Jan 17, 2024

I hope you all enjoyed Art Benjamin's MathMagic show last night. You mayrecall that he also gave a talk to our students earlier in the day on scams.Professor Benjamin was kind enough to send me the slides from his afternoonlecture. If you want to see all the scams he discussed, please use thefollowing link to download his slides: Scams

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

The annual "Great Tiger Hunt" took place this morning on TheCollege of New Jersey campus under warm, beautiful weather. This year'sevent, inspired by "The Amazing Race," challenged math campers in bothmathematical skills and historical knowledge. Ten stations were set uparound campus, each named after a distinguished Princeton scholar.

At each station, teams tackled 5 math problems and a promptabout a notable Princeton figure like the following.
In the early 1940s, I became the first woman hired as afaculty member by Princeton's physics department, breaking barriers in athen all-male institution. My groundbreaking experiments later disproved theconservation of parity in weak interactions, revolutionizing nuclear physicsand earning me the title of the "First Lady of Physics". Who am I?

To advance, teams needed to both solve the required numberof math problems correctly and identify the scholar based on theirachievements. Teams were equipped with a list of ten Princeton scholars andbrief biographies to aid in identification.

Racing across campus, teams visited stations untilcompleting all required challenges, with unsuccessful attempts resulting in"detours." None of the teams was able to finish in the allotted time, but awinner will be determined before the end of camp.

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

After The Great Tiger Hunt, we all gathered in the Decker Social Space forour first speaker of the day, Matt Weinberg of Princeton University. Matt'stopic today was Stable Matchings. He began his lecture by discussing how tomatch doctors with hospitals when doctors are looking for a job. In hishypothetical situation, he had three doctors and three hospitals. Eachdoctor ordered the hospitals in the doctor's order of preference for whichhospital to work at, and each hospital ordered the doctors in the hospital'sorder of preference of which doctors they wanted to hire. The question Mattposed is what is the optimal way of pairing the doctors and the hospitals?Dr. Weinberg went on to explain that of the several possible pairings ofdoctors and hospitals , some were stable and some were unstable. Todetermine which type of matchings were stable, Matt introduced the conceptof blocking pairs and an algorithm for establishing stable pairs. Ourstudents asked Dr.Weinberg some excellent questions throughout hislecture-he was amazed at how insightful our students were. After hislecture, he told me how much he enjoyed speaking to our students and that hewanted to return next summer.

When we returned from lunch, David Nacin from William Paterson College spoketo our students about famous sequences. He began his talk by introducingstudents to the game LUPI. Each letter in LUPI stands for a characteristicof the number needed: L = Least, U = Unique, P = Positive, and I = Integer.In this game, each group chooses a positive integer without knowing whatnumbers the other groups have chosen. The winning group is the group thatchooses the least integer that is unique among the numbers chosen by all thegroups. For example, suppose there were ten groups and the numbers chosen bythe ten groups were 3, 1, 4, 7, 1, 2, 5, 6, 4, and 2. The winning groupwould be the group that chose 3 since it is the least number that no othergroup chose. He next used colored rectangles and squares to lead thestudents to the very famous Fibonacci sequence in which each number afterthe first two numbers in the sequence is the sum of the previous two terms.Surprisingly, the Fibonacci numbers were not discovered by Fibonacci, but bythe relatively unknown Indian poet and mathematician Pingala. Pingala alsowrote about the famous triangle of numbers known as Pascal's Triangle. AsDavid noted, most of the famous mathematicians whose names are associatedwith mathematical ideas are not the ones who first wrote about them.

I will let Adam Raichel describe the final lecture of the day since I wasunable to attend it:

In the final lecture of the afternoon, presenter Nick Rauhfrom the Seattle Universal Math Museum presented a lecture on "tensegritypolyhedra." ("Tensegrity" was apparently coined by the famous architectBuckminster Fuller.) Using small dowels and rubber bands, students werechallenged to create "tensegrity units" consisting of two dowels joined ateach end by rubber bands, with an additional rubber band running the lengthof the dowels pinched between the two dowels at each end. These units couldbe combined to form larger constructions. Nick led the students through onesuch larger construction. Here is my attempt at creating the polyhedron(depending on how you look at it, either an octahedron or an icosahedron):

Nick then explained how these constructions are related tothe Platonic solids, and how most but not all of the Platonic solids can beconstructed out of triangles and squares. Some of them, such as the squareantiprism, are quite complicated!

Thank you, Adam, for your detailed description of and photos from thislecture.

Our Awards Ceremony was held in the Cromwell Hall Lounge in the evening. Ourstudents all received books that were donated by The Art of Problem Solving(AoPS), a booklet with the questions and solutions for all the tournamentrounds, and certificates. It's always a thrill for me to see the excitementof our participants as the awards are announced and to speak with some ofthe 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 AnnualMath League Summer Tournament-all participants from this summer areautomatically eligible for next summer's program.