Year: 2025
Type: article
Abstract: 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. In this paper we determine which configurations are reachable, from which other configurations, and provide bounds on the number of moves. Namely, let G(N, M) be the directed graph with vertices representing all $$M^N$$ M N configurations and edges representing possible moves. We prove that the graph G(N, M) is weakly connected, and that it is strongly connected if and only if $$M\ge N+1$$ M ≥ N + 1 (one extra room for maneuvering is both required and sufficient). For $$M\le N$$ M ≤ N , we show that the graph has a giant strongly connected component with $$\Theta (M^N)$$ Θ ( M N ) vertices and diameter $$\mathcal {O}(N^2)$$ O ( N 2 ) .
Source: Graphs and Combinatorics
Author: Davide Sclosa
Institution: University of Crete
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,190
Funder: Vrije Universiteit Amsterdam
OpenAlex ID: W4414348148