Kennedy wanted to reveal this fact, both to the American public and to the world. So, on Oct. 2, 1961, Deputy Secretary of Defense Roswell Gilpatric gave a speech in Hot Springs, Va., proclaiming great “confidence in our ability to deter Communist action or resist Communist blackmail,” owing to “a sober appreciation of the relative military power of the two sides.” The U.S. arsenal, with its “tens of thousands” of nuclear weapons, was so lethal, Gilpatric said, that “an enemy move which brought it into play would be an act of self-destruction.”

For years, Khrushchev had boasted that his factories were cranking out ICBMs “like sausages.” In fact, though, he had nothing; the missile program was in total disarray. And now the Americans were calling his bluff.

The Communist Party of the Soviet Union was about to hold its annual congress. Khrushchev was already coming under attack for backing down in Berlin, both from the Kremlin’s hard-liners and from China’s more radical Communist Party, which was competing with the Soviets for Third World allies. The balance of forces with rivals, to the east and west, was palpably shifting against him.

Khrushchev really did believe that the United States might launch a nuclear first strike against the Soviet Union. The notion wasn’t so far-fetched. During the Berlin crisis, Kennedy had ordered the Pentagon to conduct a study of whether such a strike was possible. The top secret study — a detailed, 36-page attack plan — concluded that it was very feasible. It’s unknown how much Khrushchev knew about this plan, but the Joint Chiefs wrote follow-up assessments, Kennedy himself read them and held at least one meeting in the Oval Office to discuss the issue. (I got these documents declassified for a story about the first-strike plan that I wrote for the October 2001 issue of the Atlantic.)

The Soviet premier had not given up on grabbing West Berlin, but he knew he had no leverage. If the United States did launch a nuclear strike, he wasn’t even sure that any of his missiles or bombers would be capable of retaliating.

Khrushchev did have a fair number of medium-range missiles, so he shipped them to Cuba, to put them within firing range of the United States. If he could install them without notice, then announce another gambit on Berlin (as he was planning to do in November 1962), those missiles would give him something to bargain with.

But the American U-2s spotted the missiles. And once Kennedy announced that they had, Khrushchev knew he would have to pull them out. The question was how to do so without suffering another humiliation.

As it happened, Kennedy was thinking along the same lines. The secret tapes reveal that on Oct. 18, just the third day of the crisis, Kennedy wondered aloud why Khrushchev had put the missiles in Cuba. He figured they must have been part of a bargaining gambit and that, to get them out, he might have to give Khrushchev “some out,” a way to save face. One way, he mused, might be to say, “If you pull them out, we’ll take ours out of Turkey.”

None of JFK’s advisers paid any attention to his remark. On the final day, Oct. 25, when Khrushchev proposed just such a trade, Kennedy pounced on it eagerly. “Let’s not kid ourselves,” the president is heard saying on the tapes. “Most people think that if you’re allowed an even trade, you ought to take advantage of it.” If we go to war, mounting air strikes and then an invasion on Cuba, and if the Soviets respond by grabbing Berlin, he added, “everybody’s going to say, ‘Well, this Khrushchev offer was a pretty good proposition.’ ”