## 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

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## Dantzig's unsolved homework problems

An event in George Dantzig 's life became the origin of a famous story in 1939 while he was a graduate student at UC Berkeley. Near the beginning of a class for which Dantzig was late, professor Jerzy Neyman wrote two examples of famously unsolved statistics problems on the blackboard. When Dantzig arrived, he assumed that the two problems were a homework assignment and wrote them down. According to Dantzig, the problems "seemed to be a little harder than usual", but a few days later he handed in completed solutions for the two problems, still believing that they were an assignment that was overdue.

What were the two unsolved problems which Dantzig had solved?

## 2 Answers 2

I think the two problems appear in these papers:

Read more at http://www.snopes.com/college/homework/unsolvable.asp#6oJOtz9WKFQUHhbw.99

- 3 $\begingroup$ A link to the first paper: projecteuclid.org/download/pdf_1/euclid.aoms/1177731912 $\endgroup$ – Michael Scott Asato Cuthbert Aug 27, 2014 at 15:08
- 18 $\begingroup$ Could you at least write the problems and/or results here? $\endgroup$ – BlueRaja - Danny Pflughoeft Jul 27, 2017 at 15:31
- 3 $\begingroup$ A link to the second paper: projecteuclid.org/DPubS?handle=euclid.aoms/… The problems in the two papers are complex enough that quoting them here is impractical. $\endgroup$ – Dale Mar 30, 2019 at 0:55
- 1 $\begingroup$ A link to the second paper: projecteuclid.org/euclid.aoms/1177729695 $\endgroup$ – hongsy May 25, 2020 at 9:29

In an unusual amount of detail, aimed at those with no statistical knowledge:

- ("Null hypothesis.") The new strain's average yield is 100 units.
- ("Alternative hypothesis.") The new strain's average yield is 105 units.

- A. ...the probability that we correctly keep using the old strain.
- B. ...the probability that we mistakenly switch to the new strain.
- C. ...the probability that we mistakenly keep using the old strain.
- D. ...the probability that we correctly switch to the new strain.

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

## Assumed They’d Been Assigned For Homework

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## Remembering George Dantzig: The real Will Hunting

Photo credit: oman Mager on Unsplash

- One of the iconic scenes from Good Will Hunting shows Matt Damon's character anonymously solving a nigh-impossible math problem on a blackboard at the university where he works as a janitor.
- This story, while modified for the purposes of the film, actually happened.
- George Dantzig, who would later become a famous mathematician, was late to his graduate statistics class one day when he saw two statistical problems on a blackboard that he mistook for homework.

## Good Will Hunting Scene Math Problem

## Luckily late to class

## How the story spread

## Reality Decoded

## 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.

## How many problems are waiting for us to solve them simply because we think we can’t?

## What If You Haven’t Heard The Most Important Message Of Your Life

What people don’t talk about, 2 comments ›.

wow this is amazing. Sometimes greatness are discovered by mistake. 🙂

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## TIL the mathematician George Dantzig solved two of the most famous problems of statistics, because he came into class too late to hear that they were supposed to be unsolvable

Oh jeez. I just struggled with that last semester. Large load and linear algebra took a back seat.

Senior in industrial engineering here, we basically pray to Dantzig every morning before lecture

Wow, i had no idea those were unsolvable, and simplex is so easy to apply

airline scheduling

so you're saying it doesn't actually work worth a damn?

Relevant text, from Dantzig himself:

During my first year at Berkeley I arrived late one day to one of Neyman's classes. On the blackboard were two problems which 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 to do than usual. I asked him if he still wanted the work. 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, 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 which 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.

"Seemed to be a little harder than usual..."

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Oftentimes, this is how intelligence is depicted by the media, as a trait that can only be genetically inheritable. Will was born a “genius.” Dantzig was “destined” to make mathematical breakthroughs.

Many of us may have grown up hearing consoling phrases from our parents such as, “It’s OK if you’re not good at science; not everyone is good at it.”

As Bill Gates, a strong advocate of Dweck’s work, writes in a review of her book, “My only criticism of the book is that Dweck slightly oversimplifies for her general audience. … most of us are not purely fixed-mindset people or growth-mindset people. We’re both.”

George Dantzig put in a lot of effort to solve those problems,

Will Hunting is Hollywood made

and that for every Alexander Bell, there’s an Elisha Gray.

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## PhD dissertations that solve an established open problem

I ask the moderators to consider this question as a wiki question.

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- 17 $\begingroup$ Godel's thesis comes to mind. $\endgroup$ – user2277550 Apr 24, 2018 at 7:39
- 77 $\begingroup$ Don't all PhD theses solve open problems...? $\endgroup$ – Najib Idrissi Apr 24, 2018 at 7:42
- 20 $\begingroup$ @NajibIdrissi: I understood this question to be about open problems in the narrow sense of a precise mathematical question that at least some researcher(s) have previously articulated and tried to answer , e.g. “what is the dimension of such-and-such space?”. Most theses I know only solve open problems only in a much broader sense: open-ended questions that researchers may have wondered about, like “what can we say about the homology of such-and-such space?”, or “can we develop a useful theory of homotopy-coherent diagrams in such-and-such setting?”. $\endgroup$ – Peter LeFanu Lumsdaine Apr 24, 2018 at 9:04
- 9 $\begingroup$ @NajibIdrissi: Not always: sometimes a thesis provides a new insight into an already solved problem, or a better way to solve it. Tate's thesis is the first to come to mind. $\endgroup$ – Alex M. Apr 24, 2018 at 11:27
- 6 $\begingroup$ Eric Larson proved the Maximal Rank Conjecture (a significant problem in the algebraic geometry of curves) in his MIT PhD thesis this year: math.mit.edu/research/graduate/thesis-defenses-2018.php $\endgroup$ – Sam Hopkins Apr 24, 2018 at 13:32

## 23 Answers 23

I find George Dantzig's story particularly impressive and inspiring.

While he was a graduate student at UC Berkeley, near the beginning of a class for which Dantzig was late, professor Jerzy Neyman wrote two examples of famously unsolved statistics problems on the blackboard. When Dantzig arrived, he assumed that the two problems were a homework assignment and wrote them down. According to Dantzig, the problems "seemed to be a little harder than usual", but a few days later he handed in completed solutions for the two problems, still believing that they were an assignment that was overdue. Six weeks later, Dantzig received a visit from an excited professor Neyman, who was eager to tell him that the homework problems he had solved were two of the most famous unsolved problems in statistics. Neyman told Dantzig to wrap the two problems in a binder and he would accept them as a Ph.D. thesis.

- 2 $\begingroup$ Thank you for this very interesting answer on George Dantzig's story. $\endgroup$ – Ali Taghavi Apr 24, 2018 at 7:54
- 9 $\begingroup$ For those who like the story, a Snopes article contains some interesting excerpts from an interview with George Dantzig: snopes.com/fact-check/the-unsolvable-math-problem $\endgroup$ – JohnEye Apr 24, 2018 at 10:20
- 10 $\begingroup$ That's freaking crazy. $\endgroup$ – iammax Apr 24, 2018 at 18:45
- 4 $\begingroup$ If the problems were solved concurrently, why were they published 11 years apart? $\endgroup$ – Michael Sep 25, 2018 at 20:39
- 3 $\begingroup$ @Michael --- Wikipedia explains that "Years later another researcher, Abraham Wald, was preparing to publish an article that arrived at a conclusion for the second problem, and included Dantzig as its co-author when he learned of the earlier solution." $\endgroup$ – Carlo Beenakker Sep 26, 2018 at 0:31

- 7 $\begingroup$ I’d upvote this a dozen times if I could. As I understand it, we owe the modern theory of tensor products of topological vector spaces—and hence, in particular, the theory of nuclear topological vector spaces—entirely to Grothendieck’s PhD thesis. A quick sketch can be found, for instance, in this survey chapter by Fernando Bombal. $\endgroup$ – Branimir Ćaćić Sep 30, 2020 at 0:00
- $\begingroup$ I just skimmed the Ph.D. thesis of Noam Elkies. I don't think that the result you mentioned is in there. He did solve that problem of Euler's around the same time that he wrote his Ph.D. thesis, but the solution doesn't appear in his Ph.D. thesis as far as I can tell. $\endgroup$ – Timothy Chow Nov 26, 2022 at 18:27

Godel's Completeness Theorem, was part of his PHD thesis.

.. when Kurt Gödel joined the University of Vienna in 1924, he took up theoretical physics as his major. Sometime before this, he had read Goethe’s theory of colors and became interest in the subject. At the same time, he attended classes on mathematics and philosophy as well. Soon he came in contact with great mathematicians and in 1926, influenced by number theorist Philipp Furtwängler, he decided to change his subject and take up mathematics. Besides that, he was highly influenced by Karl Menger’s course in dimension theory and attended Heinrich Gomperz’s course in the history of philosophy. Also in 1926, he entered the Vienna Circle, a group of positivist philosophers formed around Moritz Schlick, and until 1928, attended their meetings regularly. After graduation, he started working for his doctoral degree under Hans Hahn. His dissertation was on the problem of completeness. In the summer of 1929, Gödel submitted his dissertation, titled ‘Über die Vollständigkeit des Logikkalküls’ (On the Completeness of the Calculus of Logic). Subsequently in February 1930, he received his doctorate in mathematics from the University of Vienna. Sometime now, he also became an Austrian citizen.

- $\begingroup$ I don't know whether this specific question was open, but it is certainly in the family of open problems highlighted by Hilbert's 2nd Problem and the Entscheidungsproblem. $\endgroup$ – Joshua Grochow Sep 26, 2018 at 2:37
- 2 $\begingroup$ It was an open problem (posed by Hilbert and Ackermann) but only a few years earlier. It also falls out of earlier results by Skolem. $\endgroup$ – none Sep 26, 2018 at 7:11

- 1 $\begingroup$ I never heard about this very great thesis. Thank you for this answer. $\endgroup$ – Ali Taghavi May 20, 2018 at 19:12
- $\begingroup$ @AliTaghavi: You are welcome. -- And thank YOU! $\endgroup$ – Stefan Kohl ♦ Sep 11, 2022 at 18:52

In the first part of the thesis, I attack the common belief that quantum computing resembles classical exponential parallelism, by showing that quantum computers would face serious limitations on a wider range of problems than was previously known. In particular, any quantum algorithm that solves the collision problem -- that of deciding whether a sequence of $n$ integers is one-to-one or two-to-one -- must query the sequence $\Omega(n^{1/5})$ times. This resolves a question that was open for years; previously no lower bound better than constant was known. A corollary is that there is no "black-box" quantum algorithm to break cryptographic hash functions or solve the Graph Isomorphism problem in polynomial time.

There was even a second part to that thesis...

...Next I ask what happens to the quantum computing model if we take into account that the speed of light is finite -- and in particular, whether Grover's algorithm still yields a quadratic speedup for searching a database. Refuting a claim by Benioff, I show that the surprising answer is yes.

John von Neumann's dissertation seems to be an example with just the right timing.

But at the beginning of the 20th century [ in 1901, to be precise ], efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel [i.e. there was active research on the question]. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class. The axiom of foundation proposed that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel. If one set belongs to another then the first must necessarily come before the second in the succession. This excludes the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration, called the method of inner models, which later became an essential instrument in set theory. The second approach to the problem of sets belonging to themselves took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set. Van Heijenoort, Jean (1967). From Frege to Gödel: a Source Book in Mathematical Logic, 1879–1931. Cambridge, Massachusetts: Harvard University Press. ISBN 978-0-674-32450-3. OCLC 523838.

[50] Develop computer programs for simplifying sums that involve binomial coefficients. Exercise 1.2.6.63 in The Art of Computer Programming, Volume 1: Fundamental Algorithms by Donald E. Knuth, Addison Wesley, Reading, Massachusetts, 1968.

[Fase45] is the Ph.D. dissertation of Sister Mary Celine Fasenmyer, in 1945. It showed how recurrences for certain polynomial sequences could be found algorithmically. (See Chapter 4.) ... [Petk91] is the Ph.D. thesis of Marko Petkovšek, in 1991. In it he discovered the algorithm for deciding if a given recurrence with polynomial coefficients has a "simple" solution, which, together with the algorithms above, enables the automated discovery of the simple evaluation of a given definite sum, if one exists, or a proof of nonexistence, if none exists (see Chapter 8). A definite hypergeometric sum is one of the form $f(n) = \sum^{\infty}_{k=-\infty} F(n, k)$, where $F$ is hypergeometric.

- 7 $\begingroup$ Just to clarify: this is not the Rota's conjecture , but a conjecture by Rota; which apparently also is known as Rota-Heron-Welsh conjecture. $\endgroup$ – Max Horn Sep 29, 2020 at 22:08
- 1 $\begingroup$ Is this the work for which Huh was awarded a Fields medal? $\endgroup$ – Gerry Myerson Aug 28, 2022 at 1:41
- 1 $\begingroup$ @GerryMyerson It was indeed one of the things for which he was awarded a Fields medal, according to the short citation . $\endgroup$ – Timothy Chow Nov 26, 2022 at 18:31

Conjecture. (Maximal rank conjecture) Let $C \subseteq \mathbb P^r$ be a general Brill-Noether¹ curve. Then the restriction map $$H^0(\mathbb P^r, \mathcal O_{\mathbb P^r}(k)) \to H^0(C, \mathcal O_C(k))$$ has maximal rank, i.e. is injective if $h^0(\mathbb P^r, \mathcal O(k)) \leq h^0(C, \mathcal O(k))$ and surjective otherwise.

[Lar1] E. K. Larson, The maximal rank conjecture . PhD dissertation, 2018.

[Lar2] E. K. Larson, The maximal rank conjecture . Preprint, arXiv:1711.04906 .

- 2 $\begingroup$ Unfortunately, the references given in [Larson2] (which I blindly copied) are not accurate. I have been unable to find the conjecture in either [Harris] or the Severi reference given therein. I spent a few hours trying to sort this out, which got me a bit closer, but I still didn't really find the original sources. I would be grateful if someone could clear this up. $\endgroup$ – R. van Dobben de Bruyn Sep 26, 2018 at 1:21
- 2 $\begingroup$ In arxiv.org/abs/1505.05460 Jensen and Payne cite "F. Severi. Sulla classificazione delle curve algebriche e sul teorema di esistenza di Riemann. Rend. R. Acc. Naz. Lincei, 24(5):887–888, 1915." and " J. Harris. Curves in projective space, volume 85 of Séminaire de Mathématiques Supérieures. Presses de l’Université de Montréal, Montreal, Que., 1982. With the collaboration of David Eisenbud" as sources for the Maximal Rank Conjecture. (By the way, I posted this example as a comment earlier- glad to see it upgraded into an answer.) $\endgroup$ – Sam Hopkins Sep 26, 2018 at 15:13
- $\begingroup$ For anyone interested in what Brill-Noether theory is more generally, this expository article by Harris is really excellent: birs.ca/workshops/2014/14w5133/files/Osserman-reference.pdf $\endgroup$ – Sam Hopkins Sep 26, 2018 at 15:14
- $\begingroup$ @SamHopkins: Thanks, I had found those as well. But they do not settle the question of the origin of the conjecture. If I understand the paper correctly, Severi does not state the general case as a conjecture. $\endgroup$ – R. van Dobben de Bruyn Sep 26, 2018 at 23:02

Peter Weinberger's Ph.D. thesis is a superb example:

Proof of a Conjecture of Gauss on Class Number Two

See: https://en.wikipedia.org/wiki/Peter_J._Weinberger

- $\begingroup$ I couln't find this PhD thesis online. How is his strategy of proof? $\endgroup$ – user19475 Sep 27, 2018 at 3:06
- $\begingroup$ @TKe, sorry, you need to ask a specialist, I am not (too bad). $\endgroup$ – Wlod AA Sep 27, 2018 at 7:29

Since the OP mentions Gauss, this entry could be an appropriate addition to the list:

This algorithm overcomes one of the biggest headaches of $N$ -body simulations: the fact that accurate calculations of the motions of $N$ particles interacting via gravitational or electrostatic forces (think stars in a galaxy, or atoms in a protein) would seem to require $O(N^2)$ computations—one for each pair of particles. The fast multipole algorithm gets by with $O(N)$ computations. It does so by using multipole expansions (net charge or mass, dipole moment, quadrupole, and so forth) to approximate the effects of a distant group of particles on a local group. A hierarchical decomposition of space is used to define ever-larger groups as distances increase. One of the distinct advantages of the fast multipole algorithm is that it comes equipped with rigorous error estimates, a feature that many methods lack.

Zhang then asked in 1992 (conference held in 1990) the following questions:

- If $U(M)$ is virtually free, is the congruential language of $M$ a context-free language?
- Is it decidable, taking $w$ as input, whether the congruential language of $M$ is context-free?

My thesis can be found on my website .

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## USC Digital Folklore Archives

A database of folklore performances, student inadvertently solves never-before-solved math problems.

## MacTutor

... gave me thousands of geometry problems while I was still in high school. ... the mental exercise required to solve them was the great gift from my father. The solving of thousands of problems during my high school days - at the time when my brain was growing - did more than anything else to develop my analytic power.

As a teenager, I prepared some of the figures that appeared in the book.

Since its first appearance nearly half a century ago the book has gone through a number of printings and has deservedly maintained its popularity.

During my first year at Berkeley I arrived late one day to one of Neyman 's classes. On the blackboard were two problems which 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 to do than usual. I asked him if he still wanted the work. 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, 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 which 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.

My office collected data about sorties flown, bombs dropped, aircraft lost... I also helped other divisions of the Air Staff prepare plans called "programs". ... everything was planned in greatest detail: all the nuts and bolts, the procurement of airplanes, the detailed manufacture of everything. There were hundreds of thousands of different kinds of material goods and perhaps fifty thousand specialties of people. My office collected data about the air combat such as the number of sorties flown, the tons of bombs dropped, attrition rates. I also became a skilled expert on doing planning by hand techniques.

Berkeley made me an offer, but I didn't like it because it was too small. Or, to be more exact, my wife did not like it. It was a grand salary of fourteen hundred dollars a year. She did not see how we could live on that with our child David.

One of the first applications of the simplex algorithm was to the determination of an adequate diet that was of least cost. In the fall of 1947 , Jack Laderman of the Mathematical Tables Project of the National Bureau of Standards undertook, as a test of the newly proposed simplex method, the first large-scale computation in this field. It was a system with nine equations in seventy-seven unknowns. Using hand-operated desk calculators, approximately 120 man-days were required to obtain a solution. ... The particular problem solved was one which had been studied earlier by George Stigler ( who later became a Nobel Laureate ) who proposed a solution based on the substitution of certain foods by others which gave more nutrition per dollar. He then examined a "handful" of the possible 510 ways to combine the selected foods. He did not claim the solution to be the cheapest but gave his reasons for believing that the cost per annum could not be reduced by more than a few dollars. Indeed, it turned out that Stigler's solution ( expressed in 1945 dollars ) was only 24 cents higher than the true minimum per year $ 39 . 69 .

Linear programming is viewed as a revolutionary development giving man the ability to state general objectives and to find, by means of the simplex method, optimal policy decisions for a broad class of practical decision problems of great complexity. In the real world, planning tends to be ad hoc because of the many special-interest groups with their multiple objectives.

The tremendous power of the simplex method is a constant surprise to me.

If one would take statistics about which mathematical problem is using up most of the computer time in the world, then ... the answer would probably be linear programming.

[ Linear programming ] is used to allocate resources, plan production, schedule workers, plan investment portfolios and formulate marketing ( and military ) strategies. The versatility and economic impact of linear programming in today's industrial world is truly awesome.

Mathematical programming has been blessed by the involvement of at least two exceptionally creative geniuses: George Dantzig and Leonid Kantorovich .

The systematic development of practical computing methods for linear programming began in 1952 at the Rand Corporation in Santa Monica, under the direction of George B Dantzig. The author worked intensively on this project there until late 1956 , by which time great progress had been made on first-generation computers.

An impressive book, the work is very complete, its scientific level high, and its reading pleasant.

... it is interesting to note that the original problem that started my research is still outstanding - namely the problem of planning or scheduling dynamically over time, particularly planning dynamically under uncertainty. If such a problem could be successfully solved it could eventually through better planning contribute to the well-being and stability of the world.

For inventing linear programming and discovering methods that led to wide-scale scientific and technical applications to important problems in logistics, scheduling, and network optimization, and to the use of computers in making efficient use of the mathematical theory.

In recognition of his outstanding contribution to engineering and the sciences through his pioneering work in mathematical programming and his development of the simplex method. His work permits the solution of many previously intractable problems and has made linear programming into one of the most frequently used techniques of modern applied mathematics.

A member of the National Academy of Engineering, the National Academy of Science , the American Academy of Arts and Sciences and recipient of the National Medal of Science, plus eight honorary degrees, Professor Dantzig's seminal work has laid the foundation for much of the field of systems engineering and is widely used in network design and component design in computer, mechanical, and electrical engineering.

## References ( show )

- M Aigner, Diskrete Mathematik, in Ein Jahrhundert Mathematik 1890 - 1990 ( Braunschweig, 1990) , 83 - 112 .
- D J Albers and C Reid, An interview with George B. Dantzig : the father of linear programming, College Math. J. 17 (4) (1986) , 293 - 314 .
- D J Albers, G L Alexanderson and C Reid, More mathematical people. Contemporary conversations ( Boston, MA, 1990) .
- M L Balinski, Mathematical programming : journal, society, recollections, in J K Lenstra, A H G Rinnooy, K Schrijver and A Schrijver ( eds. ) , History of mathematical programming ( Amsterdam, 1991) , 5 - 18 .
- G B Dantzig, A look back at the origins of linear programming ( Chinese ) , Chinese J. Oper. Res. 3 (1) (1984) , 71 - 78 .
- G B Dantzig, Impact of linear programming on computer development, in Computers in mathematics, Stanford, CA, 1986 ( New York, 1990) , 233 - 240 .
- G B Dantzig, Linear programming. The story about how it began: some legends, a little about its historical significance, and comments about where its many mathematical programming extensions may be headed, in J K Lenstra, A H G Rinnooy, K Schrijver and A Schrijver ( eds. ) , History of mathematical programming ( Amsterdam, 1991) , 19 - 31 .
- G B Dantzig, Origins of the simplex method, in S G Nash ( ed. ) , A history of scientific computing ( Reading, MA, 1990) , 141 - 151 .
- G B Dantzig, Reminiscences about the origins of linear programming, in Mathematical programming, Rio de Janeiro, 1981 ( Amsterdam, 1984) , 105 - 112 .
- G B Dantzig, Reminiscences about the origins of linear programming, in A Schlissel ( ed. ) , Essays in the history of mathematics, American Mathematical Society, San Francisco, Calif., January 1981 ( Providence, R.I., 1984) , 1 - 11 .
- G B Dantzig, Reminiscences about the origins of linear programming, in Mathematical programming : the state of the art, Bonn, 1982 ( New York, 1983) , 78 - 86 .
- G B Dantzig, Reminiscences about the origins of linear programming, Oper. Res. Lett. 1 (2) (1981 / 82) , 43 - 48 .
- G B Dantzig, Time-staged methods in linear programming : comments, early history, future prospects, in Large scale systems, Cleveland, Ohio, 1980 ( Amsterdam-New York, 1982) , 19 - 30 .
- G B Dorfman, R The discovery of linear programming, Ann. Hist. Comput. 6 (3) (1984) , 283 - 295 .
- T H Kjeldsen, The emergence of nonlinear programming : interactions between practical mathematics and mathematics proper, Math. Intelligencer 22 (3) (2000) , 50 - 54 .
- W Orchard-Hays, History of mathematical programming systems, Ann. Hist. Comput. 6 (3) (1984) , 296 - 312 .
- Professor George Dantzig : Linear Programming Founder Turns 80 , SIAM News ( November 1994) .
- Selected publications of George B Dantzig, in Mathematical programming I, Math. Programming Stud. No. 24 (1985) , xi.

## Additional Resources ( show )

Other pages about George Dantzig:

Other websites about George Dantzig:

## Honours ( show )

Honours awarded to George Dantzig

## Cross-references ( show )

## George Dantzig: The Story of The Overlooked Genius

What did George Dantzig develop?

What is the famous “homework” story of George Dantzig?

What movie is related to George Dantzig?

Dantzig subsequently said in an interview that:

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, we 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.

## Dantzig’s early life

## Dantzig was a heartfelt statistician

## Dantzig’s advancements in military planning

## He wasn’t seen worth of Nobel Prize

## The two unsolved homework problems that George Dantzig solved

## 1. “On the Non-Existence of Tests of “Student’s” Hypothesis Having Power Functions Independent of σ”, 1940

## 2. “On the Fundamental Lemma of Neyman and Pearson”, 1951

## George Dantzig, the real Good Will Hunting

## George Dantzig’s discoveries and contributions

- The simplex algorithm : In particular, Dantzig is lauded for creating the simplex algorithm, a standard technique for resolving linear programming issues. If you have a linear objective function and linear constraints, the simplex method may help you find the best solution.
- The theory of duality in linear programming : Dantzig established a cornerstone notion in optimization theory known as the principle of duality in linear programming. The best solution to a linear programming problem can be found with the help of duality theory, which establishes a link between the original problem and its dual problem.
- Contributions to linear regression : Dantzig’s contributions to the field of linear regression are substantial. Linear regression is a statistical technique for modeling the association between a dependent variable and one or more independent variables, and Dantzig made significant contributions to this area.
- Work on the transportation problem : Dantzig also did important work in the area of transportation problems, a kind of linear programming issue that includes determining the best possible route for resources to take between different points on a map.

- Joe Holley (2005). “Obituaries of George Dantzig” .
- Donald J. Albers. (1990). “ More Mathematical People: Contemporary Conversations “
- On the Fundamental Lemma of Neyman and Pearson – Projecteuclid.org
- On the Non-Existence of Tests of “Student’s” Hypothesis Having Power Functions Independent of σ – Projecteuclid.org
- Dantzig, George (1940). “ On the non-existence of tests of “Student’s” hypothesis having power functions independent of σ “

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## IMAGES

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## COMMENTS

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...

In statistics, Dantzig solved two open problems in statistical theory, which he had mistaken for homework after arriving late to a lecture by Jerzy Neyman. [2] At his death, Dantzig was the Professor Emeritus of Transportation Sciences and Professor of Operations Research and of Computer Science at Stanford University . Early life [ edit]

When Dantzig arrived, he assumed that the two problems were a homework assignment and wrote them down. According to Dantzig, the problems "seemed to be a little harder than usual", but a few days later he handed in completed solutions for the two problems, still believing that they were an assignment that was overdue.

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.

And the professor was just overjoyed because he said, `You know, you've just solved two of the most difficult unsolved problems in statistics. We've got to write them up and submit them for...

Dantzig had no idea what Neyman was talking about until he explained that the two problems on the blackboards were famously unsolved statistical problems — not homework at all. Later on in...

2 Solving LPs: The Simplex Algorithm of George Dantzig 2.1 Simplex Pivoting: Dictionary Format ... 3 8 0 x 1,x 2,x 3 In devising our approach we use a standard mathematical approach; reduce the problem to one that we already know how to solve. Since the structure of this problem is essentially linear, we try to reduce it to a problem of solving ...

Six weeks later, Dantzig received a visit from an excited professor Neyman, who was eager to tell him that the homework problems he had solved were two of the most famous unsolved problems in statistics. He had prepared one of Dantzig's solutions for publication in a mathematical journal.

What were the two famous open problems that George Dantzig mistook for homework and solved while being a graduate student at Berkeley? All related (31) Sort Recommended Daniel McLaury Former Senior Research Engineer at Peddinghaus (2017-2018) Upvoted by Sridhar Ramesh , ABD on PhD Logic, University of California, Berkeley and Tikhon Jelvis

On the board were two problems of "unsolved" statistics that George mistook for a homework assignment. He copied them down and started working on them from home, six weeks later he turned in the work late, hoping to get at least some credit for the assignment but nothing great.

The Diet Problem GEORGE B. DaNTZIG Department of Operations Research Stanford University Stanford, California 94305-4022 This is a story about connections. ... solve it, he invented a very clever heuristic to arrive at a diet that cost only $39.93 per year (1939 prices). He did not claim it to be the cheapest solution but gave good ...

TIL the mathematician George Dantzig solved two of the most famous problems of statistics, because he came into class too late to hear that they were supposed to be unsolvable ... To make a long story short, the problems on the blackboard which I had solved thinking they were homework were in fact two famous unsolved problems in statistics ...

According to Dantzig himself, the answer is no: "If I had known that the problems were not homework but were in fact two famous unsolved problems in statistics, I probably would not have...

Six weeks later, Dantzig received a visit from an excited professor Neyman, who was eager to tell him that the homework problems he had solved were two of the most famous unsolved problems in statistics. Neyman told Dantzig to wrap the two problems in a binder and he would accept them as a Ph.D. thesis. The two problems that Dantzig solved were ...

In 1939, George Dantzig arrived late to his graduate statistics class and saw two problems on the board, not knowing they were examples of problems that had never been solved. He thought they were a homework assignment and was able to solve them. He found out the reality six weeks later when his teacher let him know and helped him publish a ...

The continuous knapsack problem may be solved by a greedy algorithm, first published in 1957 by George Dantzig,[2][3]that considers the materials in sorted order by their values per unit weight. If the sum of the choices made so far equals the capacity W, then the algorithm sets xi = 0.

The particular problem solved was one which had been studied earlier by George Stigler ( who later became a Nobel Laureate) who proposed a solution based on the substitution of certain foods by others which gave more nutrition per dollar. He then examined a "handful" of the possible 510 ways to combine the selected foods.

The two unsolved homework problems that George Dantzig solved The doctoral student George Bernard Dantzig came late to Jerzy Neyman's statistics lecture in 1939, when two homework assignments were already written on the board. He put them in writing and spent many days trying to solve them.

George Dantzig, (born Nov. 8, 1914, Portland, Ore., U.S.—died May 13, 2005, Stanford, Calif.), American mathematician who devised the simplex method, an algorithm for solving problems that involve numerous conditions and variables, and in the process founded the field of linear programming.