A distributed computing system can be operated in a fault tolerant manner using a collection of auxiliary computing devices and more main computing devices than the number of faults the system can tolerate. A quorum of all of the main computing devices can be used. In the event of a failure, an alternative quorum from a selected set of quorums, comprising at least one main computing device and some or all of the auxiliary computing devices, can be used to complete pending operations and to select a new set of quorums. Alternatively, another state machine, comprising at least one main computing device and some or all of the auxiliary computing devices, can select a new quorum comprising the currently operating main computing devices, and the new quorum can then complete pending operations and can continue to select proposals using the proposal number assigned by the other state machine.