Discrete configuration spaces of squares and hexagons

We consider generalizations of the familiar fifteen-piece sliding puzzle on the 4 by 4 square grid. On larger grids with more pieces and more holes, asymptotically how fast can we move the puzzle into the solved state? We also give a variation with sliding hexagons. The square puzzles and the hexagon puzzles are both discrete versions of configuration spaces of disks, which are of interest in statistical mechanics and topological robotics. The combinatorial theorems and proofs in this paper suggest followup questions in both combinatorics and topology, and may turn out to be useful for proving topological statements about configuration spaces.