The Optimal Number of People for Indoor and Outdoor Physical Distancing using Pigeonhole Principle

Abstract

Maintaining safe physical distances between individuals is a critical consideration in public health, emergency planning, and event management. While the urgency of physical distancing was emphasized during the COVID-19 pandemic, its relevance extends to diverse settings such as crowded concerts, emergency evacuations, and general spatial planning. Ensuring appropriate spacing reduces risks related to disease transmission, injury, and logistical inefficiencies. This paper explores how the pigeonhole principle—a foundational concept in combinatorial mathematics—can be applied to model and address such spatial constraints. The pigeonhole principle states that if more objects are placed into fewer containers, at least one container must hold multiple objects. Though intuitively understood since ancient times and referenced in early works such as the Mahabharata and writings by Jean Leurechon, it was formally named and widely applied by Peter Gustav Lejeune Dirichlet in the 19th century. Since then, the principle has played a crucial role in various disciplines, including number theory, computer science, and data analysis.

This paper introduces the pigeonhole principle through historical context and mathematical examples, demonstrating its utility in practical, real-world applications. A central focus is the application of the principle to determine optimal population densities within confined spaces while maintaining safe physical distances. The research is carried out using two complementary approaches: a theoretical framework that calculates the critical distancing threshold in segmented areas, and a computational simulation developed in Java that randomly plots individual positions to evaluate spatial constraints. By connecting a classical mathematical principle with contemporary safety challenges, this paper aims to illustrate the pigeonhole principle’s continued relevance and potential as a problem-solving tool across disciplines.