Jan 16, 2024

Our day began with our notorious Speed Round. The Speed Round consists of 60 questions that vary in difficulty from easy to challenging. Each participant must work on these 60 questions on their own and they have only 45 minutes to complete all of the questions. Each question is worth 3 points. The first few questions are relatively easy and even though the questions are not listed in their exact difficulty order, they do become progressively more difficult throughout the contest. An example of one of the easier questions would be “If 100% of x is 100, what is the value of x?” With only 45 minutes for the entire 60 questions, the average time one has per question is 45 seconds. I’m just glad no one made me take this contest!

After a short break, we met our first lecturer of the day, Pat Devlin. Dr. Devlin has taught at Yale, and recently moved to Swarthmore College. After Dr. Devlin spoke to our students last summer, I immediately asked him to return again this summer. One of my interests as a young boy was magic. Although Pat is not really a magician, his lecture had elements of magic that appealed to me and our students. The title of his lecture was “How to Use Randomness in Proofs.” He began his discussion by saying he was going to prove he was a psychic. The first thing he dd was tell the students that he could predict the number on a die he was going to throw. Amazingly, he predicted a certain number and the die matched the number he predicted. Next he told the students that he had written a number on a piece of paper and placed it in his pocket. He threw a different die andthe number shown on the die was a 1. Next he reached into his pocket, pulled out a piece of paper, unfolded it, and showed everyone that the number on the paper matched the number thrown. He then asked our participants to prove that he wasn’t a psychic by finding ways he could have performed both of these tricks without having any special mental powers. Our students discussed several ways that Pat might have done these tricks. When we was finished with his lecture, I once again asked him if he would come back in future years for our summer program. He said, “Of course I will come back. These are some exceptional kids you have here.”

After lunch, Art Benjamin of Harvey Mudd College made his first appearance of the day with a lecture to our students about scams and how not to be tricked. He began his lecture by telling us he had a 100% accurate method for predicting the score of a baseball game before it begins. He promised us he would tell us this method before he concluded his talk. Now if you are like me, you cannot wait to hear how to do this and make some money betting on these games. Of course, I also realized that there must be some trick here since no one can know the score of a baseball game before it begins. Alas, I read a little too much into his original statement which I will explain at the end of this paragraph. He next offered us a chance to win some money by playing the followinggame with him: If you give him $10, he will predict whether a fair coin will land heads or tails. If he’s right, he keeps the $10, but if he’s wrong, he gives you $15. Will you take this bet? Why or why not? This sounded like a good deal to me until I thought about it for a while. As Dr. Benjamin went on to explain that if you win the bet, you are getting back your original $10 and $5 profit. But if you lose the bet, he wins $10. Since there’s a fifty-fifty chance he wins, he ends up with profit after playing enough games.. The expected profit for him on each game is 1/2 X $10 – 1/2 X $5 = $2.50. Art Benjamin went on to discuss several other games that appeared fair on the surface, but in fact were biased in one direction. Getting back to his original statement about scores of baseball games, he proved he told us the truth originally about being 100% accurate since the scores of all baseball games before they begin are 0—0. And I thought I was going to become a wealthy man!

We concluded the afternoon with our relay rounds. While these rounds are very challenging for us to organize, they are probably the rounds our students enjoy the most. In order for a team to get points on a relay round, they must successfully solve four questions in a row. Watching the students pass their answers to the next person until the fourth person solves their problem and gives the final answer to the proctor is always exciting for us to view.

This evening we had a special treat for all of us including participants, parents, and our staff. Dr. Art Benjamin presented his world-famous MathMagic show. If you have never seen his show, you are missing out on something special. While most of his tricks involved mathematics, his first trick was a trick that Penn and Teller might have done. He held up an invisible (imaginary) deck of cards and asked a volunteer from the audience to pick one card out of the invisible deck and visualize it. After selecting this card, the volunteer was told to place it back in the deck but with the card face down so that the side facing us would not show its value. Art then reached into his pocket and took out a real deck of cards with one card face down. He asked the volunteer to tell us what card he had visualized. When the volunteer told us the card, Art turned the face down card over and revealed it was exactly the card the volunteer said he had visualized! From there, Art went on to show us a lot of impressive metal mathematics tricks. First, he showed that he could square any two-digit integer faster than four volunteers could do it on a calculator. Then he did the same thing with three-digit integers and four-digit integers. Another math magic trick he did was telling us what day of theweek any particular date fell on given any date from about 1650 or later. After showing us a number of these tricks, he asked for questions from the audience. Many of the questions asked were questions about how he performed certain tricks and he answered every such question until someone asked him how he performed the very first trick, the invisible card trick. Art refused toexplain that one saying, “I’ll be thrown out of the magician’s union if I reveal the secret to that trick.” However, he did perform a shortened version of that first trick one more time so we could try to figure it out. Adam Raichel and I discussed this after the show completed, but we still haven’t figured this one out. This is the fourth time I have seen Art’s show and like fine wine, it just keeps getting better and better with age.