In this note, we discuss a generalization to Dijkstra's Dining Philosophers problem and a distributed solution to it. We also show that the solution is deadlock-free and starvation-free and also robust, in the sense that failure of some nodes does not affect all the nodes. The results of this paper have implications for problems in the areas of resource sharing, routing in networks, processor interconnections, fault-tolerant computing, and decentralized control in distributed systems.