Finding a Battleship of Uncertain Shape
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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∈{1,2}. A ship is a subset of ℤ^d. Given a family F of ships, we say that an infinite subset X⊂ℤ^d of the cells pierces F, if it intersects each translate of each ship in F (by a vector in 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.
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