  We investigate tilings of cubiculated regions with two simply connected
floors by 2 x 1 x 1 bricks. More precisely, we study the flip connected
component for such tilings, and provide an algebraic invariant that "almost"
characterizes the flip connected components of such regions, in a sense that we
discuss in the paper. We also introduce a new local move, the trit, which,
together with the flip, connects the space of domino tilings when the two
floors are identical.
