By Dr. Zebediah Engberg
For the fifth time in as many years, the Wasatch Academy math team attended the November Harvard MIT Mathematics Tournament. This prestigious event brings in nearly 1000 students from North America and Canada. Many major cities have selective “math circles” whose students compete in this tournament, and all of the elite east-coast boarding schools attend. As usual, we felt like small fish in a big pond.
While lacking the experience, size, and prestige of other teams, our strengths included persistence and collaboration. This year, our team consisted of veterans William Wang, Nguyen Le, and Bryan Lyu, as well as newcomers Karen Liu, Jacqueline Wang, and Charlotte Cai.
We spent Friday on the Harvard and MIT campuses role-playing the lives of ivy league college students. We ate great food, toured famous institutions of learning, traveled through the Cambridge underground, and stimulated our minds with hard math. During our three hours of practice on Friday, I saw the power of teamwork as my students leveraged their strengths off of one another to problem solve. Although we lacked confidence, we made up for it with team cohesion.
Arriving on the Harvard campus Saturday morning, the energy was palpable. The Harvard Science Center was swarming with high school students, undergraduate organizers, parents, coaches, and various mathematical ambassadors. In the morning, students congregated in an enormous lecture hall for two individual rounds. The afternoon featured a team round emphasizing deep thought and creativity as well as the fast-paced intense “guts” round.
The HMMT contest problems, written by Harvard and MIT students, emphasize deep creative thinking, an open mind, and the ability to consider a problem from many perspectives. Most HMMT problems can be solved in several distinct ways, but proceeding from the problem to its solution is extremely difficult. Mathematical tools are of no help in recasting a written problem into a mathematical phenomenon — imagination is the only way to get there. For many problems, one needs to find and manipulate a pattern, to generalize a specific phenomenon, to specialize an abstract property, and to use tools from disparate parts of mathematics in tandem. One of the most accessible problems from the general round is included below.
Examples of such numbers include 58,402,917 and 48,126,357. This problem is already stated in a manner that is ripe for mathematical attack. It lies at the intersection of combinatorics (the mathematics of counting and enumeration) and number theory (the study of whole numbers). To make progress, one first needs a criterion for discerning multiples of 9 which contain exactly 8 of the 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Once such numbers are characterized in terms of possible digits, then the counting begins. The final answer comes out to be 181,440.
As is usual for our team, we excelled in the collaborative team and guts rounds. In the guts round, instead of working in classrooms, a huge lecture hall is taken over by hundreds of students split up into teams. Each team strategically stakes out a small zone in the lecture hall. Problems are given in sets of three — once a team solves all three, a student runs their solutions up to a proctor who gives the students another set of three. This repeats for 15 rounds. A massive leaderboard is projected so that teams can see their progress in real time.
Of the 150 teams, without a doubt, our team had the most interesting guts name. While our official team name was Wasatch Dream, our short guts name was the mathematical equation:
This equation, which is false in most number systems, is a common mathematical mistake which high school students enjoy making.
In the guts round, our team saw our biggest successes of the day as we solved problem after problem correctly. Although our collaborative skills were on full display in this round, we were not able to keep pace with many of the powerful teams. In the final overall results, our team was ranked 100 out of 150 teams. Our ranking was consistently higher in the collaborative rounds, though we were still far from the leaderboard. Despite not finishing close to the top, our students and I felt accomplished by the event. We thought deeply and creatively, we showed enormous passion, and we had a small taste of the highest level of high-school mathematics. Our brains were fried after the contest, yet after our 4 am start the next morning, we spent much of our plane ride home working on the problems that had stumped us in the tournament.