Mathematical surprises

As we know only few math discoveries are considered as shocking and unexpected results, in which open great gates instead of only windows, to the mathematician views. . Dave Richeson at “Division by Zero” has listed some of this kind of discoveries:

Mathematical surprises

I’m interested in compiling a list of “mathematical surprises.” The best possible example would be a mathematical discovery that no mathematician saw coming, but after it was discovered it changed mathematics in some fundamental way—Cantor’s discovery of the nondenumerability of the continuum is such an example. But I’ll settle for any surprise—Andrew Wiles surprised everyone with his proof of Fermat’s Last Theorem, the solution of the Monty Hall problem surprised many capable mathematicians, etc.

I’ve spent a couple days brainstorming and I’ve come up with the following list. Some are better than others, and they’re listed in no particular order. Please add your surprises in the comments below!