For a couple of weeks in November, the story captivated even the casual college football follower, perhaps because it did not concern allegations of child sexual abuse or recruiting violations.

Yale’s senior quarterback, Patrick Witt, the most productive passer in the program’s history, was wrestling with a decision that generated national attention and debate. Should he play in the Harvard-Yale game one more, final time, or should he attend an interview in Atlanta as a finalist for a Rhodes scholarship? Because of the schedule of the events, he could not do both.

The quarterback’s dilemma had an unlikely twist. His coach, Tom Williams, said he had faced almost identical circumstances in 1992 when he was a linebacker at Stanford. Williams said he had chosen to pursue a career in professional football at the expense of a possible Rhodes scholarship — and never regretted the decision. Witt leaned on his coach for advice, and eventually decided to play in the game. Yale was crushed, 45-7.

As it turned out, the shellacking of the Bulldogs by their storied rival in the Yale Bowl on Nov. 19 was far from the worst of it. The appealing back story — Williams’ providing counsel to his talented quarterback based on his own experience — turned out to be founded on a lie.

The coach had never been in Witt’s position. He had never been a Rhodes scholar candidate or applicant, let alone a finalist, as he had let the world believe. He had told Yale he was a candidate with an entry on his resume. His biography on the Yale website said the same thing.

On Wednesday, a day before his 42nd birthday, Williams paid with his job. Yale, which had undertaken an internal review to investigate Williams’ Rhodes scholarship assertion after a New York Times article raised doubts about its legitimacy, announced that the coach had resigned.

In a news release from the university, Williams said he had been encouraged to apply for a Rhodes scholarship while at Stanford, but never did so: “I considered the opportunity, sought advice, and was encouraged to apply by faculty members and my coach, Bill Walsh. But I did not apply.”

Williams’ story began to unravel shortly after Witt announced he would forgo the Rhodes interview to play in what is widely known as the Game. A Rhodes Scholarship Trust official, Elliott Gerson, its American secretary, told The Times he had never heard of a candidate bypassing the final interview to play in an athletic event. Gerson has been overseeing the scholarships for more than three decades.

The Times then asked the Rhodes trust to check on Williams’ claim. After a thorough investigation the trust reported that no one named Williams had even applied for a scholarship in 1991, 1992 or 1993. Asked about the trust’s findings later that day, Williams said: “I wasn’t trying to confuse anyone or make it sound different than it was. I was in the preliminary stages at Stanford and I had to decide, but the interview wasn’t official.”

The next day, Yale announced that it would investigate Williams’ claim and would not comment until its findings were completed. But before Wednesday’s announcement, Yale defensive lineman Reed Spiller said he could not envision a situation in which Williams, completing his third year as coach, would leave of his own volition.

“He’s not going to leave here voluntarily under any means — from the very beginning he’s said how excited he was to be getting this job,” said Spiller, a senior from Portsmouth, N.H. “There’s no chance he’d leave on his own.

“But that being said, I think there are a whole lot of people at this school that are overplaying the article about his Rhodes scholarship candidacy. I think that’s a moot point, really. I think that doesn’t really have anything to do with what kind of football coach he is. He’s a great guy and a great football coach, and he’s the right person for the job at this school.”

Williams, it turns out, also misrepresented his pro football credentials as part of his official Yale biography.

He had asserted that he signed with the San Francisco 49ers as a free agent in 1993 and spent 1993 on the team’s roster as a linebacker. Both claims proved to be untrue. There is no mention of him in the team’s media guide under “all-time roster,” and when contacted by The Times about Williams’ claim that he had signed with the team as a free agent, a spokesman said the team’s accounting office had no record of ever issuing a W-2 form to Williams.

Williams, Yale’s first African-American head football coach and the Ivy League’s second, took over the program in 2009. It was his first head coaching assignment. In three seasons his record was 16-14, including three losses to Harvard.

The story line for this year’s game centered on Witt’s choices: the interview or the Game. Rhodes officials took plenty of heat for not rescheduling the interview, even though the date is known well in advance and applicants are informed it is set in stone. Witt knew it.

His dilemma was well chronicled in the local and national media. There was talk of perhaps chartering a plane. But with a noon start to the game and a morning interview in Atlanta — with the possibility of a second interview in the afternoon — there was no wiggle room. At the time, Williams said of Witt: “I know it’s hard. It’s difficult to be put in that situation. … I know he’ll the make the right decision.”