Question 41
A Dutch hillwalking club with $4n$ members arranges a series of walks over a number of weekends, according to the following rules.
(a) Two walks take place each weekend – one takes place on Saturday, and the other on Sunday.
(b) Exactly $2n$ members of the club participate in each walk.
(c) On any weekend, no club member participates in both walks.
(d) After all walks are concluded, every pair of club members has participated together in the same number of walks.
Prove that after all walks are concluded, every set of three club members has participated together in the same number $t$ of walks, and that this number $t$ is divisible by $n-1$.
(a) Two walks take place each weekend – one takes place on Saturday, and the other on Sunday.
(b) Exactly $2n$ members of the club participate in each walk.
(c) On any weekend, no club member participates in both walks.
(d) After all walks are concluded, every pair of club members has participated together in the same number of walks.
Prove that after all walks are concluded, every set of three club members has participated together in the same number $t$ of walks, and that this number $t$ is divisible by $n-1$.