ra + 1 capitals is not proved to exhaust the solutions. 



A direct and exhaustive method of finding them is to seek for 

 their difference circles; that is, to seek for the perfect sets of par- 

 titions in triplets of the prime number 0>n-\- 1. 



Def — A perfect set of partitions in r-piets of N = &(?' 2 — r) + 1 

 is a system of k r-plets, 



(a, tf 2 a 3 . . . ft r ), + [a\ a\ a f 3 ... a' r ) 2 + . . . + («"• •.«£•... a" r ■ ) ki 



such that every number, 1, 2, 3, . . . N — 1, can be made by ad- 

 dition of consecutive elements of some one r-plet, the sum of the 

 elements of each r-plet being N. These partitions can be directly 

 found by the solution of a system of congruences similar to that 



