WASHINGTON — Bill Clinton found him to be cold and worrisome, but predicted he would be a tough and able leader. George W. Bush wanted to make him a friend and partner in the war on terror, but grew disillusioned over time.

Barack Obama tried working around him by building up his protégé in the Kremlin, an approach that worked for a time but steadily deteriorated to the point that relations between Russia and the United States are now at their worst point since the end of the Cold War.

For 15 years, Vladimir V. Putin has confounded American presidents as they tried to figure him out, only to misjudge him time and again. He has defied their assumptions and rebuffed their efforts at friendship. He has argued with them, lectured them, misled them, accused them, kept them waiting, kept them guessing, betrayed them and felt betrayed by them.

Each of the three presidents tried in his own way to forge a historic if elusive new relationship with Russia, only to find their efforts torpedoed by the wiry martial arts master and former K.G.B. colonel. They imagined him to be something he was not or assumed they could manage a man who refuses to be managed. They saw him through their own lens, believing he viewed Russia’s interests as they thought he should. And they underestimated his deep sense of grievance.