The legend of the “unsolvable mathematical problem”


A student mistook examples of unsolved statistical problems for a homework assignment and solved it.



A legend surrounding the “unsolvable math problem” combines one of the ultimate academic wish-fulfilment fantasies – a student not only proves to be the brightest in his class, but also beats his professor and every other scholar in his subject – with a “positive thinking” motif, which is reflected in emerges from other urban legends: When people are free to pursue goals, unhindered by perceived limitations on what they can achieve, through the combined application of native talent and hard work, they can achieve some extraordinary feats:

A young college student worked hard in a high school math class, afraid he would fail. The night before the final, he studied so long that he slept through the exam morning.

Running into the classroom a few minutes late, he found three equations written on the blackboard. The first two were fairly easy, but the third seemed impossible. He worked feverishly until he found a method that worked—just ten minutes before the deadline—and he finished the assignments when the time came.

The student handed in his exam paper and left. That evening he received a call from his professor. “Do you realize what you did at the test today?” he shouted at the student.

“Oh no,” thought the student. I must have misunderstood the problem.

“You should only solve the first two tasks,” the professor explained. “The last one was an example of an equation that mathematicians have tried unsuccessfully to solve since Einstein. I discussed them with the class before starting the test. And you just solved it!”

And this special version is all the more interesting because it’s based on a real incident!

One day in 1939, George Bernard Dantzig, a graduate student at the University of California, Berkeley, was late for a graduate statistics class and found two problems on the blackboard. Not knowing that these were examples of “unsolved” statistical problems, he thought they were part of a homework assignment, wrote them down, and solved them. (The equations tackled by Dantzig are more precisely described not as unsolvable problems but as unproven statistical theorems for which he worked out proofs.)

Six weeks later, Dantzig’s statistics professor informed him that he had prepared one of his two “homework” proofs for publication, and Dantzig was recognized as a co-author for a second paper a few years later when another mathematician independently worked out the same solution for the second problem.

George Dantzig recounted his achievement in a 1986 interview for the College Math Journal:

It happened because one day in my freshman year at Berkeley I was late for one of the following [Jerzy] Neyman’s lessons. There were two problems on the board that I assumed were given as homework. I copied them down. A few days later I apologized to Neyman for taking me so long to do the homework – the problems seemed 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 pile of papers that I feared my homework would be lost there forever. About six weeks later, on a Sunday morning around eight o’clock, [my wife] Anne and I were woken up by someone banging on our front door. It was Neyman. He rushed in, papers in hand, all excited: “I’ve just written an introduction to one of your papers. Read it so I can send it for publication immediately.” For a minute I had no idea what he was talking about. Long story short, the problems on the board that I solved thinking they were homework were actually two famous unsolved problems in statistics. That was the first inkling I had that there was something special about them.

A year later, when I started thinking about a thesis topic, Neyman just shrugged and told me to put the two problems in a folder and he would accept them as my thesis.

However, the second of the two issues was not published until after World War II. It happened like this. Around 1950 I received a letter from Abraham Wald with the last proofs of one of his works in print Annals of Mathematical Statistics. Someone had just pointed out to him that the main result in his work was the same as the second “homework” solved in my thesis. I wrote back and suggested we publish together. He simply put my name as a co-author in the galley proof.

dr Dantzig also explained how his story passed into the realm of urban legends:

The other day while taking an early morning walk, I was greeted by Don Knuth as he rode by on his bicycle. He’s a colleague at Stanford. He paused and said, “Hey, George – I was in Indiana recently and heard a sermon about you at church. Do you know that you have an influence on Christians in Central America?” I looked at him in astonishment. “After the sermon,” he continued, “the minister came up to me and asked if I knew a George Dantzig at Stanford, because that was the name of the person his sermon was about.”

The origin of this pastor’s sermon can be traced to another Lutheran pastor, Reverend Schuler [sic] the Crystal Cathedral in Los Angeles. He shared his positive thinking ideas with me 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 about the power of positive thinking. Schuler’s published version was somewhat garbled and exaggerated, but essentially correct. The moral of his sermon was that if I had known that the problem was not homework but two famous unsolved problems in statistics, I probably would not have thought positively, would have been discouraged and never solved them.

The version of Dantzig’s story published by Christian televangelist Robert Schuller contained many embellishments and misinformation that have since been circulated in urban legend-like forms of the story, such as the one cited at the top of this page: Schuller converted the erroneous homework assignment into a ten-question “final exam.” (eight of which were real and two “unsolvable”), claimed that “even Einstein was unable to unravel the mysteries” of the two additional problems, and falsely claimed that Dantzig’s professor was so impressed that he wrote “Dantzig got 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 the University of Maryland in 1936 and a master’s degree from the University of Michigan in 1937 before earning his doctorate from UC Berkeley in 1946 (interrupted by World War II). He later served in the Air Force, accepted a position as a research mathematician with the RAND Corporation in 1952, became a 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 the 1990s. In 1975 dr. Dantzig was awarded the National Medal of Science by President Gerald Ford.

George Dantzig died on May 13, 2005 at the age of 90 at his Stanford home.

sightings: This legend is used as the plot structure in the 1997 film goodwill hunt. Also one of the early scenes in the 1999 film rushmore features the main character who dreams of solving the impossible question and gaining everyone’s approval. The legend of the “unsolvable mathematical problem”

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