David Anthony Klarner (October 10, 1940March 20, 1999) was an American mathematician, author, and educator. He is known for his work in

combinatorial enumeration Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an inf ...

polyominoes A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in popul ...

, and box-packing.A procedure for improving the upper bound for the number of n-ominoes
by D. A. Klarner and

R. L. Rivest Ronald Linn Rivest (; born May 6, 1947) is an American cryptographer and computer scientist whose work has spanned the fields of algorithms and combinatorics, cryptography, machine learning, and election integrity. He is an Institute Professor ...

, Can. J. Math., Vol. XXV, No. 3, 1973, pp. 5A Finite Basis Theorem Revisited
by David A. Klarner, Stanford University, Department of Computer Science, Report Number: CS-TR-73-338, February 1973 Klarner was a friend and correspondent of mathematics popularizer

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing magic, scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writin ...

and frequently made contributions to Gardner's

Mathematical Games column Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for ''Scientific American'' magazine. During the next years, until June 1986, Gardner wrote 9 more columns, br ...

Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it, with more than 150 Nobel Pri ...

. He edited a book honoring Gardner on the occasion of his 65th birthday.
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." Gardner in turn dedicated his twelfth collection of mathematical games columns to Klarner.
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, Beginning in 1969 Klarner made significant contributions to the theory of combinatorial enumeration, especially focusing on polyominoes
Another Fine Math You've Got Me Into. . .
', By Ian Stewart, Dover Publications (January 15, 2004), p. 21, and box-packing.
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 Working with

Ronald L. Rivest Ronald Linn Rivest (; born May 6, 1947) is an American cryptographer and computer scientist whose work has spanned the fields of algorithms and combinatorics, cryptography, machine learning, and election integrity. He is an Institute Professo ...

he found upper bounds on the number of n-ominoes. 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''.''Mathematical Gems Vol. 2'', by Ross Honsberger The

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary edu ...

: The Dolciani Mathematical Expositions, p. 88, 1976. He has also published important results in group theory and number theory, in particular working on the

Collatz conjecture The Collatz conjecture is one of the most famous List of unsolved problems in mathematics, unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer ...

(sometimes called the 3x + 1 problem). The Klarner-Rado Sequence is named after Klarner and

Richard Rado Richard Rado FRS (28 April 1906 – 23 December 1989) was a German-born British mathematician whose research concerned combinatorics and graph theory. He was Jewish and left Germany to escape Nazi persecution. He earned two PhDs: in 1933 from t ...

.Klarner-Rado Sequence
Michigan State University, MSU Librarie

Biography

Klarner was born in

Fort Bragg, California Fort Bragg is a city along the North Coast (California), North Coast of California in Mendocino County, California, Mendocino County. The city is west of Willits, California, Willits, at an elevation of . Its population was 6,983 at the 2020 ...

, and spent his childhood in

Napa, California Napa is the largest city and county seat of Napa County, California, Napa County and a principal city of Wine Country in Northern California, United States. Located in the North Bay (San Francisco Bay Area), North Bay region of the Bay Area, th ...

. He married Kara Lynn Klarner in 1961. Their son Carl Eoin Klarner was born on April 21, 1969. Klarner did his undergraduate work at

Humboldt State University California State Polytechnic University, Humboldt (Cal Poly Humboldt or Humboldt) is a public university in Arcata, California. It is one of Cal Poly (disambiguation), three polytechnic universities in the California State University (CSU) sys ...

(1960–63), got his Ph.D. at the

University of Alberta The University of Alberta (also known as U of A or UAlberta, ) is a public research university located in Edmonton, Alberta, Canada. It was founded in 1908 by Alexander Cameron Rutherford, the first premier of Alberta, and Henry Marshall Tory, t ...

(1963–66), and did post-doctoral work at

McMaster University McMaster University (McMaster or Mac) is a public research university in Hamilton, Ontario, Canada. The main McMaster campus is on of land near the residential neighbourhoods of Ainslie Wood, Ontario, Ainslie Wood and Westdale, Ontario, Westd ...

Hamilton, Ontario Hamilton is a port city in the Canadian Provinces and territories of Canada, province of Ontario. Hamilton has a 2021 Canadian census, population of 569,353 (2021), and its Census Metropolitan Area, census metropolitan area, which encompasses ...

(1966–68). He also did post-doctoral work at

Eindhoven University of Technology The Eindhoven University of Technology (), Abbreviation, abbr. TU/e, is a public university, public technical university in the Netherlands, situated in Eindhoven. In 2020–21, around 14,000 students were enrolled in its Bachelor of Science, BS ...

in the Netherlands (1968-1970), at the

University of Reading The University of Reading is a public research university in Reading, Berkshire, England. It was founded in 1892 as the University Extension College, Reading, an extension college of Christchurch College, Oxford, and became University College, ...

in England working with Richard Rado (1970–71), and at

Stanford University Leland Stanford Junior University, commonly referred to as Stanford University, is a Private university, private research university in Stanford, California, United States. It was founded in 1885 by railroad magnate Leland Stanford (the eighth ...

(1971–73). He served as an assistant professor at

Binghamton University The State University of New York at Binghamton (Binghamton University or SUNY Binghamton) is a public university, public research university in Binghamton metropolitan area, Greater Binghamton, New York, United States. It is one of the four uni ...

(1973–79) and was a visiting professor at

in California (1979–80). He returned to

Eindhoven Eindhoven ( ; ) is a city and List of municipalities of the Netherlands, municipality of the Netherlands, located in the southern Provinces of the Netherlands, province of North Brabant, of which it is the largest municipality, and is also locat ...

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–Lincoln The University of Nebraska–Lincoln (Nebraska, NU, or UNL) is a Public university, public Land-grant university, land-grant research university in Lincoln, Nebraska, United States. Chartered in 1869 by the Nebraska Legislature as part of the M ...

, with a one-year break at Eindhoven in academic year 1991–92. He retired to

Eureka, California Eureka ( ; Wiyot: ; Hupa: ; ) is a city and the county seat of Humboldt County, located on the North Coast of California. The city is located on U.S. Route 101 on the shores of Humboldt Bay, north of San Francisco and south of the Oreg ...

in 1997 and died there in 1999. He was a frequent contributor to

recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research-and-application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...

and worked with many key mathematics popularizers including

, John H. Conway, Richard K. Guy,

Donald Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

Ronald Graham Ronald Lewis Graham (October 31, 1935July 6, 2020) was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He ...

, and

Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist and mathematician. He is a professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of comp ...

Organizations and awards

Klarner was a member of the

Association for Computing Machinery The Association for Computing Machinery (ACM) is a US-based international learned society for computing. It was founded in 1947 and is the world's largest scientific and educational computing society. The ACM is a non-profit professional membe ...

, the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

, the

, and the Fibonacci Association. He was awarded a

National Science Foundation The U.S. National Science Foundation (NSF) is an Independent agencies of the United States government#Examples of independent agencies, independent agency of the Federal government of the United States, United States federal government that su ...

Fellowship Award in mathematics in 1963. In 1986 Klarner received a University of Nebraska-Lincoln Distinguished Teaching Award in Computer Science. 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.

Bibliography

*
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;

Selected publications

Books

* ''The Mathematical Gardner'' (editor), Publisher: Boston : Prindle, Weber & Schmidt; Belmont, Calif. : Wadsworth International, , (electronic book) Reprinted in 1998 as ''Mathematical Recreations: A Collection in Honor of Martin Gardner'' (Dover; ), 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.

Papers

*
Polyominoes
' by Gill Barequet,

Solomon W. Golomb Solomon Wolf Golomb ( ; May 30, 1932 – May 1, 2016) was an American mathematician, engineer, and professor of electrical engineering at the University of Southern California, best known for his works on mathematical games. He most notably inven ...

, and David A. Klarner, December 2016This 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. *
The number of tilings of a block with blocks
' (with F. S. S. Magliveras), ''

European Journal of Combinatorics The ''European Journal of Combinatorics'' is an international peer-reviewed scientific journal that specializes in combinatorics. The journal primarily publishes papers dealing with mathematical structures within combinatorics and/or establishing ...

'': Volume 9 Issue 4, July 1988 *
The number of tiered posets modulo six
' ''Discrete Mathematics'', Vol. 62, Issue 3, pp. 295–297, December 1986 *
Asymptotics for coefficients of algebraic functions
' (with Patricia Woodworth), ''

Aequationes Mathematicae ''Aequationes Mathematicae'' is a mathematical journal. It is primarily devoted to functional equations, but also publishes papers in dynamical systems, combinatorics, and geometry. As well as publishing regular journal submissions on these topics ...

'', Volume 23, Issue 1, pp. 236–241, December 1981 *
An algorithm to determine when certain sets have 0-density
' ''

Journal of Algorithms Elsevier ( ) is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', '' Cell'', the ScienceDirect collection of electronic journals, '' Trends'', t ...

'', Vol. 2, Issue 1, Pages 31–43, March 1981 *
Some remarks on the Cayley-Hamilton theorem
' ''

American Mathematical Monthly ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...

'', Vol. 83, No. 5, pp. 367–369, May, 1976 *
Asymptotic bounds for the number of convex n-ominoes
' (with Ronald L. Rivest), ''

Discrete Mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...

'', Vol. 8, Issue 1, pp. 31–40, March 1974 *
A finite basis theorem revisited
' Stanford University: Computer Science Department, April 1973 *
The number of SDR's in certain regular systems
' Stanford University: Computer Science Department, April 1973 *
Selected combinatorial research problems
' (with

Václav Chvátal Václav (Vašek) Chvátal () is a Professor Emeritus in the Department of Computer Science and Software Engineering at Concordia University in Montreal, Quebec, Canada, and a visiting professor at Charles University in Prague. He has published ex ...

and Donald E. Knuth), Stanford University: Computer Science Department, June 1972 *
Sets generated by iteration of a linear operation
' Stanford University: Computer Science Department, March 1972 *
Linear Combinations of Sets of Consecutive Integers
' (with Richard Rado), Stanford University: Computer Science Department, March 1972 *
Sets generated by iteration of a linear operation
' Stanford University: Computer Science Department, March 1972 *
Packing a rectangle with congruent n-ominoes
' ''

Journal of Combinatorial Theory The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applicati ...

'', Vol. 7, Issue 2, Pages 107–115, September 1969 * ''Packing boxes with congruent figures'' (with F. Göbel), ''

Indagationes Mathematicae ''Indagationes Mathematicae'' (Latin for "Mathematical Investigations") is a Dutch mathematics journal. The journal originates from the ''Proceedings of the Royal Netherlands Academy of Arts and Sciences'' (or ''Proceedings of the Koninklijke N ...

'' 31, pp. 465–472, MR 40 #6362, 1969 * ''Some Results Concerning Polyominoes'' ''

Fibonacci Quarterly The ''Fibonacci Quarterly'' is a scientific journal on mathematical topics related to the Fibonacci numbers, published four times per year. It is the primary publication of The Fibonacci Association, which has published it since 1963. Its founding ...

'', 3, pp. 9–20, February 1965

References

External links

*
David A. Klarner fonds
University of Calgary Special Collections {{DEFAULTSORT:Klarner, David A. Mathematics popularizers Recreational mathematicians 20th-century American mathematicians California State Polytechnic University, Humboldt alumni University of Alberta alumni McMaster University alumni Academic staff of the Eindhoven University of Technology Binghamton University faculty Academic staff of the University of Calgary University of Nebraska–Lincoln faculty American number theorists Combinatorial game theorists 1940 births 1999 deaths People from Fort Bragg, California Writers from California Mathematicians from California