One of my favorites is von Neumann's randomness extractor:
Suppose you have a biased coin that comes up heads with probability $p$ and tails with $q=1-p$. How do you construct an unbiased coin from the biased one?
The answer is simple but really nice. Simply toss the bad coin twice, and discard $HH$ and $TT$. The other two events occur with equal probabilities, $pq$ and $qp$.