July 22, 2025

We had a series of fascinating lectures today as well as our Team Round and first night of our talent show.

Our day began with a return visit from Dr. Pat Devlin of Swarthmore College. His talk, “The Math of Communication,” opened with a game of “Twenty Questions” and quickly turned into a deeper exploration of logic and strategy. Students discovered that they could always guess a number between 1 and 100 in at most 7 yes/no questions—until Dr. Devlin changed the rules and allowed himself to lie once, but only once. He might not lie at all or he might lie once. With just one lie in play, the students had to think much more carefully: repeat a question and get two different answers? One must be the lie. Repeat a question and get the same answer? It must be true. With careful reasoning, the students were still able to crack the code.

Then came the second act: a magic trick using a grid of Bananagrams tiles. A volunteer flipped one tile while Dr. Devlin looked away. When he turned back, he instantly identified the changed tile. The trick? Before the volunteer acted, every row and column contained an even number of face-up tiles. Flipping one tile caused exactly one row and one column to become “odd”—revealing the intersection where the change had occurred. The room lit up when Dr. Devlin explained the math behind the trick. Students then explored what happened if two tiles were flipped—only to find that while the grid still gave clues, it was no longer possible to know exactly which tiles were changed. The “magic” fell apart, illustrating how mathematical elegance often relies on just the right assumptions.

Our second speaker today was Dr. Teddy Einstein of Swarthmore College, who posed a question that sounds simple but leads to some truly surprising mathematics: How big is infinity? Most of us assume that all infinite sets are the same size. As Dr. Einstein showed us, that assumption is far from true.

He began by discussing countable sets, those that can be paired one-to-one with the natural numbers. The set of even numbers, the set of integers, and even the rational numbers all fall into this category. While it may seem counterintuitive, these infinite sets are the same size in the mathematical sense.

Then came the twist. Dr. Einstein asked whether we could list all the real numbers between 0 and 1. Using a clever argument known as Cantor’s Diagonalization, he showed us that no matter how we try to list those numbers, we can always construct a new one that does not appear on the list. This proves that the real numbers between 0 and 1 form an uncountable set—a larger kind of infinity.

Dr. Einstein explained that this idea was first proposed by the German mathematician Georg Cantor in the late 1800s, and that Cantor’s work was initially met with disbelief and controversy. Today, Cantor is considered one of the founders of modern set theory, and his insights into infinite sets have reshaped how mathematicians think about numbers, size, and structure.

This was a fascinating and mind-stretching lecture. Our students were full of questions and walked away with a much deeper understanding of the strange and wonderful world of infinity.

After these first two lectures were completed, we went to lunch and then reported to the Social Science Building for our Team Round. For the Team Round the students on each team work together to solve 10 difficult questions in 60 minutes. These are very challenging questions that few students at this age level can solve. Each question was worth 100 points. Much to our surprise, several teams answered all the questions correctly—now this is impressive and a testament to the ability of our participants.

After the team contest round the students returned to Decker for the final lecture of the day, a presentation by Paul Ellis on "Hat Puzzles." In keeping with the teamwork theme of the afternoon, the students were asked to work together in groups to try to solve some very challenging logic puzzles. First, the students were asked to discuss the "muddy children" problem. In this scenario, a group of children is told that some of them have muddy faces, and each child can see every other child's face but has no way to know whether their own face is clean or muddy. The children are told that on the count of three, those with muddy faces should step forward, but that any child with a clean face who steps forward will be punished. If any child with a muddy face does not step forward, the process is repeated. The question at issue is: how many attempts will the perfectly logical children have to make before they get exactly the correct children stepping forward on the count of three? Students were able to come up with the correct answer by thinking through examples of groups with only 1 student, then a group with 2 students, and then a group with 3 students before a pattern emerged. If you want to know the answer for a group of n students, look at the end of this paragraph! After the muddy children problem, the students were given even more challenging problems involving prisoners in line trying to figure out whether they were wearing white or black hats, then a variation in which the hats could be any color in the rainbow. Other puzzles and their creative solutions involved trying to figure out the ages of the individuals in a group based on extremely limited information, or prisoners trying to figure out which two of 52 cards in a deck were switched without the ability to communicate. Solutions involved concepts ranging from parity (whether there are an odd or an even number of things) to mod 10 calculations (essentially the remainder when a number of things is divided by 10). The answer to the muddy children problem is: whatever number of children there are, that is the number of attempts it will take them to get it right! (So a group of 10 children would have to be given 10 attempts at stepping forward in order to guarantee that they could get it correct.)

After our busy afternoon, we had dinner and then returned to the Decker Social Space for the first night of our two talent shows. Now I know we have many talented students in our group, and about 40 of our participants signed up for the talent shows, but attrition (stage fright?) reduced the performers to eleven for our first night. Luckily, the group that did perform made up for their small quantity by their large quality. We had two singers (one performing a Chinese song and the other a Taylor Swift song), three very talented pianists, a magician ready to challenge Penn and Teller, a very gifted guitarist, a Rubik’s Cube solver who did something I’d never seen before—he used only one hand to solve Rubik’s Cube very quickly, a wonderful clarinetist, and a violinist who may soon be playing first violin with the New York Philharmonic Orchestra. Oh, did I forget to mention that one of our participants set an unofficial world’s record for the number of digits of pi recited in 60 seconds? What a night!