On his weeks, there are no computers behind the counter, no swipe cards. The front-desk staff, which includes the owner and three Russian-speaking managers, uses punch cards and ledger books. Mr. Tuberman, 75, even disconnects the phone.

Customers can check whose week it is on the bath’s website.

The old system of alternating weeks has had various effects. One has been to thoroughly confuse people who arrive with a Groupon on the wrong week. It has also preserved, at least half the time, something of the old world of the shvitz.

On David weeks, as they are called, there is a new crowd. But on Boris weeks, the regulars are still “vaguely Slavic, vaguely paunchy,” as Charles Kramer, a 62-year-old lawyer and a regular for decades, described it, patting his belly. “You could blink your eyes and be back.”

On a recent Boris week, Warren Odze, a Broadway musician, had returned to the baths for the first time since the early 1980s. Many things were as he recalled: A little kitchen in the lobby served Russian-Jewish fare, and below ground, in a narrow space not much longer than a subway car, customers sat in towels in a misty Turkish steam room and a piping hot Russian sauna, received platzas, or therapeutic thrashings with oak branches, and dipped into an ice-cold pool.

Mr. Odze also noted the changes. The price of a day pass had quadrupled, to $40. There was a juice bar, a women’s locker room, women. Three decades ago, it was different. “There was nothing healthy about it,” he recalled. “It was steaks and big bagels. ‘Goodfellas,’ but the Jewish version.” In the basement then, Mr. Odze said, “it was old men basking and groaning, and I was scared of all of them.”

Mr. Odze said the bathhouse had become “trendy.” Still, he said, there was a remnant of the period he remembered. “I was happy to see there were a lot of, as they say among us Jews, schlubs.”