July 26, 2025

The group embarked on an enlightening exploration of Princeton University itself. The morning was filled with discovery as they toured the historic campus that has nurtured some of the world's greatest minds.

The guided tour commenced at the iconic Nassau Hall, whose imposing Georgian architecture is a testament to Princeton's rich history. The guide shared an interesting detail: the architect, Robert Smith, subtly incorporated his own likeness into the front entrance—a face carved into one of the decorative elements.

As the tour progressed, students delved into the fascinating world of Collegiate Gothic architecture, a style widely adopted by Princeton in the late 19th and early 20th centuries. The guide explained how this style, while inspired by medieval Gothic architecture, was adapted to suit the needs of modern universities. Students were intrigued to learn of architects like Ralph Adams Cram, who played a pivotal role in shaping Princeton's distinctive appearance.

The Princeton University Chapel served as a stunning example of this architectural style. Students admired its towering spires, massive arched windows, and intricate stone carvings. The tour also included a visit to Firestone Library, where students learned about its extensive collection spanning over 70 miles of bookshelves. Students appreciated the concept of open stacks,which grants direct access to most library materials. They were also intrigued by the underground expansion, a creative solution that preserves the campus aesthetic while allowing the library to grow its collection.

Following the educational tour, students visited the Princeton University Store. The young mathematicians eagerly explored the aisles, selecting souvenirs to commemorate their visit. The store visit provided a perfect opportunity for students to reflect on the day's experiences and the rich mathematical heritage they had explored. Students also had up to an hour to explore the rest of the town, including the shops on Palmer Square.

After that it was back to campus for a lecture by Professor Steffen Marcus. Dr. Marcus of the mathematics department of The College of New Jersey discussed The Mountain Climber Problem. Each of our participants were provided with a two-meter piece of string so they could join in the activities Dr. Marcus had planned. He began with 1 peg in a mountain, a piece of rope, and a cat, Mauve, holding on to the rope. He asked how to loop the string around the peg so that if the peg was pulled out, Mauve would fall into the river below. The students quickly realized that no matter how they looped the string around the peg, Mauve would fall into the river. Next, Dr. Marcus put 2 pegs in the mountain and asked the students how to loop the rope around the pegs so that if either peg was pulled out, Mauve would fall into the water below. Students used their pieces of string and their fingers or pencils to simulate the pegs. After experimenting for a while, several students were able to find a way to loop the string around both pegs in such a way that no matter which peg was removed, Mauve fell into the river. From these two examples, Dr. Marcus showed the algebraic interpretation of these loops and then demonstrated how a solution to the two peg problem could be found algebraically. He next asked them to do this with 3 pegs, and several students succeeded in doing this. It turns out that the solution to this problem is really part of the study of non-Abelian groups.

After a short break, Steven Miller of Williams College gave a lecture to our participants. Professor Williams has been a long-time supporter of our summer program and an inspiration to our participants. He has devoted many hours this past year working with three of our participants from last summer, helping them develop a research paper for publication in a mathematics journal. His lecture today was titled “Egg Dropping: It Is All It Is Cracked Up To Be.” This lecture concerned finding the optimal way to find the highest floor in a building from which his special eggs can be dropped without breaking. He began with one egg to drop and then extended the problem to two eggs and more. This problem is much more difficult than it appears to be at first.