July 13, 2025

Our second day on campus began with Steven Miller of Williams College delivering two talks to our participants. Professor Williams has been a long-time supporter of our summer program and an inspiration to our participants. He has devoted many hours this past year working with three of our participants from last summer, helping them develop a research paper for publication in a mathematics journal. His first lecture today was titled “Egg Dropping: It Is All It Is Cracked Up To Be.” This lecture concerned finding the optimal way to find the highest floor in a building from which his special eggs can be dropped without breaking. He began with one egg to drop and then extended the problem to two eggs and more. This problem is much more difficult than it appears to be at first.

After a short break, Professor Miller returned for his second talk of the day entitled “Pirates and Math: Dividing the Spoils.” He began his lecture by giving us the following scenario: Five pirates find 100 gold coins and decide to divide the coins among them according to the following rules:

The pirates are numbered 1, 2, 3, 4, 5. Pirate #1 suggests a way he thinks the 100 coins should be divided among them.

If 50% or more of the pirates vote in favor of this proposal, they accept it and divide the gold as specified.

If more than 50% vote against it, pirate #1 walks the plank and dies, and then pirate #2 has his chance to make a proposal for dividing the coins under the same rules as pirate #1.
The question proposed by Professor Miller is what should pirate #1 propose for dividing the 100 coins? This is a very thought-provoking question and one which does not have a simple answer. To help us understand the solution, Professor Miller told us to first simplify the problem by changing it from 5 pirates to 1 pirate. The solution then is obvious—pirate #1 gets all the coins. Next, he had us consider the 2 pirate case. Think about that case for a few minutes and you should see that pirate #1 can propose that he get all 100 coins and pirate #2 get nothing. Since 50% of the vote is all that is needed, pirate #1 votes in favor of it and gets all 100 coins. But now things get more difficult when there are 3 pirates.

We took a break for lunch after these lectures and then returned to Decker for our final lecture of the day. Professor Paul Ellis presented our participants with the following situation: First, choose four whole numbers and place them on the vertices of a square. Next, form a second square by connecting the four midpoints of the sides of the first square. Now, find the positive difference between the numbers on the endpoints of each side of the original square and place those numbers on the respective midpoints of each side. For example, if the original square had the numbers 10, 4, 12, and 23, the midpoints (or vertices of the second square) would have the numbers 13 , 6, 8, and 11. Repeat this process repeatedly and see what happens. For our example, the next set of vertices would have the numbers 2, 7, 2, and 3. Continuing, we would get the numbers 1, 5, 5, 1, then 0, 4, 0, 4, then 4, 4, 4, 4, and finally 0, 0, 0, 0. Professor Ellis asked our students if they could find a sequence of four whole numbers that did not eventually end with 0, 0, 0, 0. Professor Ellis next considered negative integers and rational numbers, showing that the same result occurred. At the end of his lecture he introduced students to Tribonacci Numbers. What? You never heard of Tribonacci Numbers? Well, neither did I, but like most of you, I have heard of Fibonacci Numbers. Similar to Fibonacci Numbers, Tribonacci Numbers are numbers in a sequence in which each number after the third number is the sum of the 3 previous numbers. So suppose the first 3 numbers were 0, 0, 1. Then the next numbers in the sequence would be 1, 2, 4, 7, 13, 24, . . . . Dr. Ellis showed us some interesting properties of this sequence before completing his talk.

After Dr. Ellis’ talk, we all met at Bliss Hall for our first contest round of this summer. Today’s contest round was our Team Round in which the students on each team work together to solve 12 difficult questions in 90 minutes. These are very challenging questions that few students at this age level can solve. Each question was worth 100 points.

This evening was the first night of our two-evening Talent Show. One of our counselors, Haley Kooyman, flew in from California just so she could introduce our emcee, Adam Raichel, to our participants. Okay, Haley would have been here anyway to help us out, but she took special pleasure in warming up our students for Adam’s appearance. As usual, Adam did a great job introducing each act, and he even threw in a couple of jokes along the way. He’s busy tonight trying to find some jokes that will actually make us laugh tomorrow night. Of course, the purpose of our talent shows is to demonstrate that our participants are multi-talented and not just math whizzes. Adam introduced a variety of acts this evening---we had singers, pianists, dancers, a Rubik’s Cube expert, a recorder player, and a guitar player. I am always amazed at the many talents our students possess besides their mathematical talents.

After all the performers completed their acts, our guitar player returned to the front of the room to introduce our participants to a game he has developed and which is available at the App Store. It's called Pop The Lock X, and it challenges your speed and accuracy at opening locks. Speed and Accuracy...two words that are also important at Math League! You can find this game at Pop The Lock X. We all are looking forward to a demonstration of this game tomorrow since our young developer is offering a small trophy to the first participant who reaches level 30 in his game. This is just another example of the many talents our participants have.