Jan 28, 2024

After breakfast, today started with a presentation by one of Math League’s most veteran presenters, Professor Steven Miller. The presentation started with a discussion of the impracticality of Roman Numerals for mathematical operations. Students were also introduced to the Babylonian system of math, which was essentially in base 60 (we use base 10, computers use base 2). The Babylonian’s choice to organize their numerical system around the number 60 explains some otherwise odd choices in the way we measure certain things. (For example, it is the origin of the 360 degrees in circles, and it makes it more understandable that there are 60 seconds in a minute.) Why would they use 60? It is divisible by 1, 2, 3, 4, 5, 6, 10, 12…certainly it makes life easier if you don’t run into fractions when dividing! On the other hand, if you have 60 choices for x and y in your multiplication tables, there are 3600 entries to learn! How many of us know basic multiples of even 13 or 17?

Professor Miller began the next part of his talk with an intriguing proposition: he offered a $500 reward and enthusiastic letter of recommendation (and probable admission) to any college or university for any student who could come up with a solution to an open problem in mathematics: find a geometric proof that the square root of 7 is an irrational number. (An irrational number is a number that cannot be expressed as a ratio of two integers.) Professor Miller demonstrated a proof that the square root of 2 is irrational, a “proof by contradiction” that is one of oldest proofs still used in mathematics. Of course, that is an algebraic proof, and Professor Miller also showed that there is a geometric proof involving overlapping squares (Tennenbaum’s Proof). Next up were a geometric proof that the square root of 3 is irrational (using overlapping triangles) and the starts of geometric proofs that the square root of 5 is irrational (overlapping pentagons) and that the square root of 6 is irrational (more overlapping triangles). To date, however, we cannot geometrically prove that the square root of 7 is irrational, and that is the challenge! (Professor Miller notes also that it is very tough to do a proof that the square root of 6 is irrational using hexagons instead of triangles and says he will write a letter of recommendation for anyone who can do that.) The discussion closed with a mention of the sequence of numbers 1, 3, 6, 10, 15, 21, 28, … students were able to identify it as the triangle numbers (generated by adding 1, adding 2, adding 3, etc.) - but since the next triangle number is 36, which isn’t irrational, the triangle approach to proving irrational numbers will eventually break down.

Professor Miller ended his presentation with a discussion of the properties of means (arithmetic, weighted, and geometric) and of continuing fractions. It was quite an informative exchange!

Following Professor Miller's presentation, students seized the opportunity to enjoy a beautiful southern New Jersey morning by participating in the third annual "Great Tiger Hunt." Inspired by "The Amazing Race," this year's event challenged math campers to demonstrate both mathematical prowess and historical knowledge. Eight stations, each named after a distinguished Princeton scholar, were set up around campus.

At each station, teams were tasked with solving five math problems and identifying a notable Princeton figure based on given clues. For instance, one prompt inquired:
"In the early 1940s, I became the first woman hired as a faculty member by Princeton's physics department, breaking barriers in a then all-male institution. My groundbreaking experiments later disproved the conservation of parity in weak interactions, revolutionizing nuclear physics and earning me the title of the 'First Lady of Physics'. Who am I?"

To progress, teams were required to correctly answer the specified number of math problems and accurately identify the scholar. To assist with identification, teams were provided with a list of ten Princeton scholars and their brief biographies. As teams raced across campus, they visited stations until all challenges were completed. Unsuccessful attempts resulted in "detours."

The competition was fiercely contested, with two teams falling just one station short of finishing. The tiebreaker was determined by the highest number of correct answers at non-detour stations. Ultimately, a six-person hybrid team composed of members from Teams 12 and 13 emerged victorious, winning by a margin of three questions. As a reward, they will enjoy a prize of their choice tomorrow, with details to be announced.

Following the excitement of the morning's Great Tiger Hunt, the group embarked on an enlightening exploration of Princeton University itself. The afternoon 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 dinner and free time for writing, journaling, or other activities.