Milo of Croton and the sometimes failure of inductive proofs
Ranjit Bhatnagar once told me the following story: A young boy, upon hearing the legend of Milo of Croton, determined to do the same. There was a calf in the barn, born that very morning, and the boy resolved to lift up the calf each day. As the calf grew, so would his strength, day by day, until the calf was grown and he was able to lift a bull.

"Did it work?" I asked.

"No," said Ranjit. "A newborn calf already weighs like a hundred pounds."

Usually you expect the induction step to fail, but sometimes it's the base case that gets you.

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