Mathematical_Recreations-Kraitchik-2e

Mathematical Recreations
a table for 2n -1 teams is first made up. Then in each
round the 2nth team has to play the idle team given in
the first column.
The most difficult and common case is that in which the
competition is between individual players. Here we would
like to satisfy the following conditions:
(1) Each player must play the same number of rubbers.
(2) Each player must have every other player as partner
the same number of times.
(3) Each player must play against as many teams as can
be formed from the remaining n - 1 players, and must play
each of them the same number of times.
Let us denote by R .. the total number of rubbers that must
be played to satisfy these conditions, and by "Cr the combinations
of n things r at a time (that is, the number of ways of
selecting r things from a set of n things). We must form every
possible set of four players, and each such set must play 3
rubbers so that each one of the four has each of the others as
partner. Then
R =3 C _n(n-l)(n-2)(n-3)
.. .. 4 - 8 .
This may also be found by finding the number of all possible
matches between pairs of players,
Here is a table of values of R".
n
Rn
4
3
5
15
6
45
"C2 . "-2C2
2
7
105
8
210
9
378
10
630
It will be seen that the number of rubbers increases very
rapidly. For n = 40, R" = 274,170. Such a tournament is
obviously impracticable.
However, if we neglect the third condition the number can
be radically reduced. When n is of the form 4k or 4k + 1,