Principles from information theory are used to obtain the minimum of the average number of shutdowns needed with a trial-and-error testing procedure for identifying failed fuel in nuclear reactors, based on a priori failure probabilities of each assembly. Also presented is the average reduction in the number of shutdowns if other information about failure is incorporated, so the merit of this other information can be assessed quantitatively. Illustrative examples are given for applications involving the use of information from flux tilting in a pressurized water reactor, gas tagging in a liquid-metal fast breeder reactor, and burnup in the Experimental Breeder Reactor II.