## William Lowell Putnam Mathematical Competition

In 1921 **William Lowell Putnam** printed an article in the December issue of the *Harvard Graduates Magazine.* In it he wrote:-

The idealism of the undergraduate student, his eagerness to achieve something for his college, for his country, or for any cause that fills him with enthusiasm, is constantly referred to with admiration by those in charge of the universities. This unselfish impulse is recognised as one of the strongest forces in a student's life, and great results have been and are being accomplished by appealing to it. ... But it is a curious fact that no effort has ever been made to organise contesting teams in regular college studies. All rewards for scholarship are strictly individual and are given in money, or in prizes, or in honourable mentions. No opportunity is offered a student by diligence and high marks in examinations to win or help in winning honour for his college. All that is offered to him is the chance of personal reward. Little appeal is made to high ideals or to unselfish motives.

So who was William Lowell Putnam? He had been born on 22 November 1861 and studied mathematics at Harvard being a member of the class of 1882. He made a career as a lawyer and banker and in 1888 he had married Elizabeth Lowell. Elizabeth was the sister of Percival Lowell, the astronomer who founded the Lowell Observatory in Flagstaff, Abbott Lawrence Lowell, who became President of Harvard, and Amy Lowell, who became a famous poet. Now the reader may wonder why William Lowell Putnam and his wife shared the name of 'Lowell'. The answer is that they were related before marriage, being third cousins. Putnam was already handling the financial affairs of Lowell family before his marriage and he continued in this role. Putnam died in June 1924. His wife Elizabeth and her brother Abbott Lawrence Lowell, at that time President of Harvard, both shared Putnam's ideas about education and Elizabeth Putnam made provision in her will to created a trust fund known as the William Lowell Putnam Intercollegiate Memorial Fund. Two of the Putnam's sons were to be trustees of the Memorial Fund.

Elizabeth Putnam died in 1935 but moves had been made to begin competitions before her death. The first was in 1928 when Harvard and Yale competed in an English Competition. This first contest was won by Harvard but attempts to encourage Yale and Princeton to take part in a second contest met with refusals. The next competition was in fact a mathematics competition in 1933 between Harvard and West Point Juniors. Problems were set on calculus, analytic geometry, and differential equations and West Point Cadets were victorious. Again the winners were enthusiastic to continue the series and again the losers (this time Harvard) declined the challenge.

Abbott Lawrence Lowell retired as President of Harvard in 1933 and, as we noted above, Elizabeth Putnam died two years later. Now the family had a tradition of mathematics, William Lowell Putnam being a mathematics graduate as was Abbott Lawrence Lowell and one of the two sons who were now Trustees of the William Lowell Putnam Intercollegiate Memorial Fund. The two trustees approached their friend, the Harvard mathematician George David Birkhoff. The example of Hungary with the Loránd Eötvös problems established in 1882 was an example which they all felt showed the extremely positive effect of problem solving on the mathematical life of a country. All agreed that a mathematics competition was the most appropriate to organise for North America. They set up some principles from the beginning:

1. The competition would be open to 3-person teams selected by a university to represent them. It would also be open to individuals from any college in the United States or Canada.

2. The competition would be administered by the Mathematical Association of America.

3. There would be prizes and honourable mentions for several teams and individuals.

4. A graduate fellowship for Harvard would be offered to one of the top five competitors. (This was done deliberately so that other factors could be used to distinguish between the best problem solvers as to which was likely to make the vest researcher.)

Birkhoff was largely responsible for setting the first paper which was sat in 1938. As Harvard set the paper, no team from Harvard was allowed to compete. The university of Toronto won the first competition and they were asked to set the next paper and not participate. Irving Kaplansky was chosen for the Harvard fellowship - clearly a superb choice. The second competition was won by Brooklyn College, with Richard P Feynmann (MIT) an individual winner. The third competition was won by the University of Toronto, Brooklyn College won the fourth, and the University of Toronto the fifth which was in 1942. How long these two would have alternated as winners had the competition not been halted for World War II, it is impossible to tell. No competition was held in 1943, 1944 and 1945 but it was restarted in 1946.

After the restart in 1946 some changes were made to the way the competition was run. A committee set up by the President of the Mathematical Association of America set the papers, each committee serving for three years. The 1946 competition was administered by Harvard (actually by Garrett Birkhoff) and the competition was held on 1 June. With the changes to the setting of the papers, every college was allowed to compete including those from which someone was serving on the committee which set the papers. It was agreed, however, that those setting papers would not be involved in coaching students. Among those who served on the committee to set the papers was Tibor Radó, Mark Kac, Irving Kaplansky, Andrew M Gleason, L M Kelly, Leo Moser, Gian-Carlo Rota, and H S M Coxeter.

In 1948 the Mathematical Association of America resumed its administrative duties. It was decided that a Director would be appointed for a five year term. L E Bush was appointed and served for 5 terms of five years each. By 1958 the syllabus was given as follows:-

The questions will be taken from the fields of calculus(elementary and advanced)with applications to geometry and mechanics not involving techniques beyond the usual applications, higher algebra(determinants and the theory of equations), elementary differential equations and geometry(advanced plane and solid analytic geometry).

Here are two sample questions from the 1954 paper:-

1. Prove there are no integers *x*, *y* for which *x*^{2} + 3*xy* - 2*y*^{2} =122.

2. Prove that every positive rational number is the sum of a finite number of distinct terms of the series

1 +

^{1}/_{2}+^{1}/_{3}+^{1}/_{4}+^{1}/_{5}+ ... +^{1}/_{n}+ ... .

Notes. To solve the first complete we would like to complete the square but this would lead to quarters. So first multiply by 4, then complete the square to get

(2

x+ 3y)^{2}- 17y^{2}= 488.

Put *u* = 2*x* + 3*y* to get *u*^{2} - 17*y*^{2} = 488. Reduce mod 17 to now find that *u*^{2} = 12 mod 17 which is easily seen to be impossible (just check the possibilities).

The second problem has particular historical interest since the ancient Egyptians knew how to do it! In fact every rational has infinitely many different such decompositions or, thought of another way, one can omit any finite number of terms of the series and a decomposition (even infinitely many) is still possible.

The William Lowell Putnam Mathematical Competition is still an annual event with the Sixty-Fifth taking place on Saturday, 3 December, 2005. In its present format the competition takes place on the first Saturday in December every year. There are 12 problems and contestants have two sessions of three hours each in which to solve them.

A full list of the questions from the Putnam competition (with solutions!) is available HERE

JOC/EFR August 2006

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Putnam_competition.html