Discrete configuration spaces of squares and hexagons

Abstract

A fixed fraction of a large box is occupied by tiny spheres; how difficult is it for those spheres to move around in the space available in the box, and how does that difficulty depend on the chosen density? We consider a discrete version of this problem, a generalization 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 consider these questions for both sliding squares and sliding hexagons, to model very densely packed disks.