David A. Klarner

David Anthony Klarner (October 10, 1940 – March 20, 1999) was an American mathematician, author, and educator. He is known for his work in combinatorial enumeration, polyominoes,[3] and box-packing.[4][5][6]

David A. Klarner
David Anthony Klarner

(1940-10-10)October 10, 1940
DiedMarch 20, 1999(1999-03-20) (aged 58)
Alma materUniversity of Alberta
Known forCombinatorics
Klarner's Theorem[1]
Klarner-Rado Sequence[2]
Recreational mathematics
Scientific career
InstitutionsUniversity of Calgary
ThesisOn some combinatorial and probabilistic aspects of bipartite graphs
Doctoral advisorJohn W. Moon
Doctoral studentsJean Scholtz

Klarner was a friend and correspondent of mathematics popularizer Martin Gardner and frequently made contributions to Gardner's Mathematical Games column in Scientific American.[7] He edited a book honoring Gardner on the occasion of his 65th birthday.[8][9] Gardner in turn dedicated his twelfth collection of mathematical games columns to Klarner.[10]

Beginning in 1969 Klarner made significant contributions to the theory of combinatorial enumeration, especially focusing on polyominoes[11] and box-packing.[12][5] Working with Ronald L. Rivest he found upper bounds on the number of n-ominoes.[4] Klarner's Theorem is the statement that an m by n rectangle can be packed with 1-by-x rectangles if and only if x divides one of m and n.[1][13]

He has also published important results in group theory[14] and number theory, in particular working on the Collatz conjecture (sometimes called the 3x + 1 problem).[15] The Klarner-Rado Sequence is named after Klarner and Richard Rado.[2]


Klarner was born in Fort Bragg, California, and spent his childhood in Napa, California.[7] He married Kara Lynn Klarner in 1961. Their son Carl Eoin Klarner was born on April 21, 1969.[16]

Klarner did his undergraduate work at Humboldt State University (1960–63), got his Ph.D. at the University of Alberta (1963–66), and did post-doctoral work at McMaster University in Hamilton, Ontario (1966–68). He also did post-doctoral work at Eindhoven University of Technology in the Netherlands (1968-1970), at the University of Reading in England working with Richard Rado (1970–71),[17] and at Stanford University (1971–73). He served as an assistant professor at Binghamton University (1973–79) and was a visiting professor at Humboldt State University in California (1979–80). He returned to Eindhoven as a professor (1980–81), and to Binghamton (1981–82). From 1982 to 1996 he was a professor of computer science at the University of Nebraska, at Lincoln, with a one-year break at Eindhoven in academic year 1991–92. He retired to Eureka, California in 1997 and died there in 1999.[7]

He was a frequent contributor to recreational mathematics and worked with many key mathematics popularizers including Ronald L. Rivest, John H. Conway, Richard K. Guy, Donald Coxeter, Ronald Graham, and Donald Knuth.[18][8][19][11]

Organizations and awardsEdit

Klarner was a member of the Association for Computing Machinery, the American Mathematical Society, the Mathematical Association of America, and the Fibonacci Association.[7] He was awarded a National Science Foundation Fellowship Award in mathematics in 1963.[20] In 1986 Klarner received a University of Nebraska-Lincoln Distinguished Teaching Award in Computer Science.[21]

The David A. Klarner Fellowship for Computer Science was set up after Klarner's death by Spyros Magliveras a fellow professor in Computer Science at UNL.[22]


  • Asymptotically Optimal Box Packing Theorems: Klarner systems by Michael Reid, Department of Mathematics, University of Central Florida, June, 2008
  • A Lifetime of Puzzles edited by Erik D. Demaine, Martin L. Demaine, Tom Rodgers; pp. 221–225: Satterfield's Tomb, a puzzle by David A. Klarner and Wade Satterfield; ISBN 1568812450

Selected publicationsEdit


  • The Mathematical Gardner (editor), Publisher: Boston : Prindle, Weber & Schmidt; Belmont, Calif. : Wadsworth International, ISBN 0486400891, ISBN 9781468466867 (electronic book)[9]



  1. ^ a b Mathematical Gems Vol. 2, by Ross Honsberger The Mathematical Association of America: The Dolciani Mathematical Expositions, p. 88, 1976.
  2. ^ a b Klarner-Rado Sequence Michigan State University, MSU Librarie
  3. ^ The Tromino Puzzle by Norton Starr
  4. ^ a b A procedure for improving the upper bound for the number of n-ominoes, by D. A. Klarner and R. L. Rivest, Can. J. Math., Vol. XXV, No. 3, 1973, pp. 5
  5. ^ a b Klarner systems and tiling boxes with polyominoes by Michael Reid, Journal of Combinatorial Theory, Series A, Vol. 111, Issue 1, July 2005, Pages 89-105
  6. ^ A Finite Basis Theorem Revisited by David A. Klarner, Stanford University, Department of Computer Science, Report Number: CS-TR-73-338, February 1973
  7. ^ a b c d University of Calgary: Archives and Special Collections: David A. Klarner
  8. ^ a b Gardner Tribute Books The Mathematical Gardner, edited by David A. Klarner "It was quietly assembled behind the scenes, with the assistance of Ron Graham and Don Knuth, as a surprise for Martin to mark his announced retirement from his Scientific American column."
  9. ^ a b Reprinted in 1998 as Mathematical Recreations: A Collection in Honor of Martin Gardner (Dover; ISBN 0-486-40089-1), this book, edited by Klarner, was the tribute of the mathematical community to Gardner when he retired from writing his Scientific American column in 1981. Discreetly assembled for the occasion, the stature of the mathematicians submitting papers is a testament to Gardner's importance.
  10. ^ A lifetime of puzzles : a collection of puzzles in honor of Martin Gardner's 90th birthday edited by Erik D Demaine, Martin L Demaine, and Tom Rodgers, Publisher: Wellesley, Massachusetts : A K Peters, Ltd. (2008), p. 346, ISBN 1568812450
  11. ^ a b Another Fine Math You've Got Me Into. . ., By Ian Stewart, Dover Publications (January 15, 2004), p. 21, ISBN 0486431819
  12. ^ Packing a rectangle with congruent n-ominoes Journal of Combinatorial Theory, Vol. 7, Issue 2, September 1969, Pages 107-115
  13. ^ Weisstein, Eric W. "Klarner's Theorem". MathWorld.
  14. ^ A sufficient condition for certain semigroups to be free by David A Klarner, Journal of Algebra, Vol 74, Issue 1, January 1982, Pages 140-148
  15. ^ Erdős, Klarner, and the 3x + 1 Problem by Jeffrey C. Lagarias, The American Mathematical Monthly, Vol. 123, No. 8, October 2016, pp. 753-776" [This paper describes work of Erdős, Klarner, and Rado on semigroups of integer affine maps and on sets of integers they generate. It gives the history of problems they studied, some solutions, and new unsolved problems that arose from them."]
  16. ^ Carl is a Political Scientist, receiving tenure at Indiana State University and currently working at the University of Florida as a research associate.
  17. ^ Arithmetic properties of certain recursively defined sets by D. A. Klarner and R. Rado, Stanford University: Computer Science Department, March 1972
  18. ^ Election Integrity, Past, Present and Future[permanent dead link] Caltech/MIT Voting Technology Project, Participants’ Biographies
  19. ^ The Penrose Tiling at Miami University by David Kullman, Presented at the Mathematical Association of America Ohio Section Meeting Shawnee State University, October 24, 1997
  20. ^ Fellowship Awards Offered National Science Foundation 1963
  21. ^ University of Nebraska-Lincoln Distinguished Teaching Awards: Past Recipients
  22. ^ David A. Klarner Fellowship for Computer Science University of Nebraska–Lincoln: Scholarships & Aid
  23. ^ This is a 2016 revision by Barequet of the chapter of the same title originally written by Klarner for the first edition, and revised by Golomb for the second edition.

External linksEdit