When she started a math contest with a bonus question challenging students to create a new proof for the Pythagorean theorem using trigonometry, teacher Michelle Blouin Williams didn’t expect anyone to complete the task.

“I was just looking for some ingenuity,” she said, per CBS News.

Calcea Johnson and Ne’Kiya Jackson, however, blew Williams’ expectations out of the water by figuring it out in 2023. The teens were seniors at St. Mary’s Academy in New Orleans, a prestigious Catholic school for girls which has maintained a 100% acceptance rate to colleges and 100% graduation rate for 17 years, CBS News reported.

They appeared in an episode of CBS News’ “60 Minutes” on Sunday to talk about their achievement.

### How did the teens find the answer?

While they were motivated initially by the math competition’s $500 prize, an internal drive to finish what they started manifested when they reached the tricky bonus question. For two months, the high school seniors worked tirelessly to finish their proof.

CeCe Johnson, Calcea’s mother, told “60 Minutes,” “It was pages and pages and pages of, like, over 20 or 30 pages for this one problem.”

Her father, Cal Johnson, added, “Yeah, the garbage can was full of papers, which she would, you know, work out the problems and — if that didn’t work she would ball it up, throw it in the trash.”

When they finished, teachers at St. Mary’s recognized the importance of their work and submitted their proof to the American Mathematical Society for recognition at a conference in March 2023, where the students presented their work.

### What is the Pythagorean theorem and what’s a proof?

In essence, the mathematical theorem states that knowing the lengths of two sides of a right triangle enables you to figure out the length of the third using this formula: a² + b² = c².

It’s associated with Greek mathematician Pythagoras, but evidence suggests it was known earlier, in Babylon and Iron Age India, per Britannica. Its practical uses include construction and architecture, two-dimensional navigation, and surveying.

A mathematical proof is exactly what it sounds like: reasoning that proves a mathematical theorem is true. American mathematician Daniel Kane explains proofs as being like essays, but using math.

### Why is Calcea Johnson and Ne’Kiya Jackson’s work significant?

According to the “60 Minutes” episode, “there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible.”

In 1927, mathematician Elisha Loomis said as much in his book, “The Pythagorean Proposition.” Loomis argued that there could be no trigonometric proof of the theorem because it would be circular.

Stuart Anderson, a professor emeritus of mathematics at Texas A&M University–Commerce, told Scientific American, “A lot of the basic trig ‘identities’ are nothing more than Pythagoras’ theorem.”

So, because trigonometric functions are based upon the Pythagorean theorem, using them reflexively to prove the theorem would be akin to going in circles and a fundamental mathematical error, Loomis argued.

According to Scientific American, the teens refuted this in their presentation in 2023 and said that “a trigonometric identity called the law of sines didn’t depend on the Pythagorean theorem and that they could use it to prove the theorem.”

Calcea and Ne’Kiya have joined an extremely small group who’ve accomplished the same feat, including mathematician Jason Zimba, who successfully created a new proof in 2009. The two submitted their proof for final peer review this spring and continue to work on creating more proofs.

### How did the world respond to their accomplishment?

The teens were given the keys to the city of New Orleans and a commendation from the governor of Louisiana, among other public recognitions.

While their achievement “blew up,” as Ne’Kiya described it, the two students remain humble, and laughed at being called geniuses.

When news of their accomplishment broke, some people seemed to be shocked and dismissed the news as fake, St. Mary’s president Pamela Rogers said in the interview.

“They were saying, ‘Oh, they could not have done it. African Americans don’t have the brains to do it.’ ... People — have a vision of who can be successful. And — to some people, it is not always an African American female. And to us, it’s always an African American female.”

When interviewer Bill Whitaker asked why they thought there’d been such a response, Ne’Kiya said, “Probably because we’re African American, one. And we’re also women. So I think — oh, and our age. Of course our ages probably played a big part.”

“I’d like to actually be celebrated for what it is. Like, it’s a great mathematical achievement,” she continued.