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We describe a high-density storage system for physical goods in which identical conveyor modules can be plugged together to store and retrieve unit-loads or small containers. Material movement conforms to the “puzzle architecture” found in popular board games such as the 15-puzzle and Rush Hour. Control of the system is decentralized, meaning that each module contains identical operating logic that directs its behavior based on local conditions and message passing. We prove the system deadlock-free and show its performance under a wide variety of operating configurations.