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Finding a Battleship of Uncertain Shape

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 $π(F)$ 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 $π(F)$ 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 $π(F)$ in 1D.