Finding a Battleship of Uncertain Shape

TL;DR

This paper investigates the minimal density of infinite sets that intersect all translations of certain ship shapes on a grid, extending previous work from single ships to families of ships, with explicit formulas and algorithms.

Contribution

It provides a formula for the minimal piercing density when the family has two ships of size two, and identifies the hardest cases for various ship families.

Findings

Derived explicit formula for two ships of size two.

Identified toughest ship families in multiple cases.

Developed an algorithm for computing the density in 1D.

Abstract

Motivated by a game of Battleship, we consider the problem of efficiently hitting a ship of an uncertain shape within a large playing board. Formally, we fix a dimension $d\in\{1,2\}$. A ship is a subset of $\mathbb{Z}^d$. Given a family $F$ of ships, we say that an infinite subset $X\subset\mathbb{Z}^d$ of the cells pierces $F$, if it intersects each translate of each ship in $F$ (by a vector in $\mathbb{Z}^d$). In this work, we study the lowest possible (asymptotic) density $\pi(F)$ of such a piercing subset. To our knowledge, this problem has previously been studied only in the special case $|F|=1$ (a single ship). As our main contribution, we present a formula for $\pi(F)$ when $F$ consists of 2 ships of size 2 each, and we identify the toughest families in several other cases. We also implement an algorithm for finding $\pi(F)$ in 1D.