News illustrations, Ms. Mouly added, have the power to “make sense in two weeks, or two months or in 200 years as well.”

Barry Blitt, 56, is one of The New Yorker’s most prolific and fastest artists. He has done more than 80 covers since 1994, including those of Mr. Putin and the Obamas. He has resisted figuring out precisely what makes a good subject for a cover for fear of becoming too formulaic, he said.

But, buoyed by Ms. Mouly’s instruction to never censor himself, he often tries risky topics — like beheading — even if they are later rejected as too insensitive. He has also had to draw fast. Sometimes, Ms. Mouly or Mr. Remnick will present an idea with as little as a day left before the print edition is published.

“I’ll just send in a bunch of sketches, as many as I can,” he said. “Then she’ll very politely ask for more.”

His most recent cover — showing police officers pursuing a football player — came after Mr. Remnick asked for an image reflecting the N.F.L.’s troubles. The sketch took several different forms in simple black-and-white, and then in color study, before the final drawing was completed and sent to the printer.

“There’s a certain absurdity you can get with a pen and ink line, which if you rendered it another way, with photography maybe, the idea dies,” said Mr. Blitt, who works out of his studio in Roxbury, Conn.

On the wall of Ms. Mouly’s office are dozens of covers that did not quite make the cut, for reasons of taste, significance or just because of a feeling. (Tucked among them are depictions of Anna Wintour, the Vogue editor and artistic director of Condé Nast, which also owns The New Yorker.) Other rejected images, some of which are included in “Blown Covers,” a book by Ms. Mouly, include one of Lady Justice, holding her scales, but wearing a bondage ball gag, to reflect the Monica Lewinsky scandal.