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One of the essential benefits of swarm robotic systems is redundancy. In case one robot breaks down, another robot can take steps to repair the failed robot or take over the failed robot's task. Although fault tolerance and robustness to individual failures have often been central arguments in favor of swarm robotic systems, few studies have been dedicated to the subject. In this paper, we take inspiration from the synchronized flashing behavior observed in some species of fireflies. We derive a completely decentralized algorithm to detect non-operational robots in a swarm robotic system. Each robot flashes by lighting up its on-board light-emitting diodes (LEDs), and neighboring robots are driven to flash in synchrony. Since robots that are suffering catastrophic failures do not flash periodically, they can be detected by operational robots. We explore the performance of the proposed algorithm both on a real-world swarm robotic system and in simulation. We show that failed robots are detected correctly and in a timely manner, and we show that a system composed of robots with simulated self-repair capabilities can survive relatively high failure rates.