Then Ben Kwon posted the team’s first victory. He played an opening similar to the Fried Liver, but not exactly, he said, and beat an opponent rated higher than he was. He sailed across the room to his mother, Michelle Park.

Ms. Park said she had been surprised by how much there was to the chess subculture — how many tournaments, how many players. She mentioned several other tournaments going on at other schools the same weekend. “We didn’t know this existed before,” she said.

By midafternoon, things were looking better for Lower Lab. In the fifth-floor library, where students with ratings below 800 matched off, Noah Gillston, a quiet first grader with a mop of sandy hair, stood up to move his knight across the board. He was on a roll, winning his first three games.

Just before 2:30, the beginner players marched into the team room carrying trophies. More wins followed. Noah won all four games, leading his group to victory. Mr. Brain comforted a father whose son had lost some winnable games. “Those mistakes go away,” Mr. Brain told the father, before turning to the son. “I’m proud of you,” he said. “You’re doing well.”

Seven hours into the tournament, families in the less advanced groups — whose games tend to be shorter — were making their way out into the late-afternoon sunshine. Ms. Xiao awaited the return of her two sons, whose games were still in progress. “Now’s the real work,” she said: cooking dinner for five, supervising Sunday night homework and maybe stealing a quiet moment with her husband. “Then I clean up and get to bed at 12 o’clock, basically.”

Monday morning everything would start up again — the tutors, the chess puzzles, the competitive games between siblings. And in less than a month, some will go to the national championships in Nashville.

Mr. Marinis, who is planning to fly to Nashville this year (last year he and others drove), was bound for ice cream and the birthday party, his three sons in tow. His older son, Pearce, had won three of his four games, raising his rating to a tantalizing 999.

“I use these as lessons for homework,” Mr. Marinis said of the tournament. “I say, ‘If you can do what you did on Sunday, look, this math is not going to be a problem.’”