Jan 24, 2024

After breakfast, we all met at the Social Sciences building for our first contest round of this summer. Today’s contest round was our Team Round in which the students on each team work together to solve 10 difficult questions in 60 minutes. These are very challenging questions that few students at these grade levels can solve. Amazingly, FOUR of our 6-7th grade teams received perfect scores of 1000 points (each question was worth 100 points) on their contest round, and on the 8-9th grade contest round, which is significantly more advanced, we had a team with 900 points! To say we were all very impressed with these scores is putting it mildly. A great performance by our participants!

After the team round we went back to our traditional home base in Decker for the first math presentation of the day. First up was Ian Whitehead, Assistant Professor of Mathematics at the prestigious Swarthmore College. Professor Whitehead discussed Apollonian Packings, in which areas are filled in with increasingly smaller circles. 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!

Lunch today was at the Brower Student Center, the building where we had our official opening dinner. Then it was back to Decker for the next math presentation of the day. Students heard from Deven Ware, a representative of the Art of Problem Solving. Most of you are probably familiar with that company, but for those who aren’t you might want to check them out - they publish excellent workbooks for many different subjects in mathematics, and have a significant online learning presence as well.

Deven’s presentation posed an initial question to the students: what general strategies can you use for problem solving? Students came up with an impressive list of approaches: think harder, come up with a smaller, parallel version of the real problem (also known as a “toy” problem), gather information before starting (find the key idea), use resources, use the process of elimination, don’t be afraid to be wrong, reframe the question, write things down, take a break, and even act like you are talking to someone else (which was a new one for our speaker to hear from a group of students!)

Deven’s first mathematical issue to explore was - what different numbers of smaller squares can you break the entire area of a larger square into? (It seemed very fitting, given the prior presentation on filling a space with small corcles!) Students experimented with different possibilities - the easiest numbers, of course, were the perfect squares, but students then came up with solutions for 7, 10, or 13 squares. Students were encouraged to look for patterns (and to add drawing a picture to their list of problem solving techniques). They then discussed a conjecture that all even numbers except 2 would be possible - that could be achieved by starting with a big square in one corner and a small square in the opposite corner, then dividing the side areas into n-2 squares. They then made the leap to realize that by adding 3 to the possible even numbers one can get any odd number - so all numbers greater than 5 are possible! The possibility of 5 as a possible solution was left to the students as an open question to prove or disprove on their own.

Students also reasoned through the ideal strategy involved in a game of removing chips from piles of chips in several stacks, and then were left with one last problem “to go”: in a bowl of noodles, pick one noodle end point at random and connect it to another random noodle end point - if the connections complete a closed loop, take it out. What is the expected number of loops one would have at the end of such a process?

Our final speaker of the day was Professor Jay Luo, professor of cybersecurity at Rider University. Cybersecurity is certainly a hot topic given the events of the day! Professor Luo started his presentation with a question: how can we safeguard our digital data? The answer to that question started with a discussion of prime numbers and factorization. Students were challenged to do some difficult factorizations, and then to consider the difficulty involved in factoring extremely large composite numbers through trial and error.

After understanding the ease with which two extremely large primes can be multiplied, but the difficulty of taking an extreme large composite number and finding its prime factors, students came up with other examples of processes that are easy to accomplish in one direction but difficult to do in the other. Professor Luo added to that list certain animal traps, postal drop boxes, and padlocks. From those humble beginnings the conversation proceeded to cybersecurity and the way the “RSA algorithm” for encryption and decryption works using prime numbers in securing messages and other computer applications.

Professor Luo explained that there is a “public key” for encryption and a “private key” for decryption. The use of these keys involves modular arithmetic (using the remainders from division). The process gets quite complicated! Students were given some resources for further exploration, including a recent paper on the RSA algorithm here: https://arxiv.org/pdf/2308.02785v2.

After that, it was off to dinner. At this point we received some truly unexpected news: our beloved home base in Decker Hall would be unavailable to us for the rest of the program! Apparently there was a malfunction in the fire suppression system in the building, and it is not something that can be fixed quickly. All future events on the calendar that were scheduled for Decker will be relocated to Cromwell next door, the Social Sciences building where we had our team round this morning, or the Brower Student Center. We will keep you posted as the schedule is finalized, but tomorrow is already planned: the math lectures and talent show will be in Cromwell, and the contest rounds will be in the Social Sciences building as originally scheduled.

So, about that talent show! Given the unexpected disruption in our planned location, we pushed it back one day. The first day will now be tomorrow, and the second round will be on Friday. Instead, we had a movie night tonight. Students enjoyed ice cream with toppings and cookies while we watched Inside Out. It is a great animated film about emotions - I highly recommend it if you haven’t seen it!