Rabbit Hunting using Set Theory and Probability

TL;DR

This paper addresses a pursuit problem involving an invisible, moving target on a number line, proposing two algorithms—one based on set theory and the other on probability—to guarantee a hit in finite steps.

Contribution

It introduces two novel algorithms for guaranteed pursuit of an unknown moving target using set theory and probabilistic methods.

Findings

The Cantor diagonal-based algorithm guarantees a hit in finite steps.

The probabilistic algorithm also guarantees a finite-time hit with high probability.

The methods extend pursuit-evasion strategies to unknown, moving targets on a number line.

Abstract

Imagine an invisible rabbit that starts at some unknown integer point $A$ on the number line. At each time step, it hops by a fixed but unknown integer stride $B$. Both $A$ and $B$ are fixed integers, but their values are unknown. Suppose you have a magic hammer that you can throw at any integer point on the number line at each time step. When the hammer strikes the rabbit, it instantly squeals, indicating you have hit it. The problem now is to devise a strategy that guarantees your hammer will hit the rabbit in finitely many steps. We will provide two algorithms to solve this problem. The first involves Cantor's diagonal trick from set theory, and the second is a probabilistic approach.