We recast move generators for solving board games as operations on compressed sets of strings. We aim for compressed representations with space sublinear in the number of game positions for interesting sets of positions, move generation in time roughly linear in the compressed size and membership tests in constant time. To the extent that we achieve these tradeoffs empirically, we can strongly solve board games in time sublinear in the state space. We demonstrate this concept with the game Breakthrough where we empirically realize compressed representations taking roughly $n^{0.5}$ to $n^{0.7}$ space to store relevant sets of $n$ positions.