OSSINING, N.Y. — During lunch hour, the hallways of Ossining High School have a kind of barely contained chaos. Whistles bleed from the gym, students squeeze every last minute of freedom before they’re due back in class. Even the library, where Dan McQuaid sat with two of his science teachers two weeks ago, buzzes and hums.

None of this hubbub drew even the tiniest acknowledgment from these three. Instead they were there to talk about Dan’s cancer research.

A 17-year-old senior, Dan is one of 40 finalists in the nationwide Intel Science Talent Search. The winner will be announced Tuesday night in Washington, and when a reporter asked Dan about the pressure, one of his teachers, Angelo Piccirillo, stepped in protectively.

Dan, he said, has already earned distinction enough: He is the first student from Ossining High ever to reach the finals. “It’s all gravy from now on,” Mr. Piccirillo added with a smile.

That kind of gentle encouragement undoubtedly helped Dan advance to where he is. Yet this has hardly been a stress-free week for the Intel finalists, shortlisted from a group of 300 semifinalists. They have been in Washington since Wednesday and Thursday to present their projects to a judging panel and the public. The top 10 finalists will receive prizes of $20,000 to $100,000; the other 30 will receive $7,500.

And that may just be the beginning. If history is any indication, several of these young men and women will go on to greater fame: Since the science competition’s inception in 1942, as the Westinghouse Science Talent Search, seven of its alumni have won Nobel Prizes and 11 have received MacArthur “genius” awards.

The 40 finalists were culled from more than 1,700 applications, which are due in November. (The 2012 deadline was just days after Hurricane Sandy paralyzed much of the East Coast, and students affected by the storm got an extension.)

The competition is run by the Society for Science and the Public and is financed by the Intel Corporation, through its Intel Foundation. When Westinghouse ceased sponsorship 15 years ago, Intel took over, primarily “to change the conversation about young scientists in the U.S.,” said Wendy Hawkins, the foundation’s executive director.

Instead of the “endless drivel” about stereotypes of scientists as geeks and absent-minded professors, Ms. Hawkins added, “we want to focus on celebrating and supporting the life-changing work these young scientists are doing and will do throughout their careers.”

One of Intel’s first changes was to significantly increase the prize money. “Money does attract attention,” Ms. Hawkins said. “We want students like these to be just as celebrated as are the star athletes and entertainers in their schools.”

In Washington, finalists are judged by scientists from universities across the country (this reporter’s late father was a judge), whose knowledge outside their fields is sometimes outstripped by many of the finalists.

What the judges are looking for, however, is not limited to a project itself.

“Our goal is to find future leaders in science,” said the panel’s chairman, David Marker, a professor of mathematics, statistics and computer science at the University of Illinois at Chicago.

In four 15-minute interviews with groups of three judges, finalists are not only asked about their projects but are also tested on basic science knowledge. For example, Dr. Marker said, they might be asked to “diagram a plant cell and explain the functions of some of the organelles.”

And then there are the questions that no one can prepare for: “One of my favorites from a former judge was ‘Tell me about the universe,’ ” Dr. Marker said. “Another might ask them to predict the future of the newspaper industry.” The idea is to get “some indication of how they think.”

Tuesday night’s announcement will close this particular chapter of these students’ lives, but this is a story with much to come. Here are four of the finalists and their projects.