Carl Friedrich Gauss labored mightily (by hand, ca. 1795) to compute the reciprocals of the numbers from 1 to 1000. He was looking for a pattern in the length of the repeating decimal fraction 1/N. He failed — but instead found a far more precious gem, the law of quadratic reciprocity. (It had been discovered ~10 years earlier by Euler and Legendre. Gauss proved it when he was 18 years old, after working on it for a whole year.)

Or so the story goes. But what kind of a "gem" is this mysterious law? After taking a fair bit of undergraduate math I can read the equations of the quadratic reciprocity theorem, and can more or less follow the steps of its proof, if led by hand. But understand it? Not really.

As Robin Zimmermann (at a similar stage in his mathematical learning) says: They've taught you enough to pick up the magic sword — but not yet how to wield it!

TopicScience - 2001-09-27

The story highlights that many (most?) of the significant discoveries throughout history came from the apparent failure to prove or do something else entirely. The genius of the moment is to be able to see the unintended and unexpected result, and shift focus to comprehend and finish that. – Bo Leuf

(correlates: MarginAlia, PersonalResponsibility, MarryTheOne, ...)