# William Lowell Putnam Mathematical Competition

The William Lowell Putnam Mathematical Competition, often abbreviated to Putnam Competition, is an annual mathematics competition for undergraduate college students, awarding scholarships and cash prizes ranging from \$2,500 to \$250 for the top students and \$25,000 to \$5,000 for the top schools. The competition was funded in 1927 by Elizabeth Lowell Putnam in memory of her husband William Lowell Putnam (Harvard 1882), who while alive was an advocate of intercollegiate intellectual competition. The exam has been offered annually since 1938 and is administered by the Mathematical Association of America.

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## The competition today

The Putnam competition now takes place on the first Saturday in December, and consists of two three-hour sittings and a lunch break. Each competitor attempts to solve twelve problems, nearly all mathematical proofs, which can typically be solved with only basic knowledge of college mathematics but which require extensive creative thinking.

Each of the twelve questions is worth any amount from 1 to 10 points, but the most frequent scores above zero are 10 points, for a complete solution; 9 points, for a nearly complete solution; and 1 point, for the beginnings of a solution. The examination is considered to be very difficult: it is typically attempted by students specializing in mathematics, but the median score is usually one or two points out of 120 possible, and perfect scores are exceptionally rare. In 2003, of the 3615 students taking the exam, 1024 (28%) scored 10 or more points, and 42 points was sufficient to make the top 102.

At a participating college, as many students as wish to take part in the exam may enter; but the school's official team consists of three individuals whom it designates in advance. Team scoring is analogous to that used in cross-country running; a team's score is the sum of the ranks of its three team members, with the lowest team score winning. It is entirely possible, even commonplace at some institutions, for the eventual results to show that the "wrong" team was picked — i.e., that some students not on the official team outscored the official team members. The top five teams win \$25,000, \$20,000, \$15,000, \$10,000, and \$5,000 respectively, with \$1,000, \$800, \$600, \$400, and \$200 for team members.

The top five individual scorers are named Putnam Fellows and awarded \$2,500. One of them is also awarded the William Lowell Putnam Prize Scholarship of \$12,000 plus tuition for graduate study at Harvard University. Sixth through 15th place individuals receive \$1,000 and the next ten receive \$250. The names of the top 100 students are published in American Mathematical Monthly.

In December 2003 the examination was taken by 3615 students from 479 colleges. The 2004 examination was held on December 4.

Many contestants have gone on to become distinguished researchers in mathematics and other fields. A number of them have received the Fields Medal or the Nobel Prize in Physics.

## Winners

### Top-scoring teams

Since the second competition in 1939, competing teams have been ranked.

 Year First Second Third Fourth Fifth 1938 (teams were not ranked) 1939 Brooklyn College MIT Mississippi Woman's College 1940 Toronto Yale Columbia 1941 Brooklyn College UPenn MIT 1942 Toronto Yale MIT City College of New York 1946 Toronto MIT Brooklyn College Carnegie Institute of Technology 1947 Harvard Yale Columbia UPenn 1948 Brooklyn College Toronto Harvard City College of New York (tie) McGill 1949 Harvard Toronto Carnegie Institute of Technology City College of New York 1950 Caltech Harvard New York University (NYU) Toronto 1951 Cornell Harvard Cooper Union City College of New York 1952 Queen's University Polytechnic Institute of Brooklyn Harvard MIT 1953 Harvard City College of New York Cornell UC Berkeley 1954 Cornell Harvard MIT Toronto 1955 Harvard Toronto Yale Kenyon 1956 Harvard Columbia Queen's University MIT 1957 Harvard Columbia Cornell Caltech Spring 1958 Polytechnic Institute of Brooklyn Harvard Toronto University of Manitoba Fall 1958 Harvard Toronto Caltech Cornell 1959 Polytechnic Institute of Brooklyn Caltech Toronto Harvard Case Western Reserve 1960 UC Berkeley Harvard MIT Michigan State Cornell 1961 Michigan State MIT Caltech Harvard Dartmouth 1962 Caltech Dartmouth Harvard Queen's University UCLA 1963 Michigan State Brooklyn College UPenn Caltech MIT 1964 Caltech MIT Harvard Case Western Reserve UC Berkeley 1965 Harvard MIT Toronto Princeton Caltech 1966 Harvard MIT Chicago Michigan Princeton 1967 Michigan State Caltech Harvard MIT Michigan 1968 MIT Waterloo UCLA Michigan State Kansas 1969 MIT Rice Chicago Harvard Yale 1970 Chicago MIT Toronto Illinois Institute of Technology Caltech 1971 Caltech Chicago Harvard UC Davis MIT 1972 Caltech Oberlin Harvard Swarthmore MIT 1973 Caltech University of British Columbia Chicago Harvard Princeton 1974 Waterloo Chicago Caltech MIT University of British Columbia 1975 Caltech Chicago MIT Princeton Harvard 1976 Caltech Washington University in St. Louis Princeton Case Western Reserve (tie) MIT 1977 Washington University in St. Louis UC Davis Caltech Princeton MIT 1978 Case Western Reserve Washington University in St. Louis Waterloo Harvard Caltech 1979 MIT Caltech Princeton Stanford Waterloo 1980 Washington University in St. Louis Harvard Maryland, College Park Chicago UC Berkeley 1981 Washington University in St. Louis Princeton Harvard Stanford Maryland, College Park 1982 Harvard Waterloo Caltech Yale Princeton 1983 Caltech Washington University in St. Louis Waterloo Princeton Chicago 1984 UC Davis (tie) Washington University in St. Louis Harvard Princeton Yale 1985 Harvard Princeton UC Berkeley Rice Waterloo 1986 Harvard Washington University in St. Louis UC Berkeley Yale MIT 1987 Harvard Princeton Carnegie Mellon UC Berkeley MIT 1988 Harvard Princeton Rice Waterloo Caltech 1989 Harvard Princeton Waterloo Yale Rice 1990 Harvard Duke Waterloo Yale Washington University in St. Louis 1991 Harvard Waterloo Harvey Mudd Stanford Yale 1992 Harvard Toronto Waterloo Princeton Cornell 1993 Duke Harvard Miami University MIT Michigan 1994 Harvard Cornell MIT Princeton Waterloo 1995 Harvard Cornell MIT Toronto Princeton 1996 Duke Princeton Harvard Washington University in St. Louis Caltech 1997 Harvard Duke Princeton MIT Washington University in St. Louis 1998 Harvard MIT Princeton Caltech Waterloo 1999 Waterloo Harvard Duke Michigan Chicago 2000 Duke MIT Harvard Caltech Toronto 2001 Harvard MIT Duke UC Berkeley Stanford 2002 Harvard Princeton Duke UC Berkeley Stanford 2003 MIT Harvard Duke Caltech Harvey Mudd 2004 MIT Princeton Duke Waterloo Caltech

### Teams ranked by historical performance

Below is a table of teams by the number of appearances in the top five and number of titles. Where multiple teams have the same number of appearances in the top five, they are ranked by number of championships, and then listed in alphabetical order.

While some may see this as a rough gauge of the level of the undergraduate mathematical programs at various institutions, a number of factors militate against this assumption:

• Some institutions have participated for many years while others are relative latecomers.
• Some university teams actively train for the competition with faculty help and reference to past years' questions; at others, there is a student club which practices — but at many institutions, there is no formal preparation at all.
• As described earlier in this article, it is possible for the official members of a team to not score as highly as others from the same university who have not been designated members of the official team.
• Finally, and most importantly, "contest math" is seen by many as quite different from original mathematical research — the real aim of university math departments — and is not necessarily the best predictor of it.

### Putnam Fellows

Since the first competition, the top five (or six, in case of a tie) scorers on the examination have been named Putnam Fellows. Within the top five, Putnam fellows are not ranked. Only six students have been Putnam fellows four times: Don Coppersmith (1968-71), Arthur Rubin (1970-73), Bjorn M. Poonen (1985-88), Ravi D. Vakil (1988-91), Gabriel D. Carroll (2000-03), and Reid W. Barton (2001-04). Fifteen others have won the award three times.

The following table lists all Putnam fellows from 1938 to present, with the years they placed in the top five.

 George W. Mackey (Rice) 1938 Irving Kaplansky (Toronto) 1938 Michael J. Norris (College of St. Thomas) 1938 Robert W. Gibson (Fort Hays Kansas State College) 1938 Bernard Sherman (Brooklyn College) 1938, 1939 Abraham Hillman (Brooklyn College) 1939 Richard P. Feynman (MIT) 1939 William Nierenberg (City College of New York) 1939 Edward L. Kaplan (Carnegie Institute of Technology) 1939, 1940, 1941 John Cotton Maynard (Toronto) 1940 Robert Maughan Snow (George Washington University) 1940 W. J. R. Crosby (Toronto) 1940 Andrew M. Gleason (Yale) 1940, 1941, 1942 Paul C. Rosenbloom (Penn) 1941 Richard F. Arens (UCLA) 1941 Samuel I. Askovitz (Penn) 1941 Harold Victor Lyons (Toronto) 1942 Harvey Cohn (City College of New York) 1942 Melvin A. Preston (Toronto) 1942 Warren S. Loud (MIT) 1942 Donald A. Fraser (Toronto) 1946 Eugenio Calabi (MIT) 1946 Felix Browder (MIT) 1946 J. Arthur Greenwood (Harvard) 1946 Maxwell A. Rosenlicht (Columbia) 1946, 1947 Clarence Wilson Hewlett, Jr. (Harvard) 1947 William Turanski (Penn) 1947 Eoin L. Whitney (University of Alberta) 1947, 1948 W. Forrest Stinespring (Harvard) 1947, 1949 George F. D. Duff (Toronto) 1948 Harry Gonshor (McGill) 1948 Leonard Geller (Brooklyn College) 1948 Robert L. Mills (Columbia) 1948 Donald J. Newman (City College of New York) 1948, 1949, 1950 Ariel Zemach (Harvard) 1949 David L. Yarmush (Harvard) 1949 J. W. Milnor (Princeton) 1949, 1950 John P. Mayberry (Toronto) 1950 Richard J. Semple (Toronto) 1950 Z. Alexander Melzak (University of British Columbia) 1950 Arthur P. Dempster (Toronto) 1951 Harold Widom (City College of New York) 1951 Herbert C. Kranzer (New York University (NYU)) 1951 Peter John Redmond (Cooper Union) 1951 James B. Herreshoff IV (UC Berkeley) 1951, 1952, 1953 Eugene R. Rodemich (Washington University, St. Louis) 1952 Gerhard Rayna (Harvard) 1952 Richard G. Swan (Princeton) 1952 Walter L. Bailey, Jr. (MIT) 1952 Marshall L. Freimer (Harvard) 1953 Norman Bauman (Harvard) 1953 Tai Tsun Wu (Minnesota) 1953 Samuel Jacob Klein (City College of New York) 1953, 1959, 1960 Benjamin Muckenhoupt (Harvard) 1954 James Daniel Bjorken (MIT) 1954 Leonard Evens (Cornell) 1954 William P. Hanf (UC Berkeley) 1954 Kenneth G. Wilson (Harvard) 1954, 1956 Howard C. Rumsey, Jr. (Caltech) 1955 Jack Towber (Brooklyn College) 1955 David B. Mumford (Harvard) 1955, 1956 Trevor Barker (Kenyon College) 1955, 1956 Everett C. Dade (Harvard) 1955, 1957 Richard Michael Friedberg (Harvard) 1956 David M. Bloom (Columbia) 1956, 1957 J. Ian Richards (Minnesota) 1957 Richard T. Bumby (MIT) 1957 Rohit J. Parikh (Harvard) 1957 David R. Brillinger (Toronto) Spring 1958 Donald J. C. Bures (Queen's University) Spring 1958 Lawrence A. Shepp (Polytechnic Institute of Brooklyn) Spring 1958 Richard M. Dudley (Harvard) Spring 1958 Joseph Lipman (Toronto) Spring 1958, Fall 1958 Alan Gaisford Waterman (San Diego State College) Fall 1958 John Rex Forrester Hewett (Toronto) Fall 1958 Robert C. Hartshorne (Harvard) Fall 1958 Alfred W. Hales (Caltech) Fall 1958, 1959 Daniel G. Quillen (Harvard) 1959 Donald Passman (Polytechnic Institute of Brooklyn) 1959 Donald S. Gorman (Harvard) 1959 Martin Isaacs (Polytechnic Institute of Brooklyn) 1959 Stephen L. Adler (Harvard) 1959 Stephen Lichtenbaum (Harvard) 1959 Jon H. Folkman (UC Berkeley) 1960 Louis Jaeckel (UCLA) 1960 Melvin Hochster (Harvard) 1960 William R. Emerson (Caltech) 1960 Barry Wolk (University of Manitoba) 1961 Elwyn R. Berlekamp (MIT) 1961 Edward Anton Bender (Caltech) 1961, 1962 John Hathaway Lindsey (Caltech) 1961, 1962 William C. Waterhouse (Harvard) 1961, 1962 John William Wood (Harvard) 1962 Robert S. Strichartz (Dartmouth) 1962 Joel H. Spencer (MIT) 1963 Lawrence A. Zalcman (Dartmouth) 1963 Lawrence J. Corwin (Harvard) 1963 Robert E. Greene (Michigan State) 1963 Stephen E. Crick, Jr. (Michigan State) 1963 Barry B. MacKichan (Harvard) 1964 Fred William Roush (North Carolina) 1964 Roger E. Howe (Harvard) 1964 Rufus Bowen (UC Berkeley) 1964 Vern Sheridan Poythress (Caltech) 1964 Andreas R. Blass (University of Detroit) 1965 Barry Simon (Harvard) 1965 Daniel Fendel (Harvard) 1965 Lon M. Rosen (Toronto) 1965 Robert Bowen (UC Berkeley) 1965 Marshall W. Buck (Harvard) 1966 Robert E. Maas (Santa Clara) 1966 Robert S. Winternitz (MIT) 1966 Theodore C. Chang (MIT) 1966 Richard C. Schroeppel (MIT) 1966, 1967 David R. Haynor (Harvard) 1967 Dennis A. Hejhal (Chicago) 1967 Don B. Zagier (MIT) 1967 Peter L. Montgomery (UC Berkeley) 1967 Dean G. Huffman (Yale) 1968 Gerald S. Gras (MIT) 1968 Neal Koblitz (Harvard) 1968 Gerald A. Edgar (UC Santa Barbara) 1968, 1969 Don Coppersmith (MIT) 1968, 1969, 1970, 1971 Alan R. Beale (Rice) 1969 Steven Winkler (MIT) 1969 Robert A. Oliver (Chicago) 1969, 1970 Jeffrey Lagarias (MIT) 1970 Jockum Aniansson (Yale) 1970 Steven K. Winkler (MIT) 1970 Arthur Rubin (Purdue, Caltech) 1970, 1971, 1972, 1973 Dale Peterson (Yale) 1971 David Shucker (Swarthmore) 1971 Robert Israel (Chicago) 1971 Michael Yoder (Caltech) 1971, 1972 Arthur Rothstein (Reed College) 1972 David Vogan (Chicago) 1972 Dean Hickerson (UC Davis) 1972 Ira Gessel (Harvard) 1972 Angelos J. Tsirimokos (Princeton) 1973 Matthew L. Ginsberg (Wesleyan) 1973 Peter G. De Buda (Toronto) 1973 David J. Anick (MIT) 1973, 1975 Grant M. Roberts (Waterloo) 1974 James B. Saxe (Union) 1974 Karl C. Rubin (Princeton) 1974 Philip N. Strenski (Armstrong State) 1974 Thomas G. Goodwillie (Harvard) 1974, 1975 Ernest S. Davis (MIT) 1975 Franklin T. Adams (Chicago) 1975 Christopher L. Henley (Caltech) 1975, 1976 David J. Wright (Cornell) 1976 Nathaniel S. Kuhn (Harvard) 1976 Paul M. Herdig (Case Western Reserve) 1976 Philip I. Harrington (Washington University, St. Louis) 1976 Steven T. Tschantz (UC Berkeley) 1976, 1978 Adam L. Stephanides (Chicago) 1977 Michael Roberts (MIT) 1977 Paul A. Vojta (Minnesota-Minneapolis) 1977 Stephen W. Modzelewski (Harvard) 1977 Russell D. Lyons (Case Western Reserve) 1977, 1978 Mark R. Kleiman (Princeton) 1978 Peter W. Shor (Caltech) 1978 Randall L. Dougherty (UC Berkeley) 1978, 1979, 1980 Charles H. Walter (Princeton) 1979 Mark G. Pleszkoch (Virginia) 1979 Miller Puckette (MIT) 1979 Richard Mifflin (Rice University) 1979 Daniel J. Goldstein (Chicago) 1980 Laurence E. Penn (Harvard) 1980 Michael Raship (Harvard) 1980 Eric D. Carlson (Michigan State) 1980, 1982, 1983 Adam Stephanides (Chicago) 1981 Robin A. Pemantle (UC Berkeley) 1981 Scott R. Fluhrer (Case Western Reserve) 1981 David W. Ash (Waterloo) 1981, 1982, 1983 Michael J. Larsen (Harvard) 1981, 1983 Brian R. Hunt (Maryland, College Park) 1982 Edward A. Shpiz (Washington University, St. Louis) 1982 Noam D. Elkies (Columbia) 1982, 1983, 1984 Gregg N. Patruno (Princeton) 1983 Benji N. Fisher (Harvard) 1984 Daniel W. Johnson (Rose-Hulman) 1984 Richard A. Stong (Washington University, St. Louis) 1984 Michael Reid (Harvard) 1984, 1987 Everett W. Howe (Caltech) 1985 Keith A. Ramsay (Chicago) 1985 Martin V. Hildebrand (Williams) 1985 Douglas S. Jungreis (Harvard) 1985, 1986 Bjorn M. Poonen (Harvard) 1985, 1986, 1987, 1988 David J. Zuckerman (Harvard) 1986 Waldemar P. Horwat (MIT) 1986 David J. Grabiner (Princeton) 1986, 1987, 1988 David J. Moews (Harvard) 1986, 1987, 1988 Constantin S. Teleman (Harvard) 1987 John S. Tillinghast (UC Davis) 1987 Jeremy A. Kahn (Harvard) 1988 Ravi D. Vakil (Toronto) 1988, 1989, 1990, 1991 Andrew H. Kresch (Yale) 1989 Christo Athanasiadis (MIT) 1989 Colin M. Springer (Waterloo) 1989 Sihao Wu (Yale) 1989 William P. Cross (Caltech) 1989 Jordan Lampe (UC Berkeley) 1990 Raymond M. Sidney (Harvard) 1990 Eric K. Wepsic (Harvard) 1990, 1991 Jordan S. Ellenberg (Harvard) 1990, 1992 Joshua B. Fischman (Princeton) 1991 Xi Chen (Missouri-Rolla) 1991 Samuel A. Kutin (Harvard) 1991, 1992 Jeffrey M. Vanderkam (Duke) 1992 Serban M. Nacu (Harvard) 1992 Adam M. Logan (Princeton) 1992, 1993 Craig B. Gentry (Duke) 1993 Wei-Hwa Huang (Caltech) 1993 J. P. Grossman (Toronto) 1993, 1994, 1995 Kiran S. Kedlaya (Harvard) 1993, 1994, 1995 Lenhard L. Ng (Harvard) 1993, 1994, 1995 William R. Mann (Princeton) 1994 Jeremy L. Bem (Cornell) 1994, 1996 Sergey V. Levin (Harvard) 1995 Yevgeniy Dodis (NYU) 1995 Dragos N. Oprea (Harvard) 1996 Ioana Dumitriu (NYU) 1996 Robert D. Kleinberg (Cornell) 1996 Stephen S. Wang (Harvard) 1996 Daniel K. Schepler (Washington University, St. Louis) 1996, 1997 Ovidiu Savin (University of Pittsburgh) 1997 Patrick K. Corn (Harvard) 1997 Samuel Grushevsky (Harvard) 1997 Mike L. Develin (Harvard) 1997, 1998 Ciprian Manolescu (Harvard) 1997, 1998, 2000 Ari M. Turner (Princeton) 1998 Nathan G. Curtis (Duke) 1998 Kevin D. Lacker (Duke) 1998, 2001 Christopher C. Mihelich (Harvard) 1999 Colin A. Percival (Simon Fraser) 1999 Davesh Maulik (Harvard) 1999 Derek I.E. Kisman (Waterloo) 1999 Sabin Cautis (Waterloo) 1999 Abhinav Kumar (MIT) 1999, 2000 Pavlo Pylyavskyy (MIT) 2000 Alexander B. Schwartz (Harvard) 2000, 2002 Gabriel D. Carroll (UC Berkeley, Harvard) 2000, 2001, 2002, 2003 George Lee, Jr. (Harvard) 2001 Jan K. Siwanowicz (City University of New York) 2001 Reid W. Barton (MIT) 2001, 2002, 2003, 2004 Deniss Cebikins (MIT) 2002 Melanie E. Wood (Duke) 2002 Ralph C. Furmaniak (Waterloo) 2003 Ana Caraiani (Princeton) 2003, 2004 Daniel M. Kane (MIT) 2003, 2004 Vladimir V. Barzov (MIT) 2004 Aaron C. Pixton (Princeton) 2004

### Elizabeth Lowell Putnam Award winners

Since 1992, the Elizabeth Lowell Putnam Award has been available to be awarded to a female participant with a high score. It is not awarded every year.

 Dana Pascovici (Dartmouth) 1992 Ruth A. Britto-Pacumio (MIT) 1994 Ioana Dumitriu (NYU) 1995, 1996, 1997 Wai Ling Yee (Waterloo) 1999 Melanie E. Wood (Duke) 2001, 2002 Ana Caraiani (Princeton) 2003, 2004

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