Jan 26, 2024

Today 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 “What is the least positive integer that has exactly 5 divisors?” With only 45 minutes for the entire 60 questions, the average time one has per question is 45 seconds. I personally have written speed rounds for our contest, and it would be a close call for me to get through every question accurately in that amount of time!

Next up was the first lecture of the day, presented by Michael Wehar. His lecture was titled Checking Divisibility Using Finite Automata. Finite automata are represented by simple diagrams similar to flow charts. As an initial exercise, students discussed the problem of checking divisibility by a fixed number, and then attempted a general solution to this problem using finite automata. For example, they developed a way to check divisibility by 3 of a 20-digit number with ease. Michael’s overarching thesis was that finite automata are simple, powerful, and fun mathematical structures that can be used to help solve problems. I found Michael’s final statement of his presentation particularly appropriate to our program, as he encouraged students to make friends with others who share mathematical ability, as he met some of his oldest friends in that way. I hope our students are way ahead of you on that score, Michael!

After our lunch break, it was time for our relay rounds. While these rounds are very challenging for us to organize, they are the rounds many of our students enjoy the most. The teams are divided into 3 smaller groups, and in order for a group to get points on a relay round, each individual in the group must successfully solve a question until four consecutive questions have all been solved correctly. Faster groups get more points for the final correct answer. 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. So is hearing the cheers from the students when they fond out their group got the correct final answer!

After the relay, David Nacin from William Paterson College spoke to our students about famous sequences. He began his talk by introducing students to the game LUPI. Each letter in LUPI stands for a characteristic of the number needed: L = Least, U = Unique, P = Positive, and I = Integer. In this game, each group chooses a positive integer without knowing what numbers the other groups have chosen. The winning group is the group that chooses the least integer that is unique among the numbers chosen by all the groups. For example, suppose there were ten groups and the numbers chosen by the ten groups were 3, 1, 4, 7, 1, 2, 5, 6, 4, and 2. The winning group would be the group that chose 3 since it is the least number that no other group chose.

Next the students were challenged to use colored rectangles and squares (2x1 purple tiles and 1x1 red tiles) to fill a board of dimensions 1xn (and short and long notes to fill a line of music) to lead the students to the very famous Fibonacci sequence in which each number after the first two numbers in the sequence is the sum of the previous two terms.

Surprisingly, the Fibonacci numbers were not discovered by Fibonacci, but by the relatively unknown Indian poet and mathematician Pingala. Pingala also wrote about the famous triangle of numbers known as Pascal's Triangle. As David noted, most of the famous mathematicians whose names are associated with mathematical ideas are not the ones who first wrote about them.

Next students filled a 1 x n board with rectangles of dimensions 1 x 2 and 1 x 3. Looking at the number of ways that could be accomplished, we saw that the total number of possible ways to fill for length n is equal to the sum of ways to fill rectangles with lengths n - 2 and n - 3 respectively. This led to the numbers in the Padovan sequence, defined by P(n) = P(n - 2) + P(n - 3).

After that, it was off to dinner and to prepare for Talent Show Day 2!

Finally, in the evening we had our second night of the annual Talent Show. We enjoyed performances by numerous piano players, singers, flute players, and violinists, as well as less commonly seen talents on our stage such as a saxophone player, a trombone player, an erhu player (a traditional Chinese vertical stringed instrument played with a bow), and a magician. Of course, back despite popular demand was your faithful emcee, yours truly, providing a little levity between acts.

That’s all for today!