July 16, 2025

It was a beautiful day on campus this morning, the sun was shining, and our participants were ready for our annual scavenger hunt. John Hagen, our supervisor of counselors, organized this event, so I will let him describe it.

Our first event of the day was our third annual Scavenger Hunt that we called "Hunt of the Ivies", held under gorgeous skies and mid-70s sunshine. Modeled after "The Amazing Race," this campus-wide challenge pushed our campers to apply not just mathematical insight, but also strategic teamwork and problem-solving under time pressure. Each team started at a different Ivy League-themed station and made their way across campus. At each station, teams had to solve a set of math problems in order to move forward; in some cases, they were sent on a "detour," where they had to solve an additional set of math problems to continue.

One team managed to complete all of the stations and even solved the Grand Ivy Final Challenge, a remarkable feat that no team had ever accomplished before. We invite you to try the final question for yourself:

Imagine that all the numbers from 1 to 100 inclusive are written on a blackboard. At every stage, you are allowed to erase two numbers that appear on the board (let's call the numbers you erased x and y) and in place of the two erased numbers, write the number x + y + xy. Repeat this operation until only a single number remains. What single number will remain?

The winning team utilized a strategy discussed by Professor Miller from Sunday's pirate lecture. Would you have been able to solve it?

Thank you, John, for providing all of us with this summary of Hunt of the Ivies-we all appreciate the time and energy you spent developing this scavenger hunt.

After our scavenger hunt, we all gathered in the Decker Social Space for our first speaker of the day, Teddy Einstein of Swarthmore College. Dr. Einstein's topic today was titled "How Big Is Infinity?" Now I'm sure many of you assume that all infinities are the same size, but incredible as it may seem at first, there are actually different sizes of infinity as Georg Cantor, the great German mathematician, proved over 100 years ago. At the time, other mathematicians could not believe this result and Cantor's work was not accepted. However, eventually it became the foundation of modern set theory,

For his lecture, Dr. Einstein (no relation to Albert!) asked the following question: How could it be possible for a hotel with infinitely many rooms be full and yet still have room for more guests? In his talk, he explained this amazing paradox and other fun facts about infinite sets. In particular, Dr. Einstein showed students how to compare the sizes of two infinite sets. While most of us intuitively think that all infinite sets are the same size, Dr. Einstein showed us this is not the case.

When we returned from lunch, our participants were introduced to Dr. Ralph Gomez of Swarthmore College. Professor Gomez's topic was Stepping into Other Dimensions of Space. As he explains this:

The idea of "dimension" is used a lot in superhero movies usually as a kind of porthole into another universe. Moreover, many video games are often called "2D" or "3D". The "D" stands for dimension but what does that really mean mathematically? The notion of dimension is a very important idea used in mathematics and science. In this talk, we will explore the mathematical concept of dimension of a space and try to construct spaces that go beyond "3D". We will try to mathematically "see" shapes that live beyond 3D. If we indeed venture into this higher dimensional realm, let us hope that we can return!"

We had a short break after Dr. Gomez completed his talk and then Dr. Paul Ellis returned for his second visit with our students. His topic today was The Futurama Theorem. For those of you unfamiliar with it, Futurama is an animated science fiction television series that has been broadcast since 1999. It follows the adventures of Phillip J. Fry who was frozen in 1999 and revived on December 31, 2999. Incredible as it may seem, in one episode of this long-running series, a real mathematical theorem now known as The Futurama Theorem is developed. In this episode titled The Prisoner of Benda, a brain swapping machine is invented that allows two people to swap brains. Unfortunately, this machine does not allow these two people to trade their brains with each other more than once. However, they can trade their brains with a third person.

Our Awards Ceremony was held in the Cromwell Hall Lounge in the evening. Our students all received books that were donated by The Art of Problem Solving (AoPS), a booklet with the questions and solutions for all the tournament rounds, and certificates. It's always a thrill for me to see the excitement of our participants as the awards are announced and to speak with some of the parents of our students before and after the ceremony.

Best of luck to all our participants in their future mathematical studies. And I hope to see many of you back here next summer at our Fourteenth Annual Math League Summer Tournament-all participants from this summer are automatically eligible for next summer's program.