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Lennon-McCartney is likely one of the most famous songwriting credits in music. John Lennon and Paul McCartney wrote lyrics and music for almost 200 songs and The Beatles have sold hundreds of millions of albums. The story goes that the two Beatles agreed as teenagers to the joint credit for all songs they wrote, no matter the divide in work.

Over the years, Lennon and McCartney have revealed who really wrote what, but some songs are still up for debate. The two even debate between themselves - their memories seem to differ when it comes to who wrote the music for 1965's "In My Life."

If the songwriters' memories (perhaps tainted by the mind-altering era they were writing in) have failed, how can this mystery ever be solved? Well, we can get by with a little help from math.

Mathematics professor Jason Brown spent 10 years working with statistics to solve the magical mystery. Brown's the findings were presented on Aug. 1 at the Joint Statistical Meeting in a presentation called "Assessing Authorship of Beatles Songs from Musical Content: Bayesian Classification Modeling from Bags-Of-Words Representations."

Stanford University mathematician Keith Devlin breaks down how Brown figured it out. Read Devlin's edited conversation with NPR's Scott Simon below and listen at the audio link.

Scott Simon: I don't understand "bags-of-words representations," although, I've sometimes been called a bag of words by some.

Keith Devlin: We'll get to that. First of all, just to say that this is really serious stuff in terms of what was done. The three co-authors of this paper - there was someone called Mark Glickman who was a statistician at Harvard. He's also a classical pianist. Another person, another Harvard professor of engineering, called Ryan Song. And the third person was a Dalhousie University mathematician called Jason Brown. And you may recall back to 2008, you and I talked about him. He figured out how The Beatles created that striking opening chord in "Hard Day's Night." And when we did that piece, I actually said he was working on who wrote the music for "In My Life." And 10 years later, here we are talking about the discovery. It took him a long time, but he's now got it.

" Bags of words"? What are they?

It actually goes back to the 1950s. It's used by the computer scientists who created spam filters. What you do is you take a piece of text, and you ignore the grammar, you ignore the word order, and you just regard it as a collection of words. And once you've done that, you can count the frequencies of the different words in the bag of words. To do it for music, you had to get little snippets, and the way they did that was the team analyzed, I think, about 70 songs from Lennon and McCartney, and they found there were 149 very distinct transitions between notes and chords that are present in almost all Beatles songs. And those transitions will be unique to one person or the other person.
So they'd be bags of notes and chords.
Bags of notes and chords, pairs of notes and chords. Those are the little items, and you just count them.
Part of the confusion is that Paul McCartney said he wrote the music. John Lennon said Paul McCartney wrote a section of music. So what did this trio of mathematicians detect?
Cutting to the chase, it turns out Lennon wrote the whole thing. When you do the math by counting the little bits that are unique to the people, the probability that McCartney wrote it was .018 - that's essentially zero. In other words, this is pretty well definitive. Lennon wrote the music. And in situations like this, you'd better believe the math because it's much more reliable than people's recollections, especially given they collaborated writing it in the '60s with an incredibly altered mental state due to all the stuff they were ingesting.
I know what you are saying.
I would go with mathematics.
Keith, alright, I ask you - what about the artistic process of collaboration? Isn't it possible they were such close and accomplished collaborators that they inhaled a little bit of each other's technique and Lennon could write like McCartney and McCartney like John Lennon?
For sure. And that's why it's hard for the human ear to tell the thing apart. It's also hard for them to realize who did it and this is why actually the only reliable answer is the mathematics because no matter how much people collaborate, they're still the same people, and they have their preferences without realizing it. Lennon would use certain kinds of things over and over again. So would McCartney. It was the collaboration. Those two things come together that works, but they were still separate little bits. The mathematics isolates those little bits that are unique to the two people.
Web intern Emily Abshire contributed to this story.

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A Songwriting Mystery Solved: Math Proves John Lennon Wrote 'In My Life'

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Lennon-McCartney is likely one of the most famous songwriting credits in music. John Lennon and Paul McCartney wrote lyrics and music for almost 200 songs and The Beatles have sold hundreds of millions of albums. The story goes that the two Beatles agreed as teenagers to the joint credit for all songs they wrote, no matter the divide in work.

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John Lennon - Live in Madison Square Garden (1972 )

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John Lennon: The Full 1975 Harris interview. Beatles Break-up, reunion, George Martin, NYC & more

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Who was John Lennon? Everything You Need to Know

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John Lennon was a prominent English musician and one of the co-founders of the rock band The Beatles. This biography of John Lennon provides detailed information about his childhood, life, achievements, works & timeline.

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John Lennon left behind an indelible legacy of music that evoked a range of emotions. Songs like "Real Love" (with the Beatles), "Jealous Guy" and "Imagine," to name a few, were a testament to his musical genius.

On December 8, 1980, Lennon was fatally shot in front of his New York City home by crazed fan Mark David Chapman. Only two months earlier, on October 9, 1980, Lennon had celebrated his 40th birthday with his wife, Yoko Ono, and five-year-old son, Sean (who coincidentally shares the same birthday). The occasion was uniquely momentous for Lennon, as Ono had surprised both her husband and son with a sky-written message above their Manhattan apartment building: "Happy Birthday John & Sean - Love Yoko." That November, Lennon and Ono had released the collaborative project Double Fantasy, and only weeks before his tragic death, Lennon had been overjoyed by the news that the record had gone gold.

In an interview with Rolling Stone on December 5 - just three days before he was killed - Lennon had shared some eerily prophetic words of wisdom: "Give peace a chance, not shoot people for peace. All we need is love. I believe it."

Lennon was in good spirits on the morning of December 8, 1980, but sadly, his life would be tragically cut short before the day's end. Below, we present a timeline of the legendary musician's final hours in detail.

11 a.m. - Rolling Stone magazine photographer Annie Leibovitz arrives at Lennon's apartment for a photo shoot. Around the same time, Lennon returns home after getting a haircut - reportedly in a 1950s Teddy Boy style - at an Upper West Side barbershop near the Dakota, the building where he and Ono live. During the shoot, Leibovitz wants to take a photograph of Lennon and Ono together, both nude, but digresses after Ono says she's uncomfortable with being completely nude. She takes a single Polaroid of a completely clothed Ono lying on her back and a naked Lennon, who's kissing and embracing his wife in a fetal-like position. Upon looking at the shot, the three know that they've created something profound. Incredibly impressed with the image, the couple tells Leibovitz, "You've captured our relationship exactly."

(The now-iconic photo would go on to be printed on Rolling Stone's cover on January 22, 1981. In 2005, it's named the "top cover of the previous 40 years" by the American Society of Magazine Editors.)

12 p.m. - Paul Goresh, a friend of Lennon's, makes small talk outside of the Dakota with a stranger he'd encountered there minutes earlier: 25-year-old Chapman. Chapman, who's holding a copy of Double Fantasy, tells Goresh that he's from Hawaii and is awaiting Lennon's exit from the building in hopes that he'll autograph his album.

Photo: Bettmann/CORBIS

12:40 p.m. - Employees from San Francisco's RKO Radio, including famed radio personality Dave Sholin, arrive at the Dakota for an interview with Lennon to promote Double Fantasy. A smiling Lennon "does a little jump in the air" and in a welcoming fashion announces, "Well, here I am, folks, the show's ready to begin!" He speaks candidly about the lost revolutionary focus of the 1960s and his advocacy for world peace and feminism. "Maybe in the '60s we were naïve and like children and later everyone went back to their rooms and said, 'We didn't get a wonderful world of flowers and peace... The world is a nasty horrible place because it didn't give us everything we cried for,'" he said. "Right? Crying for it wasn't enough." When the interview comes to a close, Lennon remarks, "I consider that my work won't be finished until I'm dead and buried and I hope that's a long, long time."

During the interview, his son Sean, accompanied by nanny Helen Seaman, returns home after spending time on Long Island.

4:30 p.m. - Lennon and Ono have exited the Dakota with the RKO Radio crew and are awaiting a ride to take them to the Record Plant studio in Midtown where they plan to work on a new single, "Walking On Thin Ice." During this time Chapman, his Double Fantasy copy still in hand, approaches Lennon and extends the album. When the former Beatle asks the stranger if he wants him to sign it, Chapman nods timidly. Standing nearby and camera in hand, Goresh takes a couple of photos of the two men. RKO's limo arrives shortly thereafter, and Sholin offers to give Lennon and Ono a lift to the studio.

5 p.m. - Lennon and Ono arrive at Record Plant, where they begin working with producer Jack Douglas on their new song, the lyrics of which would later prove strangely prescient: Walking on thin ice/ I'm paying the price/ For throwing the dice in the air/ Why must we learn it the hard way/ And play the game of life with your heart? ... I may cry some day/ But the tears will dry whichever way/ And when our hearts return to ashes/ It'll be just a story/ It'll be just a story. Lennon plays a guitar piece for the track-his last musical recording-before the jovial couple exits the studio.

10:50 p.m. - Husband and wife leave the studio and head home to the Dakota. After exiting their limo, Lennon makes eye contact with the awkward young man he'd met hours earlier: Chapman. With his now-signed album in hand, he's after more than an autograph this time. Seconds later, Chapman pulls out a .38 handgun and fires five shots at the musician, hitting Lennon four times in the back and chest. Lennon somehow manages to continue walking, eventually collapsing in the front vestibule of the Dakota. Strewn around him are a number of cassettes that he'd been holding.

A terrified Ono enters the building screaming, "John's been shot!" Seconds later, building worker Jay Hastings alerts the police. Officer Steve Spiro arrives at the scene within minutes and Lennon is taken to Roosevelt Hospital, near Central Park on 59th Street.

11:15 p.m. - After doctors in Roosevelt's emergency room spend several frantic minutes trying to save Lennon, Ono is approached by the head ER doctor. He informs her that the medical staff was unable to resuscitate her husband. Lennon is officially pronounced dead. Ono is hysterical for several minutes afterward. "She refused to accept or believe that," Lynn recalled in a 2005 interview with the Washington Post. "For five minutes, she kept repeating, 'It's not true. I don't believe you. You're lying.'" After regaining her composure, Ono asks Lynn to wait to publicly announce the news so that she can tell Sean.

ABC's Howard Cosell is first to break the news that Lennon has died, interrupting a Monday Night Football game between the New England Patriots and Miami Dolphins. Cosell also announces that a special edition of Nightline will air 30 minutes after the game, providing additional information on the musician's death.

Photo: NY Daily News Archive via Getty Images

Following the report - and continuing for several days thereafter - thousands of Lennon fans unite near the Dakota to mourn the late musician. The public response to Lennon's murder is noted as the first of its magnitude since the assassination of President John F. Kennedy. Vigils are held in numerous locations worldwide.

Events are held around the world each December to commemorate the rock icon's death and everlasting impact.

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John Lennon's Death: A Timeline of Events

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John Lennon left behind an indelible legacy of music that evoked a range of emotions. Songs like "Real Love" (with the Beatles), "Jealous Guy" and "Imagine," to name a few, were a testament to his musical genius. On December 8, 1980, Lennon was fatally shot in front of his New York City home by crazed fan Mark David Chapman.

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Receding silver hair tied back in a stringy pony-tail; thick glasses perched on his thin nose, a wiry old man steps creakily into the spotlight, blinking into the vast space where an audience roar their love. It is John Lennon, veteran rocker, political activist, acclaimed author and winner of the Nobel Peace Prize for his role as figurehead of the anti-war movement that did so much to heal global wounds during the first turbulent decade of the 21st century.

What would he say, what would he play on this, the occasion of his 75th birthday at Madison Square Garden? Would the spotlight come up on a white piano, with some sharp words about how this song remains as true as when he wrote it in 1971, while the opening chords to Imagine ring out? Might he cheekily congratulate new Labour leader Jeremy Corbyn on the size of his marrow and strum Working Class Hero on acoustic guitar? Perhaps he would invite "an old girlfriend" to share the moment, with Paul McCartney entering stage righ t, so they could play the first song they wrote together back in 1957, One After 909, thrilling to the simple pleasures of throwaway lyrics and perfect harmony: "I said move over once, move over twice, come on baby don't be cold as ice." Or perhaps this imaginary concert would open with some unwritten classic from the second half of Lennon's career that we have never heard and never will.

Credit: REX

It is impossible to know what Lennon, born on this day in 1940, might be doing had he lived because he was such a singular character, who always went his own unpredictable way. One of his greatest achievements was surely to break the mould of pop stardom and leave something in its place that was more malleable and individualised, encouraging those in his wake to create their own templates for showbusiness careers. His supreme talent, extreme fame and tragic end have led to him being lionised, deified and mythologised but his music and personality still resonate so strongly because of his art of honesty, his instinct for cutting through the veil and revealing the real complexities and contradictions of the human condition. Lennon was the first truly modern pop star, where there was no distinction between his art and his life.

The Beatles changed everything, as actors and catalysts in the Sixties pop cultural revolution. And for all the talent of that extraordinary band, Lennon was their leader, the rebellious, creative firebrand who ignited the spark. Everyone who knew him pre-fame attests to his charisma, intelligence, humour and boundary-breaking sense of adventure, qualities that the English education system failed to nurture but, in the white hot furnace of rock and roll, facilitated the fastest evolving, most audaciously accomplished career in pop history.

Following the first rush of glorious harmonic songcraft, Lennon was responsible for some of the most florid, baroque psychedelic pop ever heard, tracks like Strawberry Fields Forever, Lucy In The Sky With Diamonds and I Am The Walrus, in which new musical palettes underpinned wild, surrealist lyrics. But if we were to speculate about music he might be making now, it would be unlikely to be trendy, experimental or even particularly grand in scale. 1966's Tomorrow Never Knows may have been a big influence on techno via the Chemical Brothers 1996 homage Setting Sun, but I doubt Lennon would be interested in the synthetic rush of EDM, nor would you find him engaged by the formulaic polish and trivial content of today's pop. He might have appreciated the way Ed Sheeran and Taylor Swift craft pop songs out of real emotional situations but he was searching for something that dug deeper.

Lennon really found his particular groove when he started paring things down, musically and lyrically. Early classics like Help, Girl and In My Life opened up into the almost brutal honesty of Yer Blues and I Want You (She's So Heavy). He is the first of the truly confessional singer-songwriters. Has there ever been a song as guiltily apologetic as Jealous Guy? As painfully vulnerable as Mother? As angrily demanding as Gimme Some Truth?

He might have taken to hip hop, with its wordiness and attack, responding to the bravado, braininess and peculiar vulnerability of Kanye West and tough truth telling of Jay Z. But at heart Lennon was a rocker, drawn to music that reflected his teenage obsessions, so apparent on his joyous 1975 album Rock N Roll. He would have liked Nirvana and The White Stripes, although Oasis might have been deemed too indebted to his own back catalogue. His idealistic nature and love of anthems would have endeared him to U2 and Coldplay and he would surely have jumped enthusiastically into the chorus line of Do They Know It's Christmas and turned up with his guitar at Live Aid.

Credit: 2015 Kevin Mazur/Kevin Mazur

I wonder if he would have taken to Twitter, such a robust forum for comment, humour and campaigning? Social media might have been the perfect medium for him. Lennon was determined to reveal himself as a human being, flaws and all.

Some biographers have dwelt on his hypocrisies, his temper, cruelty and early misogynism, but Lennon was harder on himself than any critic could ever be. He could have been an important character online, wrestling with issues of self-exposure and public shaming.

Had Lennon lived, the Beatles would have got back together, as every band seems to in the end. They were a gang from youth, with enduring affection.

There would have been at least one more album, and you have to believe it would have been great because there was too much talent and pride in their ranks to accept anything less. But I doubt they would have become a vintage touring act. Lennon did not have the showbusiness impulse that has kept McCartney on the road. Five quiet years as a house husband in the late Seventies seemed to suit him, and he is more likely to have been an occasional public figure, rather than a perpetual presence like his friends Bob Dylan and Elton John.

Credit: GEORGE KONIG

There would have surely been more books. And there would have been more music, not all of it worth listening to. Lennon's solo career was erratic and impulsive, and some albums are truly terrible, like 1972's Some Time In New York City and 1975's Shaved Fish. But just as 1980's Double Fantasy is a touching evocation of the mundane joy of domestic family existence, who wouldn't want to hear Lennon's occasional dispatches from middle and old age?

This is what the world was robbed of, on that terrible day in New York, December 8, 1980, when Lennon was gunned down. Yet in 20 years of music making, he accomplished so much that he can barely be missed, because he remains ever present in pop culture. Perhaps, at 75, we would find him, as he imagined himself, enjoying a quiet retirement with Yoko Ono. "I hope we're a nice old couple living off the coast of Ireland or something like that, looking out at our scrapbook of madness," he said in 1970.

As he sang that same year, in his sonorous myth-busting masterpiece God, "the dream is over".

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John Lennon: what would he be like at 75?

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Receding silver hair tied back in a stringy pony-tail; thick glasses perched on his thin nose, a wiry old man steps creakily into the spotlight, blinking into the vast space where an audience roar their love.

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John Lennon and Bob Dylan´s Favorite Subject

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John Lennon and Bob Dylan´s Favorite Subject

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Physics

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