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Solution to Warm-Up Puzzle:
Even if three primary devices failed, the backups could perform all functions along with the remaining three primaries. To arrive at that answer, consider that there are really two cases.
Divide the primary devices into three pairs: A–B, C–D and E–F. Notice that two backups cover each pair. If the three failures are from three different pairs, then backup 1 could handle the A–B failure, backup 2 could handle the C–D failure and backup 3 could handle the E–F failure. If the failures come from two different pairs, then the pair in which both primaries fail gets covered by the two backups that cover that pair. The remaining backup can cover any of the other four devices.