My favorite example of mathematics that blends intuition and rigor comes from the works of Archimedes of Syracuse: the mechanical method used to intuit results such as the area of a parabola, followed by the method of exhaustion used to prove the result rigorously.
In the mechanical method, Archimedes imagined, for example, a section of a parabola balancing with a triangle, using the law of the lever (which he also discovered) to derive the area necessary to achieve said balance. He then used inscribed and circumscribed polygons to prove upper and lower bounds on the area thus derived, with (in modern parlance) the two bounds converging in the limit n → ∞, thereby establishing the result. The rigorous method of exhaustion (due originally to Archimedes' predecessor Eudoxus) is effectively equivalent to integral calculus (~2000 years ahead of its time), but guessing the right answer would in many cases have been difficult or impossible without the non-rigorous mechanical method.
Incidentally, the mechanical method itself might have been lost to history had it not been for the discovery of the Archimedes Palimpsest in a medieval prayer book [1], which contains the only known copy of the work describing it. Often called simply The Method, it takes the form of a letter from Archimedes to Eratosthenes, the chief librarian of the Library of Alexandria. When I had occasion to see some original pages of the palimpsest last year (at The Huntington Library in San Marino, adjacent to Pasadena, California), I was struck by the collegial tone of the letter, whose genuine human warmth was instantly recognizable even across two millennia.
In the mechanical method, Archimedes imagined, for example, a section of a parabola balancing with a triangle, using the law of the lever (which he also discovered) to derive the area necessary to achieve said balance. He then used inscribed and circumscribed polygons to prove upper and lower bounds on the area thus derived, with (in modern parlance) the two bounds converging in the limit n → ∞, thereby establishing the result. The rigorous method of exhaustion (due originally to Archimedes' predecessor Eudoxus) is effectively equivalent to integral calculus (~2000 years ahead of its time), but guessing the right answer would in many cases have been difficult or impossible without the non-rigorous mechanical method.
Incidentally, the mechanical method itself might have been lost to history had it not been for the discovery of the Archimedes Palimpsest in a medieval prayer book [1], which contains the only known copy of the work describing it. Often called simply The Method, it takes the form of a letter from Archimedes to Eratosthenes, the chief librarian of the Library of Alexandria. When I had occasion to see some original pages of the palimpsest last year (at The Huntington Library in San Marino, adjacent to Pasadena, California), I was struck by the collegial tone of the letter, whose genuine human warmth was instantly recognizable even across two millennia.
[1]: http://en.wikipedia.org/wiki/Archimedes_Palimpsest