USMA-Harvard Math Competition

by BG (Ret) David Arney and George Rosenstein

The Harvard-United States Military Academy Mathematics Competition of 1933:
Genesis of the William Lowell Putnam Mathematical Competition

Abstract:

The predecessor of the first national Putnam Competition was a mathematics contest between Harvard and the United States Military Academy. The details of this contest from William Lowell Putnam's original idea for academic competition between schools in 1921 through the examination day in 1933 and later consequences of the competition are provided. Much of this history was compiled from original letters between the principle organizers found in a historical file of the Department of Mathematics, United States Military Academy (USMA). The role of the Mathematical Association of America, which has been involved in all 50 national Putnam Competitions, in this 1933 contest is also explained. A copy of the 22 question examination is provided.

Background

In an article in the Harvard Graduates' Magazine [1] in December 1921, William Lowell Putnam suggested the development of academic competitions between teams of undergraduate students of different schools in regular college studies. He believed that the motive of winning laurels for their college in team competition would provide students with more interest in their studies. He concluded the article with: "It seems probable that the competition which has inspired young men to undertake and undergo so much for the sake of athletic victories might accomplish some result in academic fields." The merits of his suggestion were shared by Mrs. Putnam and her brother, A. Lawrence Lowell, President of Harvard. Mrs. Putnam established a trust fund in her will to support such competition [1].

The first experiment of such academic competition supported by the Putnams was held in 1928 in the subject of English between Harvard and Yale. The winner, Harvard, won a prize of $5000. While Harvard and the Putnams wanted to repeat this contest, potential adversaries, Yale and Princeton, declined to compete in another English examination, as did Cambridge University in Economics.

Mrs. Putnam and Mr. Lowell finally found a willing competitor at a luncheon at Lowell's house after the ArmyHarvard football game on November 5, 1932. Mrs. Putnam volunteered funding for a mathematics competition between these same two schools. Herbert Robbins, an eventual competitor for Harvard, recalled Lowell's comments to his guests from West Point that even though Army "could trounce Harvard in football, Harvard could just as easily win any contest of a more academic nature." Upon returning to West Point, Major General D. Connor, Superintendent of the United States Military Academy, took exception to Lowell's comments and in a letter to President Lowell challenged Harvard to such a competition in the spring of 1933. In the letter, Connor wrote: "There is one matter that I have had in my mind ever since my visit and that is the mathematical contest your sister contemplated. I have no particular interest in the financial part of it, but I would very much like to test out method of teaching mathematics against that of your institution. I, frankly, think our method is superior to yours and would like to try it out.

Just two days later President Lowell responded: "Your challenge is a very interesting one which we will be glad to accept."

Preparations for the Competition

The leaders of the two schools then turned over the responsibility for arranging the details of the contest to the heads of their Departments of Mathematics, Lieutenant Colonel Harris Jones, later the Dean at USMA, and Professor William C. Graustein. The two Professors of Mathematics exchanged information about their curricula, pedagogy, and students in order to establish fair rules for the competition. Although they had never met one another, Jones1 wife was a friend of Graustein' 5 sisters and the Joneses had visited Graustein's parents in Cambridge several times. Therefore, their correspondence showed both mutual professional and personal respect for one another.

In a January 12, 1933 letter, Jones described the four semesters of mathematics taught at West Point. There were no other mathematics courses because all cadets took common courses for all four years. The freshmen class spent about 8 hours per week in mathematics class, and the sophomores were in mathematics class for 4 hours per week. The subjects studied included algebra from Wells' Advanced Course in Algebra, solid geometry from Phillips and Fisher, Elements of Geometry, trigonometry from Crockett, Elements of Plane and Spherical Trigonometry, analytic geometry from textbooks by Ziwet and Hopkins, Analytic Geometry, and by Smith and Gale, Elements of Analytic Geometry, differential calculus from Granville, Smith, and Longley,Elements of Differential and Integral Calculus, and integral calculus from Murray's Integral Calculus. The upper third of the class also took a brief, one-month study in the method of least squares from Bartlett's Least Squares at the end of their two-year mathematics requirement.

Graustein outlined Harvard's courses in a letter on January 16, 1933. The Harvard mathematics courses met for only 3 hours per week. The textbooks used in the courses for freshmen were Osgood and Graustein Plane and Analytic Geometry and Osgood'sIntroduction to Calculus. The sophomore level of instruction was under the guidance of tutors with the subjects of analytic geometry and algebra being covered.

While several important issues were resolved in these first letters, such as the competitors would be sophomores, the topics examined would be analytic geometry and calculus, and the test would be given in May 1933, a couple of potential problems surfaced. West Point wanted 20 to 25 men per side while Harvard preferred 10. Both teams wanted to be the home team and host the competition. The issue of the number of contestants would take quite a while to resolve. It can only be surmised that USMA thought it had the advantage in depth of quality students while Harvard believed it had several star performers. The enrollment of that sophomore class at USMA was 313, while Harvard had an enrollment of around 500 in their freshman mathematics classes and 150 in the sophomore-level mathematics. 35 to 40 men in the sophomore class at Harvard were concentrating in mathematics, and the team members were expected to be selected from that group.

The original plan was to have a third member from an impartial institution help the two department heads. In a January 18 letter, Graustein responded to Jones's earlier suggestion that Professor Woods of Massachusetts Institute of Technology join them in establishing the rules for the competition and write and grade the examination. In response, Graustein wrote: "The man I have in mind as the third member of the committee is the President of the Mathematical Association of America, the body which stands for collegiate mathematics in this country. Of course, if Professor E. T. Bell, of Pasadena, were still in office, the proposition would not be feasible since it would be impossible for the committee to get together for a meeting. But a new President, Professor Arnold Dresden, was elected at the beginning of the year, and fortunately, he is at Swarthmore. The committee might then meet in New York. Perhaps, if we had the docket sufficiently well prepared in advance, one meeting would suffice." Graustein then added: "Dresden ... seems almost 'ordained' for the job."

An influenza epidemic at West Point delayed Jones's next correspondence to Graustein until February 18, 1933. In this letter, Harris answered curricular questions raised by Graustein. In a February 22 letter, Harris lowered the suggested number of participants to 15 and as a concession to achieve that request offered to have his team travel to Harvard for the test on May 20. Jones suggested one three-hour test claiming Graustein suggestion of six hours would "convert an interesting contest into dull drudgery and kill the enthusiasm of the competition." Jones whole-heartily endorsed Professor Dresden as "obviously the man to make up the examinations and grade them if he will consent to do so."

In a letter written on February 27, Graustein definitely established the number of contestants per side as 10 by stating "that is the number that both Mrs. Putnam and Mr. Lowell have talked about in their conversations with me." He also made clear Mrs. Putnam s support by writing: "Under the circumstances, there was no expectation that the Military Academy would bear any part of the expenses. Mrs. Putnam is giving a certain sum for the purpose of the competition. The expenses will be taken from this sum and the balance given to the winning institution. Whether Mrs. Putnam will want to recognize in any way the members of the winning team, I do not as yet know."

Graustein invited Dresden to participate as a member of the competition committee in a letter dated February 27, 1933. He gave Dresden the background of the proposed examination and what issues had been already decided. Then he extended the invitation by writing: "You have been unanimously and enthusiastically selected and I hope (and pray) that nothing will stand in your way of serving with us." Dresden accepted the invitation made initially by Graustein and later tendered by Jones, Connor, and Lowell.

On March 3, Graustein once again asked Jones to consider the test be given in two sessions of three hours each. The second part could be slightly more challenging than the first. He also suggested twice as much calculus as analytic geometry in keeping with the ratio of the time spent on those topics in the curricula of the two schools. Also in keeping with the curricula, Graustein thought that there should be some theoretical questions on the test. His final request was for formula cards, logarithm tables, and Pierce'sIntegral Tables (Short Table of Integrals) to be allowed as references. His justification for the last reference was given in the letter. He wrote: "And, to be quite frank, our men would feel lost without integral tables, unless we spent a great deal of time training them to get along without them, and this seems hardly feasible." Graustein concession for these requests was that Harvard would be happy to travel to West Point since "the cadets all came to Cambridge last fall" for the football game.

By March 8 all the major issues for administering the examination were resolved. Jones agreed to a two-part test and the use of Peirce's Integral Tables as long as Harvard could bring enough copies for the cadet competitors to use since West Point did not use this book. The scoring was also agreed to be that used in cross-country meets, with first-place counting one, second counting two, etc. The team with the lowest sum for their 10 participants would be declared the winner.

The committee of three professors, Graustein, Jones, and Dresden, met at the Harvard Club in New York at 1:00 PM, Saturday, March 25 to decide the remaining details and establish the scope of the examination which Dresden would write later. Since they couldn't agree on whose formula card should be used, it was decided that Dresden would include necessary formulas with each question. The list of topics Jones took back to West Point for the cadets to study from was as follows:

I. Plane Analytic Geometry

1. Various forms of equation of straight line
2. Distance from a point to a line
3. Relative positions of lines, parallelism, perpendicularity, angle between lines.
4. Areas of rectilinear figures.
5. Conics - Derivation of equations in rectangular and polar coordinates.
6. Relative positions of straight lines and conics. Tangent to a point on the curves.
7. General equation of the second degree. Simplification by translations and rotation of coordinate axes.
8. Locus problems involving applications of the above.

II. Solid Analytic Geometry:

1. Equations of straight lines and planes.
2. Relative position of straight line and plane.
3. Distance from point to a plane or line.
4. Equations of spheres, cylinders, cones, surfaces of revolution.
5. Elementary properties of quadric surfaces.

III. Calculus:

1. Elementary theorems on limits.
2. Differentiation of a product, quotient, sine, logarithm.
3. Differentiation of algebraic, trigonometric, exponential and logarithmic functions.
4. Tangents and normals to curves.
5. Slope of curves, polar coordinates.
6. Maximum and minimum values of functions of one variable.
7. Curve tracing.
8. Velocity, acceleration, rates.
9. Radius and center of curvature.
10. Evolutes.
11. Definition and use of differentials.
12. Rolle's Theorem.
13. Taylor's Theorem, functions of one variable.
14. Series - tests for convergence, expansion of functions, integration.
15. Partial derivatives, total differential, applications of normal line and tangent plane to surfaces.
16. Definite integral, fundamental theorem.
17. Evaluation of indefinite integrals.
18. Areas, rectangular and polar coordinates.
19. Volumes and areas, surfaces of revolution.
20. Arc length, rectangular and polar coordinates.
21. Volumes by thin cross-sections.
22. Mean values.
23. Applications to mechanics, center of gravity, moment of inertia, radius of gyration, attraction, fluid pressure, work.
24. Elementary differential equations, first order, linear equations with constant coefficients, orthogonal trajectories, simple equations of higher order.

While writing the examination, Dresden became dissatisfied with their original plan to include the formulas with the examination questions. In letters to his two colleagues on May 5, He expressed concern that "giving a particular formula would be an indirect way of indicating a method of solution." He preferred the use of Harvard's formula card and suggested in the letters to Graustein and Jones that Harvard send copies to West Point for use by all contestants.

The cadet magazine, The Pointer, carried a humorous article about the preparations for the contest being carried out that spring at West Point. The author was one of the competitors and writes: "there has been introduced an activity which is more in keeping with the true purpose of an educational institution such as West Point. One refers to the squad of mathematics." The author of the article wonders about the examination writer: "one suspects that he will seize upon this opportunity to put forward all the pet problems which have been preying on his mind for many years." Then the grading is mentioned: "Wait, wait a minute, please--there will be a penalty. Someone has omitted his constant of integration." Finally, along with a humorous song, cheer, and line-up, the team statement is given: "We are really series about this contest. We really mean it. We're just dyne to meet those dumb Harvard guys, and we're determinant to win. We all hope to make our integral signs."

The actual preparation for the USMA team was taken quite seriously by the staff and students. From March 15 to May 20, the team was given special treatment usually only allowed for athletic teams by being excused from parade 3 days a week, excused from intramural athletics, and drilled in extra mathematics two afternoons a week. In addition, Lieutenant Cliff Robinson of the Department of Mathematics was excused from his afternoon duties of conducting marching drill and intramural athletics so he could help coach the team.

The Harvard team was confident of victory and took a lighter approach in their preparations. Robbins recalled that: "The Harvard Mathematics Department assigned Professor Marston Morse to coach the team, and we met with him about four times during the fall and winter. It was assumed that our Harvard intellects would easily carry the day, and our meetings with Morse were spent in general conversation rather than problem-solving."

By May 6, the arrangements for the trip were complete. The Harvard contingent would consist of 13 students (3 alternates) and Professor D.V. Widder since Graustein couldn't make the trip. This group left Boston on a noon train on Thursday, May 18 and arrived at West Point at 8:46 PM that evening. They were bussed by the Academy to the West Point Hotel. They slept at the hotel for two nights at the rate offered visiting athletic teams of $1.50 per man per day. Their meals were taken for free at the cadet mess. The first test was taken the next morning, Friday, May 19, from 8:45 AM to 11:45 AM. The 12-man (2 alternates) West Point team was excused from afternoon classes on Friday. Contestants from both teams went over to Lieutenant-Colonel and Mrs. Harris Jones's home for tea that afternoon. The second and more difficult test was given the next day, Saturday, May 20, at the same time as the first test. The Harvard team left for home at 1:20 PM Saturday.

This academic competition was well publicized in the sports section of the New York Times. The subtitle of the "Sports of the Times" column by John Kieran on May 18 read "The Coordinate Clash, or Block that Abscissa!" The article was a humorous analogy of the mathematics competition with a football game. It began with a poem called "A Logarithmic Lilt" which read:

"The Harvard horde is plotting, under cover of the dark, A fight to make the Crimson Chord subtend the Army arc. The Coefficient Corps has drilled with sharpened pencil tips And plans to drive the enemy away from the ellipse. The Harvard cry is 'Break the square and take the cube away!

While at the Point 'Abscissa' is the watchword of the day. And high upon the turret top the sentry turns his head And hears the Cambridge legion come with logarithmic tread. 'Advance and give the cosine!' rings the challenge through the air.

The Crimson host advances-and we hope the fight is fair." Later in the article, Kieran mentions the role of Lieutenant C. P. Nicholas, later head of the Mathematics Department, and Lieutenant Robinson as the Army coaches of the Analytics and Calculus, respectively. Other articles in newspapers continued the football analogy as their headlines read "Army Meets Harvard in Mathematical 'Go'," Squads at West Point Begin Contest in Calculus and Analytic Geometry," and "Harvard and West Point Line up on the Geometry Field."

The Test

Dresden wrote his examination from the outlines and topics given to him from Graustein and Jones. Each of the two three-hour tests consisted of 11 problems. The first test was more elementary and as Harvard contestant, Herbert Robbins stated: "The problems were rather cut and dried, technical integrations and the like, with little call for originality." Six of the eleven questions covered analytic geometry, two questions involved differential calculus, and the remaining three tested integral calculus. The second test was probably more demanding for all the competitors. It contained one question on logarithms and three analytic geometry problems. Six questions tested differential calculus and one involved integral calculus. The heading on both examinations read: "Participants in this examination are requested to give the full details of the work done in answer to the questions. Only TEN questions are to be answered. The use of formula card, of Pierce's Table of Integrals and of a table of logarithms, is permitted." The questions contained in the two tests and their solutions are provided in the appendices.

This test is obviously not as difficult or challenging as the first National Putnam Examination given on 16 April 1938. Nevertheless, it probably was quite a challenge for the sophomores from these two schools. From the very beginning of the planning, Graustein and Jones had a formidable test in mind for their students. Graustein's January l8 letter to Jones stated: "The second examination could be made a real test of the men's capacity. Of course, I do not mean to imply that the first examination should be a routine affair; that, too, should be a test of the ability of the men to apply their mathematics to new problems." To our knowledge, there were no raw individual scores ever published by Dresden. The rumor prevalent on the West Point team before the test was given was that a score of 20 percent would be excellent. Later, rumors raised the level of an excellent score to 60 or 70 percent.

The Results

The headline of a New York Times article read: "Army 'Mathletes' Defeat Harvard 98-112; Cadet Smith is First in Calculus Affray." The article quoted Dresden as saying "The papers bore evidence of the fact that the students worked very hard on this examination and that they are to be complimented for the serious way in which they prepared for it." The article was written from a press release provided at West Point by M.P. Echols.

The contestants in order of merit were as follows:

The Harvard Crimson covered the results in an article on June 5. The headlines read: "Crimson Bow to West Point Mathematicians" and "Harvard Mathematics Team Out figured by West Pointers." It was a close contest, and probably Army's advantage as home team spelled the difference. If one Harvard man had moved up just 7 places, the Crimson would have won. However, as Robbins reflected: "Back at Harvard we found out, to our shame, that we lost the competition to Army."

General Connor reflected the Academy's pride in his letter of congratulations to Jones on June 1, 1933. He wrote: "I feel that we have accumulated a very distinct advantage from the contest, in that every man who competed for the team was given a much more thorough knowledge of mathematics than he ever would have without the preparation incident to the contest. I congratulate you very sincerely upon the results which for the first time give me concrete backing to the belief that I have always held, that is, that our instruction methods at West Point were up to date and on a par with those of other similar institutions."

The Army Chief of Staff and previous Superintendent at the Academy, General Douglas MacArthur, heard the good news and wrote to General Connor: "Will you please express to the members of the team my personal congratulations on their splendid victory. It is a tribute not only to them personally but equally to the system in vogue at West Point and the instructors and professors who have evolved and carried out the details thereof."

The members of the West Point team were awarded certificates, medals, and mathematics books. They all wrote personal letters to Mrs. Putnam thanking her for supporting the competition and expressing enthusiasm for more contests. However, with Mr. Lowell's retirement from the position of President of Harvard in 1933 and Mrs. Putnam s fading health (she died in 1935), the Harvard-USMA competition was not repeated.

From this one competition, the inspiration for academic achievement sought by William Putnam in 1921 was manifested at both schools. Over two years later, the descriptions of the West Point contestants in their 1935 yearbook contained numerous references to the mathematics competition and listed their involvement with the Mathematics Squad as one of their achievements along with their athletic and club participation. The motivation to participate of the first place finisher, George Rosse Smith, who was also a football player and competed for a Rhoades scholarship, was quite novel. His yearbook description read: "It was, I believe, the inducement of a Snicker Bar that persuaded George to go into the Mathematical Contest with Harvard and take first place." All the members of the Army team continued their high academic achievement by having class standings in the top 20 of their graduating class. Seven of the ten spent over 20 years in the Army serving in two wars. Most later became professors and obtained their Ph.D.. None were mathematicians; all were engineers.

The 8th place finisher, Herbert Robbins of Harvard, later obtained a Ph.D. in Mathematics from Harvard and became the Higgins Professor of Mathematics Statistics at Columbia University. Despite being on the losing side, he states: "I never would have studied more than a year of mathematics, much less have become for a time a mathematician, were it not for my experience with the Putnam competition." The third-place finisher, R. D. Sard, obtained a Ph.D. in Physics at Harvard and later became a professor at Massachusetts Institute of Technology.

It is well known that after this competition Harvard played a key role in keeping William Putnam's dream alive. After their mother's death, Putnam's sons George and August consulted with George Birkhoff of the Harvard Mathematics Department. Birkhoff along with others in that department wrote the first national examination in 1938. He also set up some of the continuing principles for the competition: teams should consist of three people selected by the competing schools, the test should be administered by the Mathematics Association of America, and prizes should be distributed to several top teams and individuals. The secretary of the MAA in 1938, Professor W. P. Cairns of Oberlin College, established and published the final rules for the first test. Through the efforts of Birkhoff, Cairns, and many others, and the experience garnered from the Harvard-USMA Competition of 1933, the annual William Lowell Putnam Mathematics Competition came into existence in 1938 and has prospered in the 50 years since.

• Birkhoff, G., "The William Lowell Putnam Mathematics Competition: Early History," American Math. Monthly, 72 (1965) p. 469-483.
• Bush, L.E., "The William Lowell Putnam Competition: Later History and Summary of Results,"
• Putnam, William Lowell, "A Suggestion for Increasing the Undergraduate Interest in Studies," Harvard Graduate Magazine, Dec 1921.
• Robbins, Herbert, "Recollections of the First Putnam Examination"
• Gleason, A.M., Greenwood, R.E., Kelly, L.M., The William Lowell Putnam Mathematics Competition Problems and Solutions: 1938-1964, MAA, 1980.