With five added helper modules, all facet-connected square-grid configurations become mutually reachable under pivot moves, using quadratically many moves.
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Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The $O(1)$ Musketeers
With five added helper modules, all facet-connected square-grid configurations become mutually reachable under pivot moves, using quadratically many moves.