Shape formation by hybrid programmable matter

\(n\) passive hexagonal tiles (rhombic dodecahedra in 3D) occupy nodes of a lattice and form a connected structure. A single active agent, a deterministic finite automaton, starts on a tile, must always stand on or next to a tile, sees its node and the adjacent nodes, and in each round lifts a tile, places the tile it carries, or moves to an adjacent node; it carries at most one tile, and the tiles (with the carried tile counted at the agent's node) must stay connected. It must rearrange the tiles into a target shape (a line, a triangle, or an intermediate structure such as a block or a tree). Time counts the agent's rounds.

Measures: Time, rounds; in each round the agent performs one Look-Compute-Move step.