Dr. Schrieffer, a graduate student of Dr. Bardeen’s, came up with the final piece of the puzzle in early 1957. He was in New York City at the time attending a physics conference when he had his crucial insight while riding the subway.

What he came up with was a mathematical expression that described how the pairs of electrons coalesced into one large clump, allowing them to move together without scattering — that is, without generating electrical resistance. By his own account, he had begun to “scribble down” the solution while on the train.

That December the full explanation for the electrons’ behavior was published in the journal Physical Review with a simple title, “Theory of Superconductivity.”

The explanation is now known as B.C.S. theory, after the initials of the last names of the three scientists who wrote the article: Dr. Bardeen; Leon N. Cooper, a postdoctoral researcher in Dr. Bardeen’s group; and Dr. Schrieffer.

During a celebration of the 50th anniversary of the paper, in 2007, Dr. Schrieffer described his insight with an analogy to a line of ice skaters, arm in arm. “If one skater hits a bump,” he said, the skater is “supported by all the other skaters moving along with it.”

The three received the Nobel Prize in Physics in 1972.

John Robert Schrieffer was born on May 31, 1931, in Oak Park, Ill., to John H. and Louise (Anderson) Schrieffer. His father was a pharmaceutical salesman who became a citrus grower in Florida.

After studying electrical engineering at the Massachusetts Institute of Technology for two years, Dr. Schrieffer switched to physics and graduated with a bachelor’s degree in 1953. He completed his doctorate at the University of Illinois in 1957.