There was not much fitting those requirements on the market at the time, but the search ended in the summer of 2010 in Cobble Hill at 229 Baltic Street, whose original listing, at $4.2 million, was once featured as House of the Day on Brownstoner.com. After a price cut to $3.995 million, the house went for $3.8 million.

It was a sale much like many others in that desirable enclave, except for one thing: the buyer was a school, Saint Ann’s, which had decided to purchase a residence as part of the lure for a new headmaster, Vincent Tompkins, who was then the deputy provost at Brown University.

Many of the elite schools that provide housing bought it years ago, when real estate was not nearly so expensive. So the Saint Ann’s search was a major undertaking, with the board of trustees contributing generously to the purchase, said Peter Darrow, president of the board.

Buying the house not only solved the problem of where Mr. Tompkins, who has three children, would live, but also made the school more competitive with its peers in attracting heads of school, and gave it a place for formal and informal gatherings for faculty members, alumni, parents and students.

“We weren’t just looking for a kind of social space; we were looking for an added dimension that would make his tenure as head of school more successful,” Mr. Darrow said. “The thought that he could have a place that was his place, in terms of where he lived, and he could entertain faculty, give seminars, we just thought that would enhance his leadership.”

While it is not uncommon for private schools to provide housing or a subsidy for headmasters, it does not appear to be the norm, according to a survey by the National Association of Independent Schools for 2010-11: 283 of the 1,261 schools that completed the survey, or 22.4 percent, reported on-campus housing for the head of school, while 171, or 13.6 percent, reported offering a housing allowance.

In New York, where elite schools often create their campuses with a series of buildings in a neighborhood, more schools offer an allowance than housing: just 6 of the 70 New York City schools in the survey, or 8.6 percent, reported offering on-campus housing, while 22 schools, or 31.4 percent, reported offering a subsidy.

“In New York City, the cost of real estate is so much higher that it might factor into a potential headmaster’s decision to take a job,” said Myra A. McGovern, the association’s senior director of public information. “In terms of using it as a recruitment tool, it might be particularly valuable there.”

The old Board of Education thought that way too for a time, making a controversial purchase of a five-bedroom, four-story brownstone at 80 State Street for $995,000 in 1990 and spending $250,000 to renovate it to attract Joseph A. Fernandez, the Miami schools chief, to the job of chancellor.

The house passed to his successors — except Harold O. Levy, who insisted on a housing stipend and stayed in his apartment on the Upper West Side while the city rented out the town house — until Mayor Michael R. Bloomberg took over the school system in 2002. The next year, the city sold the building at auction for $2.4 million. According to Marge Feinberg, a spokeswoman for the Education Department, the department no longer offers housing or a stipend for the chancellor.

In the case of Saint Ann’s, the headmaster’s residence shows how far the school — founded by Stanley Bosworth, who said he enjoyed “scandalizing and scaring away the bourgeoisie” — has traveled from its roots in a church basement. Although the school has retained much of its freewheeling artistic and intellectual sensibility, eschewing grades and emphasizing creativity, it, like the brownstone Brooklyn neighborhoods it draws from, has come to reflect more of that bourgeoisie Mr. Bosworth disdained.

At the same time, the school has undergone a rocky transition, focusing in recent years on more traditional concerns like organizational structure, professional management and money. Its last headmaster, Larry Weiss, quintupled the endowment to $5 million before he left last year amid dissatisfaction with his leadership style, which some described as being too reserved.

For all of the change, though, part of the appeal of the house was its neighborhood, which “hasn’t been taken apart like the other neighborhoods,” said Joan Goldberg, a former Saint Ann’s parent and the Brown Harris Stevens broker who sold the property. “It has no through-streets from the B.Q.E. or the bridge, so that cuts down on the traffic. It’s retained its really nice quality of mom-and-pop shops.”

And part of the appeal was the house itself, divided into a garden apartment and upper triplex at the time of the sale. At 24 feet across, it is wide and spacious, running 48 feet deep on the garden and parlor floors and 40 feet deep on the top two levels on a 100-foot lot. It has central air-conditioning, four wood-burning fireplaces, marble mantles and views of the sky from its many windows.

“It looks rather modest from the outside, so it’s really a surprise when you come in,” said Ms. Goldberg, who described its style as Greek Revival with a few Anglo-Italianate features, probably built around 1850 or 1860. “There are so many gorgeous houses in that area. This is one of the most beautiful.”