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The Legend of the 'Unsolvable Math Problem'

A student mistook examples of unsolved math problems for a homework assignment and solved them., david mikkelson, published dec 3, 1996.

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A legend about the "unsolvable math problem" combines one of the ultimate academic wish-fulfillment fantasies — a student not only proves himself the smartest one in his class, but also bests his professor and every other scholar in his field of study — with a "positive thinking" motif that turns up in other urban legends: when people are free to pursue goals unfettered by presumed limitations on what they can accomplish, they just may manage some extraordinary feats through the combined application of native talent and hard work:

A young college student was working hard in an upper-level math course, for fear that he would be unable to pass. On the night before the final, he studied so long that he overslept the morning of the test. When he ran into the classroom several minutes late, he found three equations written on the blackboard. The first two went rather easily, but the third one seemed impossible. He worked frantically on it until — just ten minutes short of the deadline — he found a method that worked, and he finished the problems just as time was called. The student turned in his test paper and left. That evening he received a phone call from his professor. "Do you realize what you did on the test today?" he shouted at the student. "Oh, no," thought the student. I must not have gotten the problems right after all. "You were only supposed to do the first two problems," the professor explained. "That last one was an example of an equation that mathematicians since Einstein have been trying to solve without success. I discussed it with the class before starting the test. And you just solved it!"

And this particular version is all the more interesting for being based on a real-life incident!

One day in 1939, George Bernard Dantzig, a doctoral candidate at the University of California, Berkeley, arrived late for a graduate-level statistics class and found two problems written on the board. Not knowing they were examples of "unsolved" statistics problems, he mistook them for part of a homework assignment, jotted them down, and solved them. (The equations Dantzig tackled are more accurately described not as unsolvable problems, but rather as unproven statistical theorems for which he worked out proofs.)

Six weeks later, Dantzig's statistics professor notified him that he had prepared one of his two "homework" proofs for publication, and Dantzig was given co-author credit on a second paper several years later when another mathematician independently worked out the same solution to the second problem.

George Dantzig recounted his feat in a 1986 interview for the College Mathematics Journal :

It happened because during my first year at Berkeley I arrived late one day at one of [Jerzy] Neyman's classes. On the blackboard there were two problems that I assumed had been assigned for homework. I copied them down. A few days later I apologized to Neyman for taking so long to do the homework — the problems seemed to be a little harder than usual. I asked him if he still wanted it. He told me to throw it on his desk. I did so reluctantly because his desk was covered with such a heap of papers that I feared my homework would be lost there forever. About six weeks later, one Sunday morning about eight o'clock, [my wife] Anne and I were awakened by someone banging on our front door. It was Neyman. He rushed in with papers in hand, all excited: "I've just written an introduction to one of your papers. Read it so I can send it out right away for publication." For a minute I had no idea what he was talking about. To make a long story short, the problems on the blackboard that I had solved thinking they were homework were in fact two famous unsolved problems in statistics. That was the first inkling I had that there was anything special about them. A year later, when I began to worry about a thesis topic, Neyman just shrugged and told me to wrap the two problems in a binder and he would accept them as my thesis. The second of the two problems, however, was not published until after World War II. It happened this way. Around 1950 I received a letter from Abraham Wald enclosing the final galley proofs of a paper of his about to go to press in the Annals of Mathematical Statistics. Someone had just pointed out to him that the main result in his paper was the same as the second "homework" problem solved in my thesis. I wrote back suggesting we publish jointly. He simply inserted my name as coauthor into the galley proof.

Dr. Dantzig also explained how his story passed into the realm of urban legendry:

The other day, as I was taking an early morning walk, I was hailed by Don Knuth as he rode by on his bicycle. He is a colleague at Stanford. He stopped and said, "Hey, George — I was visiting in Indiana recently and heard a sermon about you in church. Do you know that you are an influence on Christians of middle America?" I looked at him, amazed. "After the sermon," he went on, "the minister came over and asked me if I knew a George Dantzig at Stanford, because that was the name of the person his sermon was about." The origin of that minister's sermon can be traced to another Lutheran minister, the Reverend Schuler [sic] of the Crystal Cathedral in Los Angeles. He told me his ideas about thinking positively, and I told him my story about the homework problems and my thesis. A few months later I received a letter from him asking permission to include my story in a book he was writing on the power of positive thinking. Schuler's published version was a bit garbled and exaggerated but essentially correct. The moral of his sermon was this: If I had known that the problem were not homework but were in fact two famous unsolved problems in statistics, I probably would not have thought positively, would have become discouraged, and would never have solved them.

The version of Dantzig's story published by Christian televangelist Robert Schuller contained a good deal of embellishment and misinformation which has since been propagated in urban legend-like forms of the tale such as the one quoted at the head of this page: Schuller converted the mistaken homework assignment into a "final exam" with ten problems (eight of which were real and two of which were "unsolvable"), claimed that "even Einstein was unable to unlock the secrets" of the two extra problems, and erroneously stated that Dantzig's professor was so impressed that he "gave Dantzig a job as his assistant, and Dantzig has been at Stanford ever since."

George Dantzig (himself the son of a mathematician) received a Bachelor's degree from University of Maryland in 1936 and a Master's from the University of Michigan in 1937 before completing his Doctorate (interrupted by World War II) at UC Berkeley in 1946. He later worked for the Air Force, took a position with the RAND Corporation as a research mathematician in 1952, became professor of operations research at Berkeley in 1960, and joined the faculty of Stanford University in 1966, where he taught and published as a professor of operations research until the 1990s. In 1975, Dr. Dantzig was awarded the National Medal of Science by President Gerald Ford.

George Dantzig passed away at his Stanford home at age 90 on 13 May 2005.

Sightings:   This legend is used as the setup of the plot in the 1997 movie Good Will Hunting . As well, one of the early scenes in the 1999 film Rushmore shows the main character daydreaming about solving the impossible question and winning approbation from all.

Albers, Donald J. and Constance Reid.   "An Interview of George B. Dantzig: The Father of Linear Programming."     College Mathematics Journal.   Volume 17, Number 4; 1986   (pp. 293-314).

Brunvand, Jan Harold.   Curses! Broiled Again!     New York: W. W. Norton, 1989.   ISBN 0-393-30711-5   (pp. 278-283).

Dantzig, George B.     "On the Non-Existence of Tests of 'Student's' Hypothesis Having Power Functions Independent of Sigma."     Annals of Mathematical Statistics .   No. 11; 1940   (pp. 186-192).

Dantzig, George B. and Abraham Wald.   "On the Fundamental Lemma of Neyman and Pearson."     Annals of Mathematical Statistics .   No. 22; 1951   (pp. 87-93).

Pearce, Jeremy.   "George B. Dantzig Dies at 90."     The New York Times .   23 May 2005.

By David Mikkelson

David Mikkelson founded the site now known as snopes.com back in 1994.

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The Northern Star Magazine Online

Student solves unsolvable problems, student solves ‘unsolvable’ statistical problems.

Mr. George Bernard Dantzig, a doctoral candidate at the University of California (USC), Berkeley in 1939, arrived late for his graduate-level statistics class and found two problems written upon the blackboard. Not knowing that they were examples of ‘unsolvable’ statistical problems, he mistook them for a homework assignment, jotted them down and solved them. The equations that he solved are actually more accurately described best as unproved statistical theorems, rather than unsolvable problems.

Assumed They’d Been Assigned For Homework

In 1986, George recalled the event in a College Mathematics Journal interview: “It happened because during my first year at Berkeley, I had arrived late one day for a Jerzy Neyman class. On the blackboard there were two problems. I assumed they’d been assigned for homework, so I copied them down. A few days later I apologized to Neyman for taking so long, but the problems seemed to be harder than usual. I asked him if he still wanted them. He said yes, and told me to put them on his desk.

All Excited

About six weeks later, around eight o’clock on a Sunday morning, we were woke by someone banging on our front door. It was Neyman. He rushed in with the papers in hand, all excited. “I’ve just written an introduction to one of your papers. Please read it so I can send it off right away for publication.” For a minute I had no idea what he was talking about. To make a long story short, the problems on the blackboard that I had solved, thinking that they were homework, were in fact two famous unsolvable math problems in statistics. That was the first inkling I had that there was anything special about them.”

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This Guy Says He Solved the Most Controversial Open Problem in Math

He has the (600-page) proof. But other mathematicians have their pitchforks.

• Math proofs can go through many iterations and attempts before they're correct.
• The abc conjecture dates to the 1980s and is an extension of Fermat's last theorem .

Has one of the major outstanding problems in number theory finally been solved? Or is the 600-page proof missing a key piece? The verdict isn’t in yet, but the proof, at least, will finally appear in a peer-reviewed journal.

However, there’s just one catch: the mathematician himself, Shinichi Mochizuki, is one of the journal’s seniormost editors.

For those outside of academic mathematics, it’s hard to explain both how weirdly dramatic this situation has been and just how huge a successful proof of the abc conjecture would be. Nature compares it to the 1994 proof of Fermat’s last theorem , which was a gigantic landmark in math—and, at 26 years old, the most recent one on the same level of achievement.

More From Popular Mechanics

Both proofs also involve a unique algebraic category referred to as diophantine problems. These are equations that people seek to find integer solutions for, like the special cases of the Pythagorean theorem called Pythagorean triples . When you’re studying an equation and you’re only interested in solutions that are whole numbers, this is a diophantine problem.

The abc conjecture has some commonalities with the Pythagorean theorem and other diophantine problems, involving a relationship between an a and b added together to a resulting c . Can these numbers be taken to high exponents and still have a demonstrable relationship? This is what mathematicians have been trying to prove since mathematicians first observed it in the mid 1980s. And, in fact, the conjecture is an extension of Fermat’s last theorem.

More from Nature:

The abc conjecture expresses a profound link between the addition and multiplication of integer numbers. Any integer can be factored into prime numbers, its ‘divisors’: for example, 60 = 5 x 3 x 2 x 2. The conjecture roughly states that if a lot of small primes divide two numbers a and b , then only a few, large ones divide their sum, c .

Mochizuki first published a Michener-novel-length proof of the abc conjecture in 2012, when he unceremoniously dumped 500 pages online and said he’d proven it. But this isn’t anyone’s first rodeo. Previous public attempts to prove the conjecture have been shown to have errors. That isn’t unusual in the process of proving complex and landmark ideas, where different scientists often iterate one new step at a time based on what their colleagues are doing.

When Mochizuki’s proof first appeared, other mathematicians reeled at both the idea of a proof of the abc conjecture and the baffling obscurity of the work itself. Mochizuki had invented a phantom scaffolding of abstract notions that shadow real mathematical ideas and notation in order to hang his very long proof upon that scaffold. In a way, trying to decipher the proof first required learning an entirely new system and notation.

To date, no one has fully understood this proof enough to validate it and communicate its structure and logical flow to others. Mochizuki himself is reclusive and hasn't really helped to illuminate his obscure mechanisms.

A few years ago, mathematicians were upset to learn that the proof was to be published in a peer-reviewed journal, and in 2018, two prominent fellow mathematicians said they were sure the proof was wrong.

The rumors of publication were false then, or perhaps rescinded after the outcry from mathematicians. But now, the proof will indeed appear in some kind of special issue of Publications of the Research Institute for Mathematical Sciences (RIMS), following what they say is the same rigorous peer review they would do for anyone. It’s not just the scandalous proof at issue here—it’s the fact that Mochizuki is RIMS ’s chief editor.

Perhaps having the 500-page behemoth in print will revive the debate and bring any flaws to the surface, conclusively, once and for all. It’s hard to say for sure when the mathematics ideas themselves are so far from the norm as to seem like outsider math . Anyone who has fielded submissions for technical or mathematical publication has received gigantic, complicated documents whose authors insist they’ve proven something huge.

The truth is rarely, well, so straightforward.

Caroline Delbert is a writer, avid reader, and contributing editor at Pop Mech. She's also an enthusiast of just about everything. Her favorite topics include nuclear energy, cosmology, math of everyday things, and the philosophy of it all. 

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