  If taken seriously, the advice in the title leads to interesting
combinatorics. Consider N people moving between M rooms as follows: at each
step, simultaneously, the smartest person in each room moves to a different
room of their choice, while no one else moves. The process repeats. Let G(N,M)
be the directed graph with vertices representing the M^N possible
configurations and edges representing possible moves. We classify the pairs
(N,M) such that every configuration can be reached from every other. More
precisely, we prove that the graph G(N,M) is strongly connected if and only if
M>=N+1 (one extra room for maneuvering is both required and sufficient). In the
case M<=N, we explicitly describe strongly and weakly connected components.
Proofs are either impossibility arguments, or explicit algorithms based on
subgroups generated by derangements.
