# Unsolvable Math Problems Homework Market

George Bernard Dantzig | |
---|---|

Gerald R. Ford awarded George B. Dantzig at the National Medal of Science Awards Ceremony, 1976 | |

Born | (1914-11-08)November 8, 1914 Portland, Oregon |

Died | May 13, 2005(2005-05-13) (aged 90) Stanford, California |

Citizenship | American |

Alma mater | Bachelor's degree:University of Maryland Master's degree: University of Michigan Doctor of Philosophy: University of California, Berkeley |

Known for | Linear programming Simplex algorithm Dantzig-Wolfe decomposition principle Generalized linear programming Generalized upper bounding Max-flow min-cut theorem of networks Quadratic programming Complementary pivot algorithms Linear complementarity problem Stochastic programming |

Awards | John von Neumann Theory Prize(1975) National Medal of Science in Mathematical, Statistical, and Computational Sciences (1975) Harvey Prize(1985) Harold Pender Award(1995) |

Scientific career | |

Fields | Mathematics Operations research Computer science Economics Statistics |

Institutions | U.S. Air Force Office of Statistical Control RAND Corporation University of California, Berkeley Stanford University |

Doctoral advisor | Jerzy Neyman |

Doctoral students | Robert Fourer Alfredo Noel Iusem Ellis L. Johnson Thomas Magnanti Roger J-B Wets Yinyu Ye |

Influences | Wassily Leontief John von Neumann Marshal K. Wood |

Influenced | Kenneth J. Arrow Robert Dorfman Leonid Hurwicz Tjalling C. Koopmans Thomas L. Saaty Paul Samuelson Harry M. Markowitz Philip Wolfe |

**George Bernard Dantzig** (; November 8, 1914 – May 13, 2005) was an American mathematical scientist who made important contributions to operations research, computer science, economics, and statistics.

Dantzig is known for his development of the simplex algorithm,^{[1]} an algorithm for solving linear programming problems, and for his other work with linear programming. 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]}

Dantzig was the Professor Emeritus of Transportation Sciences and Professor of Operations Research and of Computer Science at Stanford.

## Life[edit]

Born in Portland, Oregon, George Bernard Dantzig was named after George Bernard Shaw, the Irish writer.^{[3]}^{[4]} His father, Tobias Dantzig, was a BalticGerman mathematician and linguist, and his mother, Anja Dantzig (née Ourisson), was a French linguist of Jewish origin. Dantzig's parents met during their study at the University of Paris, where Tobias studied mathematics under Henri Poincaré, after whom Dantzig's brother was named.^{[4]} The Dantzigs immigrated to the United States, where they settled in Portland, Oregon.

Early in the 1920s the Dantzig family moved from Baltimore to Washington. His mother became a linguist at the Library of Congress, and his father became a math tutor at the University of Maryland, College Park. Dantzig attended Powell Junior High School and Central High School; one of his friends there was Abraham Seidenberg, who also became a professional mathematician.^{[4]} By the time he reached high school he was already fascinated by geometry, and this interest was further nurtured by his father, challenging him with complicated problems, particularly in projective geometry.^{[2]}^{[4]}

George Dantzig received his B.S. from University of Maryland in 1936 in mathematics and physics, which is part of the University of Maryland College of Computer, Mathematical, and Natural Sciences. He earned his master's degree in mathematics from the University of Michigan in 1938. After a two-year period at the Bureau of Labor Statistics, he enrolled in the doctoral program in mathematics at the University of California, Berkeley, where he studied statistics under Jerzy Neyman.

With the outbreak of World War II, Dantzig took a leave of absence from the doctoral program at Berkeley to join the U.S. Air Force Office of Statistical Control. In 1946, he returned to Berkeley to complete the requirements of his program and received his Ph.D. that year.^{[3]} Although he had a faculty offer from Berkeley, he returned to the Air Force as mathematical advisor to the comptroller.^{[4]}

In 1952 Dantzig joined the mathematics division of the RAND Corporation. By 1960 he became a professor in the Department of Industrial Engineering at UC Berkeley, where he founded and directed the Operations Research Center. In 1966 he joined the Stanford faculty as Professor of Operations Research and of Computer Science. A year later, the Program in Operations Research became a full-fledged department. In 1973 he founded the Systems Optimization Laboratory (SOL) there. On a sabbatical leave that year, he headed the Methodology Group at the International Institute for Applied Systems Analysis (IIASA) in Laxenburg, Austria. Later he became the C. A. Criley Professor of Transportation Sciences at Stanford, and kept going, well beyond his mandatory retirement in 1985.^{[3]}

He was a member of the National Academy of Sciences, the National Academy of Engineering, and the American Academy of Arts and Sciences. Dantzig was the recipient of many honors, including the first John von Neumann Theory Prize in 1974, the National Medal of Science in 1975,^{[5]} an honorary doctorate from the University of Maryland, College Park in 1976. The Mathematical Programming Society honored Dantzig by creating the George B. Dantzig Prize, bestowed every three years since 1982 on one or two people who have made a significant impact in the field of mathematical programming.

Dantzig died on May 13, 2005, in his home in Stanford, California, of complications from diabetes and cardiovascular disease. He was 90 years old.^{[2]}

## Work[edit]

Freund wrote further that "through his research in mathematical theory, computation, economic analysis, and applications to industrial problems, Dantzig has contributed more than any other researcher to the remarkable development of linear programming".^{[6]}

Dantzig's seminal work allows the airline industry, for example, to schedule crews and make fleet assignments. Based on his work tools are developed "that shipping companies use to determine how many planes they need and where their delivery trucks should be deployed. The oil industry long has used linear programming in refinery planning, as it determines how much of its raw product should become different grades of gasoline and how much should be used for petroleum-based byproducts. It is used in manufacturing, revenue management, telecommunications, advertising, architecture, circuit design and countless other areas".^{[2]}

### Mathematical statistics[edit]

An event in 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.^{[4]}^{[7]}

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.^{[2]}^{[4]} He had prepared one of Dantzig's solutions for publication in a mathematical journal.^{[8]} As Dantzig told it in a 1986 interview in the *College Mathematics Journal*:^{[9]}

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.

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.^{[4]}^{[10]}

This story began to spread and was used as a motivational lesson demonstrating the power of positive thinking. Over time Dantzig's name was removed, and facts were altered, but the basic story persisted in the form of an urban legend and as an introductory scene in the movie *Good Will Hunting*.^{[7]}

### Linear programming[edit]

Linear programming is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. Linear programming arose as a mathematical model developed during World War II to plan expenditures and returns in order to reduce costs to the army and increase losses to the enemy. It was kept secret until 1947. Postwar, many industries found its use in their daily planning.

The founders of this subject are Leonid Kantorovich, a Russian mathematician who developed linear programming problems in 1939, Dantzig, who published the simplex method in 1947, and John von Neumann, who developed the theory of the duality in the same year.

Dantzig's original example of finding the best assignment of 70 people to 70 jobs exemplifies the usefulness of linear programming. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the number of particles in the universe. However, it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the Simplex algorithm. The theory behind linear programming drastically reduces the number of possible optimal solutions that must be checked.

In 1963, Dantzig’s *Linear Programming and Extensions* was published by Princeton University Press. Rich in insight and coverage of significant topics, the book quickly became “the bible” of linear programming.

## Publications[edit]

Books by George Dantzig:

- 1953.
*Notes on linear programming*. RAND Corporation. - 1956.
*Linear inequalities and related systems*. With others. Edited by H.W. Kuhn and A.W. Tucker. Princeton University Press. - 1963.
*Linear programming and extensions*. Princeton University Press and the RAND Corporation. pdf from RAND - 1966.
*On the continuity of the minimum set of a continuous function*. With Jon H. Folkman and Norman Shapiro. - 1968.
*Mathematics of the decision sciences*. With Arthur F. Veinott, Jr. Summer Seminar on Applied Mathematics 5th : 1967 : Stanford University. American Mathematical Society. - 1969.
*Lectures in differential equations*. A. K. Aziz, general editor. Contributors: George B. Dantzig and others. - 1970.
*Natural gas transmission system optimization*. With others. - 1973.
*Compact city; a plan for a liveable urban environment*. With Thomas L. Saaty. - 1974.
*Studies in optimization*. Edited with B.C. Eaves. Mathematical Association of America. - 1985.
*Mathematical programming : essays in honor of George B. Dantzig*. Edited by R.W. Cottle. Mathematical Programming Society. - 1997.
*Linear programming 1: Introduction*. G.B.D. and Mukund N. Thapa. Springer-Verlag. - 2003.
*Linear programming 2: Theory and Extensions*. G.B.D. and Mukund N. Thapa. Springer-Verlag. - 2003.
*The Basic George B. Dantzig*. Edited by Richard W. Cottle. Stanford Business Books, Stanford University Press, Stanford, California.^{[11]}

Book chapters:

- Dantzig, George B. (1960), "General convex objective forms", in Arrow, Kenneth J.; Karlin, Samuel; Suppes, Patrick,
*Mathematical models in the social sciences, 1959: Proceedings of the first Stanford symposium*, Stanford mathematical studies in the social sciences, IV, Stanford, California: Stanford University Press, pp. 151–158, ISBN 9780804700214.

Articles, a selection:

- Dantzig, George B. (June 1940). "On the Non-Existence of Tests of 'Student's' Hypothesis Having Power Functions Independent of σ".
*The Annals of Mathematical Statistics*.**11**(2): 186–92. doi:10.1214/aoms/1177731912. JSTOR 2235875. - Wood, Marshall K.; Dantzig, George B. (1949). "Programming of Interdependent Activities: I General Discussion".
*Econometrica*.**17**(3/4): 193–9. doi:10.2307/1905522. JSTOR 1905522. - Dantzig, George B. (1949). "Programming of Interdependent Activities: II Mathematical Model".
*Econometrica*.**17**(3): 200–211. doi:10.2307/1905523. JSTOR 1905523. - Dantzig, George B. (1955). "Optimal Solution of a Dynamic Leontief Model with Substitution".
*Econometrica*.**23**(3): 295–302. doi:10.2307/1910385. JSTOR 1910385.

## See also[edit]

## Notes[edit]

**^**Gass, Saul I. (2011). "George B. Dantzig".*Profiles in Operations Research*. International Series in Operations Research & Management Science.**147**. pp. 217–240. doi:10.1007/978-1-4419-6281-2_13. ISBN 978-1-4419-6280-5.- ^
^{a}^{b}^{c}^{d}^{e}Joe Holley (2005). "Obituaries of George Dantzig". In:*Washington Post*, May 19, 2005; B06 - ^
^{a}^{b}^{c}Richard W. Cottle, B. Curtis Eaves and Michael A. Saunders (2006). "Memorial Resolution: George Bernard Dantzig". Stanford Report, June 7, 2006. - ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}Albers, Donald J.; Alexanderson, Gerald L.; Reid, Constance, eds. (1990). "George B. Dantzig".*More Mathematical People*. Harcourt Brace Jovanovich. pp. 60–79. ISBN 978-0-15-158175-7. **^**National Science Foundation – The President's National Medal of Science**^**Robert Freund (1994). "Professor George Dantzig: Linear Programming Founder Turns 80". In:*SIAM News*, November 1994.- ^
^{a}^{b}"The Unsolvable Math Problem". Snopes. June 28, 2011. **^**Dantzig, George (1940). "On the non-existence of tests of "Student's" hypothesis having power functions independent of σ".*The Annals of Mathematical Statistics*.**11**(2): 186–192. doi:10.1214/aoms/1177731912.**^**Allende, Sira M.; Bouza, Carlos N. (2005). "Professor George Bernard Dantzig, Life & Legend"(PDF).*Revista Investigación Operacional*.**26**(3): 205–11.**^**Dantzig, George; Wald, Abraham (1951). "On the Fundamental Lemma of Neyman and Pearson".*The Annals of Mathematical Statistics*.**22**: 87–93. doi:10.1214/aoms/1177729695. Retrieved 14 October 2014.**^**Todd, Michael J. (2011). "Review:*The Basic George B. Dantzig*, by Richard W. Cottle".*Bull. Amer. Math. Soc. (N.S.)*.**48**(1): 123–129. doi:10.1090/S0273-0979-2010-01303-3.

## Further reading[edit]

- Cottle, Richard; Johnson, Ellis; Wets, Roger (March 2007). "George B. Dantzig (1914–2005)"(PDF).
*Notices of the American Mathematical Society*.**54**(3): 344–62. - "Professor George Dantzig: Linear Programming Founder Turns 80",
*SIAM News*, November 1994 - O'Connor, John J.; Robertson, Edmund F., "George Dantzig",
*MacTutor History of Mathematics archive*, University of St Andrews . - Dantzig, George B. (1990). "The Diet Problem".
*Interfaces*.**20**(4): 43–7. doi:10.1287/inte.20.4.43. JSTOR 25061369. - Cottle, Richard W. (2005). "George B. Dantzig: a legendary life in mathematical programming".
*Mathematical Programming*.**105**(1): 1–8. doi:10.1007/s10107-005-0674-4. ISSN 0025-5610.

## External links[edit]

### CLAIM

A student mistook examples of unsolved statistics problems for a homework assignment and solved them.

True### RATING

True### ORIGIN

A legend about the “unsolvable math problem” combines one of the ultimate academic wish-fulfillment 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 which 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, 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 statistic professor notified him that he had prepared one of his two “homework” proofs for publication, and Dantzig was given credit on another 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 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, 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 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 ) 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, was awarded the National Medal of Science by President Gerald Ford.

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

This legend is used as the setup of the plot in the 1997 movie . 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.

FeedbackSources

Fact Checker:David Mikkelson

Published:4 December 1996

Updated:20 August 2017

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.

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