Jan 19, 2024

Today started with a presentation from Professor Ralph Gomez of Swarthmore College about dimensions. Professor Gomez is a very dynamic speaker, who kept the students very engaged with his high-energy, highly interactive approach to teaching! The students were introduced to a hypothetical cat, Mathy. If Mathy lives in a 1-dimensional world, Mathy’s address can be specified by exactly one number. For example, it could be said that Mathy lives at “1” and it would be perfectly clear where Mathy lives. In a 2-dimensional space Mathy’s address would require 2 numbers to pinpoint, for example (1,2). From there students were led through a discussion of n-dimensional spaces, what can go wrong if the specifications of a location lack one or more of the required dimensions, and how dimensions are used in applications such as video games. Particular attention was paid to the problem of visually representing 4-D objects, and students were challenged to draw a 4-D cube, also known as a hypercube.

Our second 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 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. Students came up with climbing a mountain vs descending, or mixing two ingredients vs separating them out. 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. Students were left with a final question to think about after the presentation: how can you securely deliver the prime number “key” to the end user of an encrypted communication so that they can use it in decryption?

After a break for lunch, students participated in a presentation from Keith Calkins, a veteran math educator and physics PhD who has helped Math League with editing questions and has presented to our students many times in the past. The presentation started with each student choosing a favorite number and then discussing the properties of the chosen numbers. (Counting numbers? Whole? Rational? Real?) The group explored different ways to think about and even to express numbers, for example in Roman numerals or in base 2. Bases in particular were discussed, including base e (an irrational base) and base -3 (a negative base)!