Answer 4—Six combinations must be tried in order to guarantee opening the lock. A geometric argument goes like this: Think of the 16 possible switch settings as binary numbers labeling the 16 corners of a 4-D hypercube. With one nonfunctional switch, it is necessary to try at least one end of each of the 32 edges of the hypercube. Each corner tests four edges, and there are two sets of eight corners that accomplish this with no redundancy, the set with even parity and the set with odd parity. If there are two nonfunctioning switches, then one corner on each of the 24 faces of the hypercube must be tested. Although one corner tests six faces, it is impossible to find four corners that don't have some redundancy in the faces tested. In fact, no matter how you do it, there will be at least two faces untested, and they won't share a single corner. It takes two more tests to finish the job.

Contributor: Eddie Insam

Published August 2003