The minimum size required of a solitaire army

The solitaire army is a one-person peg jumping game where a player attempts to advance an "army" of pegs as far as possible into empty territory. The game was introduced by John Conway and is also known as "Conway's Soldiers". We consider various generalizations of this game in different 2D geometries, unify them under a common mathematical framework, and find the minimum size army capable of advancing a given number of steps.