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We present the first universal reconfiguration algorithm for transforming amodular robot between any two facet-connected square-grid configurations usingpivot moves. More precisely, we show that five extra "helper" modules("musketeers") suffice to reconfigure the remaining $n$ modules between any twogiven configurations. Our algorithm uses $O(n^2)$ pivot moves, which isworst-case optimal. Previous reconfiguration algorithms either require lessrestrictive "sliding" moves, do not preserve facet-connectivity, or for thesetting we consider, could only handle a small subset of configurations definedby a local forbidden pattern. Configurations with the forbidden pattern do havedisconnected reconfiguration graphs (discrete configuration spaces), and indeedwe show that they can have an exponential number of connected components. Butforbidding the local pattern throughout the configuration is far fromnecessary, as we show that just a constant number of added modules (placed tobe freely reconfigurable) suffice for universal reconfigurability. We alsoclassify three different models of natural pivot moves that preservefacet-connectivity, and show separations between these models.

Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The $O(1)$ Musketeers