  We present a version of the domino shuffling algorithm (due to Elkies,
Kuperberg, Larsen and Propp) which works on a different lattice: the hexagonal
lattice superimposed on its dual graph. We use our algorithm to count perfect
matchings on a family of finite subgraphs of this lattice whose boundary
conditions are compatible with our algorithm. In particular, we re-prove an
enumerative theorem of Ciucu, as well as finding a related family of subgraphs
which have 2^{(n+1)^2} perfect matchings. We also give three-variable
generating functions for perfect matchings on both families of graphs, which
encode certain statistics on the height functions of these graphs.
