Rabbit Hunting using Set Theory and Probability

TL;DR

The paper addresses locating an invisible rabbit that starts at an unknown integer $A$ and hops by an unknown integer $B$ on the number line, using either a set-theoretic diagonal strategy or a probabilistic search. The set-theoretic method encodes all possible $(A,B)$ sequences into a countable lattice and guarantees hit by diagonalizing through a bijection $\mathbb{Z}^2 \to \mathbb{N}$; the probabilistic method selects random hammer positions with an increasing horizon $h(n)$, ensuring hit with probability 1 when $\sum 1/h(n)$ diverges (with $h(n)=\lfloor n\log n\rfloor$ as a natural choice). Despite almost-sure success, the expected hitting time is infinite under the proposed probabilistic scheme, as shown via Raabe's test. The paper discusses generalizations to quadratic trajectories and touches on higher dimensions and real-valued starting parameters, highlighting the distinct natures of the two approaches. These results illuminate connections between combinatorial enumeration, probability, and group structure in search problems on the integers.

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.

Figures (1)

• Figure 1: A bijection from $\mathbb{Z}^2 \longrightarrow \mathbb{N}$.