Problem:
I have three opaque jars filled with coins. One contains only nickels; one contains only dimes; one contains some nickels and some dimes. Also, the three jars are labeled "Nickels," "Dimes," and "Nickels and Dimes," but it is known that all of the jars are labeled incorrectly. You are allowed to choose one jar and look at one coin randomly drawn from it. How can you determine which jar is which?
Solution:
Since the jar labeled "Nickels and Dimes" is mislabeled, it contains either only nickels or only dimes. This suggests drawing a coin from this jar.
If the coin drawn is a nickel, then the "Nickels and Dimes" jar actually contains only nickels. Then, the "Dimes" jar (because it is also mislabeled) must contain nickels and dimes, and the "Nickels" jar contains only dimes. If, on the other hand, the coin drawn is a dime, then the "Nickels and Dimes" jar contains only dimes; this forces the "Nickels" jar to contain coins of both types, and the "Dimes" jar then contains nickels. Thus, either way, it is possible to identify all three jars correctly.