Not long ago, I was talking about daily life in the N.F.L. with Ryan Fitzpatrick, the veteran Tennessee Titans quarterback, who studied economics at Harvard. “It’s such a physical game,” he said. “You see three-hundred-pounders hitting each other, and people think of the physicality. When people see the game, they think we’re meatheads; they think of the way jocks acted in high school. But we spend more time studying than we do on the field.”

During the period of more than a year that I spent with the New York Jets coaching staff while writing a book, I came to understand what Fitzpatrick was talking about. Football is a grand spectacle—never more so than in the playoffs, which begin this weekend—and it depends on layers of sophisticated tactics that are not immediately apparent. Winning certainly requires imposing your athletic will on an opponent; that part of the game is easy to see. Yet victories also redound to players who can outthink their adversaries. Because there are so few football games in a season, football players generally don’t learn about members of other teams by playing against them, the way baseball and basketball players do. Until they face another team—and, in a given year, they won’t see most of those outside their own division—N.F.L. players are unlikely even to be able to name most of its members. Football players must master the opposition conceptually. In addition to the raw speed and strength that professional football requires, the game involves more mental preparation than any other team sport.

In developing a game plan, coaches typically break down everything that happened in the opponent’s past four games to granular levels of “tendencies”—down, distance (to a first down), field position, and time remaining on the game clock. Once assembled, this research fills many pages of the game-plan binders players are given on Wednesday to prepare them for Sunday. (Teams have also begun to use iPads.) The binders are dense with intricate drawings and written instructions. They are often as thick as a left tackle’s fist.

The crucial portion of the game plan is a selection of new plays and modifications to old ones the coaches have created for the current opponent. N.F.L. coaches are deft and obsessive probers of game film; they live to devise. The problem is that there’s a limit to how much fresh information most players can absorb before each Sunday. Marv Levy, who coached the Buffalo Bills to four consecutive Super Bowls in the early nineties, told me he always fell back on something the legendary Notre Dame coaching innovator Knute Rockne once said: “I never ask if a player has the will to win. I ask if he has the will to prepare.”

Levy also wanted to know if players had a brain for football. Since every N.F.L. roster possesses talent sufficient to defeat any given opponent, one of the most coveted qualities in football players is what N.F.L. personnel men call “football intelligence.”

What kind of mind is ideally suited to football? Pat McInally studied history at Harvard, Class of 1975. He was the sort of undergraduate who, out of curiosity, visited the law-school classes of Clark Byse, the professor who was an inspiration for the Charles Kingsfield character in “The Paper Chase.” McInally was also an All-American receiver and punter. He went on to spend ten years as a wide receiver and All-Pro punter for the Cincinnati Bengals. But as far as the N.F.L. was concerned, McInally became a legend when he sat down to take the Wonderlic test and earned a perfect score—the only player ever to do so.

The Wonderlic is a fifty-question examination that tests the ability to answer increasingly difficult questions under time pressure—in this case, a limit of twelve minutes. There might be questions about pattern recognition, numerical sequences, word definition (What is the difference between “flammable” and “inflammable”?), trigonometry, and logic. “I thought it was funny we had to take an intelligence test,” McInally, now a collector of rare books and real estate who coaches at the Brethren Christian High School, in Huntington Beach, said. “I just blew through the thing.”

Among active players, Fitzpatrick is reputed to have the highest recorded Wonderlic score, a forty-nine. Or perhaps it was a forty-eight. Wonderlic scores are not made public, and while Fitzpatrick confirms that he answered forty-nine questions, skipping one that “didn’t make sense,” his score is a mystery even to him: “I have been told multiple scores, so I am unsure at this point.” Fitzpatrick has never found much of a relation between his academic talents and his football skills, but, he said, he knew that his ability to solve problems on the clock would distinguish him in the eyes of the N.F.L. draft directors. So he approached the Wonderlic with ferocity, completing it in nine minutes. His performance helped him enter the league—the average score for quarterbacks is in the high twenties. (For wide receivers, the mean is about ten points lower.) During Jets game-plan installation meetings, I saw how it gave the defensive coaches a source of motivation: they called Fitzpatrick “Big Brain,” and told their players they needn’t worry, because, under pressure, what would he do—throw a book at them?

More than any other position, playing quarterback requires mastering a farrago of detail, and then sifting through it while staring at eleven large people eager to break your face. The best N.F.L. quarterbacks, like Tom Brady, Drew Brees, and Peyton Manning, have reputations as keen, obsessive students of opposing defenses, whose schemes they decode in real time. And yet, what does it say that the great model of lethally consistent play, Peyton, scored a twenty-eight on the Wonderlic while his more erratic brother, Eli, scored a thirty-nine?

One theory some in the N.F.L. hold is that the highest-scoring quarterbacks are too rigidly scholarly, prisoners of research who don’t handle in-game adjustments well, while those whose scores are very low simply can’t handle a high volume of preparation.

Oliver Luck was twice an Academic All-American quarterback at West Virginia University, spent five years in the N.F.L., went on to law school, and is now the athletic director at his alma mater. His son, Andrew, (Stanford Class of 2012, architectural design; Wonderlic, thirty-seven) is the Indianapolis Colts’ excellent second-year quarterback. “Football intelligence to me is situational awareness,” Oliver Luck told me. “The variables in football are so many. Every play is a decision and you do it at full speed. Life involves more thought.” (If there is a dark undercurrent to a discussion of bright football players, it has to do with life after the scrum and the long-term effects that hits to the head can have on the brain.)

That said, Oliver Luck thinks that there have been certain moments post-football when his aptitude for the game has been helpful to him. “I remember distinctly sitting for the Texas bar exam after I finished law school,” he recalled. “There were maybe five hundred people in there. People were sighing and groaning. A guy one table away from me suddenly lost it. I wanted to tell him, ‘Suck it up! You can do it!’ The way I would in the huddle. I was focussed. I knew how to work through that test.”