60 On a Tactical Theorem relating to the Triads of Fifteen Things. 



but it is very easy to show that there is no solution which has a 

 i) without 3. I wish to show that there is in every solution at 

 least a 6 without 3. This being so, there will be (if they all 

 exist) 3 classes of solutions, viz. those which have at most (1) a 

 6 without 3, (2) a 7 without 3, (3) an 8 without 3. I believe 

 that the first and second classes exist, as well as the third, which 

 is known to do so. 



The proposition to be proved is, that given any system of 35 

 triads involving all the duads of 15 things; there are always 6 

 things which are a 6 without 3, that is, they are such that no 

 triad of the 6 things is a triad of the system. This will be the 

 case if it is shown that the number of distinct hexads which can be 

 formed each of them containing a triad of the system is less than 



/15. 14. 13. 12. 11. 10 K „ : __ ■ \ „ K ._ ,. 



( -= ^ — ~ — -. — = — „ =5 . 7 . 11 . 13= )5005,theentirenum- 



\ 1 . 2. 3. 4.5. 6 / 



ber of the hexads of 15 things. Now joining to any triad of the 



system a triad formed out of the remaining 12 things (there are 



_— - — — - — -=4.5.11 = 220 such triads), we obtain in all 

 1 . £ . o 



(220 x 35 = ) 7700 hexads, each of them containing a triad of 



the system. But these 7700 hexads are not all of them distinct. 



For, first, considering any triad of the system, there are in the 



system 16 other triads, each of them having no thing in common 



with the first-mentioned triad. (In fact if e. g. 123 is a triad 



of the system, then the system, since it contains all the duads, 



must have besides 6 triads containing 1, 6 triads containing 2, 



and 6 triads containing 3, and therefore 35 — 1—6 — 6 — 6 = 16 



triads not containing 1, 2, or 3.) Hence we have ( — '- — = ) 280 



hexads, each of them composed of two triads of the system ; and 

 since each of these hexads can be derived from either of its two 

 component triads, these 280 hexads present themselves twice over 

 among the 7700 hexads. 



Secondly, there are in the system seven triads containing each 

 of them the same one thing, e. g. 



123, 145, 167, 189, 1.10.11, 1.12.13, 1.14.15, 



containing each of them the thing 1. That is, we have 



ft 5 -) 



21 pairs such as 123, 145 containing the thing 1, 



and therefore (15 x 21 = )315 pairs such as a/3y, aSe. And for 

 any such pair, combining with a/3j8e any one of the remaining 

 10 things, we have 10 hexads aj3yBe^ each of them derivable from 

 either of the triads a/3y, a8e ; that is, we have (315 x 10= ) 3150 

 hexads which present themselves twice over among the 7700 



