The Year Bethel Made the Sweet 16
… In Mathematics
It is fitting in Bethel’s 125th year to look back at some of its most notable accomplishments over those years. The performance of the 1964 Bethel Putnam Team, as a high point in the long history of mathematical distinction at Bethel, is one of those notable accomplishments. The story of that team and Bethel’s excellence in mathematics is worth taking the time to understand and to celebrate. To appreciate this story, we need first to develop some understanding of the Putnam itself. The rise of Bethel toward an elite mathematics program began with the teaching career of Dr. Arnold M. Wedel and culminated in the 1964 Putnam. But a superior mathematics program has continued due to the work of dedicated and talented professors, many hardworking students and support from Bethel presidents and academic deans who believed in the value of this program.
The Putnam
The William Lowell Putnam Mathematical Competition (the Putnam
for short) is an annual competition for undergraduate college students. It was established through Harvard University.In 1938 it was opened to all undergraduate students in U.S. and Canadian colleges and universities. Originally there were fewer than 50 schools and a couple hundred students competing. By the mid-1960s, there were about 200 schools and 1000 students. Currently there are more than 4,000 participants representing over 500 colleges and universities.
The Putnam is considered the most prestigious university-level math examination in the world. The competition is a six-hour test consisting of 12 10-point questions. The problems on this test usually require a deep mathematical understanding, but not necessarily advanced college mathematics. Solving a Putnam problem usually requires some creative insight in how to begin to approach the problem.
How hard are the questions? Let’s just say that 40 to 50 percent of the participants (almost all of whom are excellent math students) score zero points! A score of 50 points (5 problems correct) usually ranks in the top 100 individuals, which is a very impressive accomplishment. Several of the greatest 20th-century mathematicians and other scientists are among those who have finished at or near the top on the Putnam.
Students compete individually, but if a university or college wants to field a Putnam team as an institution, it must choose a three-member team, in advance. The school’s score is the sum of the ranks of the three team members. It turns out that it is often difficult for team sponsors to correctly predict the best performers among their Putnam contestants. For example, in 1959, Harvard had four of the top five contestants, but finished fourth in the team competition!
The institutions that have dominated the competition include the very best academic universities. The universities with the most top-five finishes are Harvard, MIT, Caltech, Princeton and Waterloo/Toronto. Only a few liberal arts colleges have ever placed near the top, since they are competing with all the biggest and best public and private universities.
When David (Uncle Davy) Richert joined the Bethel faculty in 1906, he brought with him a passion for mathematics and a belief in excellence that has lasted through the decades. He passed that enthusiasm on to his students. One of them, Arnold M. Wedel joined the mathematics faculty at Bethel in 1951, immediately after earning his Ph.D. in mathematics from what is now Iowa State University. It was Dr. Wedel who first introduced Bethel students to national mathematics competitions.
In fact, the ascent of the Bethel math program toward its peak performance of 1964 was constructed primarily by Dr. Wedel and several gifted and hardworking students.
Bethel’s first Putnam competitor was Samir Khabbaz. Samir was a brilliant student who had been at Bethel for two semesters when Arnold encountered him in a calculus class in the fall of 1951. Dr. Wedel recognized that Samir had exceptional abilities and in the fall of 1953, he had Samir take the Putnam. Khabbaz earned a score that ranked him 74th in the nation, an outstanding individual score! Samir went on to earn a Ph.D. at the University of Kansas and to teach and do research in algebraic topology for many years at Lehigh University.
From 1956 through 1965, Arnold worked with professors at the University of Kansas to establish and maintain the Kansas Intercollegiate Mathematics Contest, which was a three-hour math test open to all private colleges in Kansas. It was not as difficult as the Putnam, but it was still a very challenging test and was a good substitute/preparation for competing in the Putnam. Most of the private colleges in Kansas competed. Bethel won the competition six times in its ten years! One year Bethel even had the top five individual scores!
By 1959, Dr. Wedel felt he had a set of students who could compete at the national level. That fall, he fielded the first Bethel Putnam team, consisting of George Dick, Alfred Habegger and Nabil Khabbaz. (Dick had won the Kansas Intercollegiate Math Contest three times; Habegger later became a member of the English faculty at KU; and Nabil was the younger brother of Samir Khabbaz.) The team finished 48th in the country. Bethel teams continued to be competitive over the next four years.
Then, in 1964, Bethel’s team placed 14th in the nation.
The top five teams that year were Caltech, MIT, Harvard, Case Western Reserve and UC-Berkeley. It is likely that only a handful of schools the size of Bethel have ever finished that high in the Putnam. This ranking is comparable to a Bethel College basketball team making it to the NCAA Division I basketball Sweet 16.
Imagine the local and national amazement if that ever occurred.
The members of that greatest
Bethel Putnam team were: Ken Graber, Don Quiring and Elias Toubassi. Also competing for Bethel that year were Silas Law, Gary Lyndaker and Robert Pankratz.
Putnam practices were remembered for their fun, challenge, early morning hours (often starting at 6 on Saturday mornings), and doughnuts which Dr. Wedel brought to inspire
the team. During the week team members were assigned problems to bring to the next session to present. Sometimes, we just worked as a team to solve problems that Dr. Wedel would propose. The goal was to master a set of mathematical tools and to build skills that would help us see how to tackle a problem. It was also critical to learn how to write up a solution so that each step was clear and each built a logical progression from the beginning to the final step.
Ken Graber was a junior majoring in physics and mathematics. After graduating from Bethel, Graber earned an MA in public administration and management from the University of San Francisco. His career included 12 years as a Social Worker and Systems Manager with the San Francisco Department of Social Services. He spent 20 years doing system design and management with electronic data systems.
Don Quiring, who ranked in the top 100 on the Putnam that year, was also a junior. After earning his mathematics degree, with a physics minor, he was awarded an NSF scholarship for graduate school. He went on to earn a Ph.D. in mathematics from the University of New Mexico. Quiring taught mathematics at the university level for about 10 years and then spent 21 years as an engineer doing applied mathematics and mathematical modeling for Boeing.
Elias Toubassi, who was also a junior, graduated from Bethel in mathematics, with a minor in physics. He earned a Ph.D. in mathematics at Lehigh University, where he studied under his uncle, Samir Khabbaz. Elias served on the mathematics faculty at The University of Arizona for 27 years. He did research in Abelian group theory, taught and led a restructuring of their introductory mathematics curriculum.
Silas Law, a junior, also graduated with mathematics and physics majors. He earned an MA in mathematics and a Ph.D. in civil engineering at the University of Oklahoma. After some years of teaching and doing research, he went into private business.
Gary Lyndaker was only a freshman in 1964. He was fortunate to be part of the Bethel Putnam teams that finished in the top 64 in 1964, ’65 and ‘66. After graduating with a math degree, he worked in many varied vocations, but spent about 25 years in the field of information technology. He retired in 2010, following eight years as the CIO for the Missouri Department of Mental Health.
Robert Pankratz, a senior, majored in mathematics and later earned an MS in modern European History. He taught high school in Colorado for many years and then moved to Homer, Alaska, where he has run his own construction company.
Bethel’s Putnam achievements didn’t end in 1964. Between 1959 and 1994 (Dr. Wedel’s years as mathematics chairman), Bethel placed in the in the top 64 teams seven times, in the top 32 twice, including the Sweet 16
finish in 1964. Finishing in the top 64 put Bethel’s teams among the very best colleges and universities in North America.
Individual performances by Bethel students include a total of at least 23 contestants in the top 400, of which 11 finished in the top 200. Bethel students finished in the top 100 six times. Besides Samir Khabbaz and Don Quiring, Gary Lyndaker accomplished this in 1966, Andrew Rich did it in 1976 and Jon McCammond did it twice, in 1985 and 1987.
Dr. Richard Rempel was also part of Bethel’s mathematics faculty for many years, from 1965 until 2007. Like Arnold Wedel, Richard was deeply involved with the Putman teams and their success over the years. The three professors - Uncle Davy
Richert, Arnold Wedel and Richard Rempel — chaired the Bethel mathematics department with distinction for almost 100 years. Other notable mathematics faculty who supported the Bethel Putnam team during Wedel’s tenure included Paul Harms, who assisted in 1964, and Robert Neufeld.
But the strength of Bethel’s math department isn’t just measured by the Putnam. In recent years, Bethel's programming team has delivered strong performances in the ACM regional programming competition. This team, led by Professor Karl Friesen is challenged to use mathematical and programming skills to solve complex and tricky
problems. Bethel’s teams have out performed many major Midwestern universities. Graduating math students have also gone on to excel in graduate mathematics studies and in a wide variety of careers including teaching, medicine, actuarial science, industry research, computer science and even in the fields of peace studies, law and the modeling of human languages.
According to Dr. Keith Sprunger’s recently published history of Bethel College, the school’s original charter declared that Bethel’s purpose was to teach ‘all the higher branches of learning’
(10). Excellence in education became one of Bethel’s primary goals over the years, especially strengthened during the presidency of Edmund G. Kaufman in the 1930s and 1940s. Academic excellence perhaps reached a second peak in the early 1960’s under the leadership of President Vernon Neufeld and Dean Albert Meyer.
In the past, many of Bethel’s mathematics majors also majored or minored in one of the sciences; primarily physics, but also chemistry and biology. In recent years the STEM (Science, Technology, (pre)Engineering and Mathematics) disciplines have worked closely together on several initiatives. Science students, including those above plus Psychology, have published undergraduate research papers and some have been nationally recognized.
The loss of a physics major and reduction of some computer science classes has been a blow to the math department at Bethel. There is some hope that a physics major or minor can be rebuilt if enrollment can be increased sufficiently. But there is a new
opportunity to emphasize application of mathematics in biology and psychology, as well as chemistry and even other disciplines. Command of mathematical skills is a strong asset in a growing number of careers. President Perry White has shown through his enthusiastic support of STEM initiatives that he intends to assure that the sciences and mathematics remain strong components of the Bethel College academic disciplines.
Now, in celebration of Bethel’s 125th anniversary, we look back proudly on one of Bethel College's great academic achievements. We can thank Uncle Davy Richert, Arnold M. Wedel, Richard Rempel and others for a proud mathematical legacy. We look forward with confidence that Bethel will extend this mathematical — and Putnam — tradition. With the direction of current professors Karl Friesen and Lisa Thimm, working with the quality students who choose Bethel, maybe we’ll see another set of Bethel Putnam headlines again soon.
The answer is: n2 — n + 2
It is clear that the formula holds for n = 1, since in that case there would be 2 regions.
For all n > 1 (more than 1 great circle) we will use Euler’s formula for polyhedra:
E + 2 = V + F
where E is the number of edges, V the number of vertices and F the number of faces (in this case the regions will be the faces.) Euler's formula is true for a sphere provided V > 1, so it is true in our problem whenever n > 1.
Each great circle intersects every other great circle at 2 vertices. So the total number of vertices is twice the number of ways to choose 2 circles. Since we can choose all pairs of circles in (n(n — 1)) / 2 different ways, the number of vertices is n(n — 1). So:
V = n(n — 1) = n2 — n
There are 4 edges that meet at each vertex. So, 4V counts each edge twice and thus: E = 2V. Using the formula for V above, we have:
E = 2(n2 — n)
Solving Euler’s formula for F, substituting from the equations above, and simplifying gives the result:
F = E + 2 — V = 2(n2 — n) + 2 — (n2 — n) = n — n + 2
Therefore, the formula holds for all n > 0.