96 ON A TACTICAL THEOREM RELATING TO THE TRIADS OF FIFTEEN THINGS. [323
them each of 3 out of the 7 things, and the remaining 28 each of 1 out of the
7 things, and 2 out of the 8 things: or attending only to the 8 things, there are
28 triads each of them containing a duad of the 8 things, but there is no triad
consisting of 3 of the 8 things. More briefly, we may say that in the system there
is an 8 without 3, that is, there are 8 things such that no triad of them occurs in
the system.
I believe, but am not sure, that in all the solutions which have been given of
the school-girl problem there is an 8 without 3.
Now, considering the more simple problem, there are of course solutions which
have an 8 without 3 (since every solution of the school-girl problem is a solution of
the more simple problem): but it is very easy to show that there is no solution which
has a 9 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
the entire number of the hexads of 15 things. Now joining to any triad of the system
12.11.10
1 . 2.3
a triad formed out of the remaining 12 things (there are
= 4.5.11 = 220
such triads), we obtain in all (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, con
sidering 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
21 pairs such as
123, 145 containing the thing 1, and therefore (15 x 21 =) 315 pairs such as a@y, a8e.
