In constructing a possible proof, theorists follow a method described thusly:
Like a sonnet or an aria, a mathematical proof has a distinct form and set of conventions. It begins with axioms, or accepted truths, and employs a series of logical statements to arrive at a conclusion. If the logic is deemed to be watertight, then the result is a theorem. Unlike proof in law or science, which is based on evidence and therefore subject to qualification and revision, a proof of a theorem is definitive.
Grigori Perelman, who may have finally cracked the code, is said in the story to have offered a proof that is "unorthodox... astonishingly brief for such an ambitious piece of work."
In addition to the nature of the solution the piece talks at length about the personalities behind the competition to be first with a proof, jealousy in the academy and the importance of the riddle itself.