July 15, 2025

Our day began with our notorious Speed Round. The Speed Round consists of 60 questions that vary in difficulty from easy to challenging. Each participant must work on these 60 questions on their own and they have only 45 minutes to complete all of the questions. Each question is worth 3 points. The first few questions are relatively easy and even though the questions are not listed in their exact difficulty order, they do become progressively more difficult throughout the contest. An example of one of the easier questions would be “What is the largest prime factor of 2025?” With only 45 minutes for the entire 60 questions, the average time one has per question is 45 seconds. I’m just glad no one made me take this contest!

After a short break, we met our first lecturer of the day, Pat Devlin. Dr. Devlin has taught at Yale, and now teaches at Swarthmore College. After Dr. Devlin spoke to our students two summer ago, I immediately asked him to return again. The title of his lecture was “The Math of Communication.” He began his discussion by asking our students if they were familiar with the game “Twenty Questions?” A few did know the game, but most did not. If you are not familiar with it, it goes like this: One person thinks of something (maybe school or a dog or football). The other people then try to determine what the first person is thinking of by asking a series of yes/no questions to the person. Dr. Devlin then told our participants that he was thinking of an integer from 1 to 10 and asked them to ask him a series of yes/no questions that would lead them to finding his number. He explained that they could just ask him, “Is it 1?,” Is it 2?,” Is it 3?,” . . . , “Is it 9?” As soon as the answer to any of these questions was “yes,” they would know his number. This process might take as many as 10 questions. He then asked our students to find a more efficient method for finding what number he was thinking of. When they came up with a faster method for 10 integers, he then increased the possible numbers to the integers from 1 to 100. It soon became clear that by progressively halving the numbers, one could determine his number in no more than 7 steps. Dr. Devlin then made this game more difficult by changing the rules and allowing himself to lie once. In other words, one of his yes/no answers might not be true. However, he was only allowed to lie once, so if anyone asked him the same question two times, Dr. Devlin’s answer would be a lie if he gave different responses each time—once yes, and once no. The difficult of the game is greatly increased once one lie is allowed. He finished his talk by asking the students to think about how many questions would be needed to always find the number if there were 100 possible numbers he might be thinking of and he could lie once.

After lunch, Ian Whitehead, Assistant Professor of Mathematics at the prestigious Swarthmore College, discussed Apollonian Packings, in which areas are filled in with increasingly smaller circles. Like most of you, this was a topic I knew nothing about before Dr. Whitehead began his talk. The lecture started with a review of the subject, which started over 2000 years ago with Apollonius of Perga and continued through Rene Descartes (who wrote about it in letters to Princess Elizabeth of Bohemia!) and into the present. Images of Apollonian packings can be found in many places, including some Shinto shrines in Japan and Kandinsky paintings.

The discussion started with the concept of curvature, which is the reciprocal of the radius of the circle. Smaller circles thus have higher curvatures. Curvatures can also be zero (for a straight line) or negative (for a circle outside of an inner circle that is being discussed). Students were then challenged to figure out patterns in the curvatures of smaller circles in arrangements adjacent to larger circles. Then our attention turned to tangent circles, starting with groupings of three tangent circles and asking whether a fourth circle tangent to all 3 would be possible. It turns out that two such circles are possible, one “inside” circle and one “outside” circle. That discovery led to a discussion of the “rule of triangles,” which says that the relationship among the curvatures is: Inside tangent circle curvature + Outside tangent circle curvature = 2 X “Triangle”of curvatures, or x + y = 2 X (a + b + c). Students were challenged to calculate the curvatures of various circles in several Apollonian packings when given three of the curvatures as a starting point. Some of them were trickier than you might imagine!

We concluded the afternoon with our relay rounds. While these rounds are very challenging for us to organize, they are probably the rounds our students enjoy the most. In order for a team to get points on a relay round, they must successfully solve four questions in a row. Watching the students pass their answers to the next person until the fourth person solves their problem and gives the final answer to the proctor is always exciting for us to view. Once again, speed counts since the faster a team successfully completes a relay round, the more points they earn. Some of you may be wondering why the scores from our Speed Round are not posted. As a matter of fact, many years ago one young Chinese student asked me that exact question. As I explained to this student, anticipation is one of the joys of life. We will hold off announcement of the Speed Round scores until the Awards Ceremony is completed. Parents and family members are invited to join us tomorrow at 7 PM in the Cromwell Hall Lounge to celebrate the achievements of our students during their time with us.

This evening we had our annual movie night. No, Adam Raichel did not pause the film to tell some of his corny jokes! Instead, we were all treated to Spy Kids.

I hope to see many of you tomorrow for our awards ceremony.