Alex R's answer reminds me of another sort of clever "strategy" problem. The way I've heard it phrased is as follows (with apologies to the vegetarians in the audience):
You are given 1000 cups filled with water, exactly one of which (unknown to you) is laced with a lethal poison that is guaranteed to kill within 24 hours if ingested. You are given ten rats with which to determine the poisoned cup. What is the minimum amount of time needed to be certain?
One naive approach would be to break up the cups into groups of 100 to be fed to each rat (i.e. rat 1 gets cups 1 through 100, rat 2 gets 101 through 200 etc). After 24 hours one rat will be dead and you have narrowed the poisoned cup to one in 100. Repeat two more times: you have your poisoned cup identified within 72 hours.
There is in fact a much better solution, which identifies the cup in the minimal 24 hours. Since $1000~<~2^{10} = 1024$, assign each cup a number expressed in binary. Then give the water in that cup to those rats for which there is a '1' in the binary representation (so, for example, cup 17 = 10001 is fed to rats 1 and 5). After 24 hours, you can simply read off the number of the poisoned cup by the numbers of the dead rats.
You can impress your audience with the remarkable (some might say brutal) efficiency of this procedure: to identify one in a million cups you would need only 20 rats (ignoring, of course, any deaths caused by overhydration...)
