Jan 18, 2024

Our first lecturer today might have looked familiar to our students, because we were lucky enough to have a second lecture from yesterday’s afternoon lecturer, Nick Rauh, from the Seattle Universal Math Museum!

Today Nick taught the student’s “Grundy’s Game.” The students paired off for the game and started with a pile of 8 chips between them. Players took turns splitting any pile they liked into two piles, but the two piles had to be of unequal size. Whichever player was the last to be able to split a pile was the winner. Nick challenged the students to try to come up with the optimal strategy for the game. Questions raised included: What's a strategy for winning? Does it matter who makes the first move? What happens if we start with 9 chips? 10 chips?

Once the students came up with the optimal strategies for both 8-chip and 9-chip variations of the game, the group discussed what general rules could be derived. For instance, piles of either 1 or 2 chips cannot be divided, so getting down to piles of that size is the key to the game. Extrapolating up from there, students were able to see how a pile of 7 would always mean the end of the game for the player who had to make the next move. Eventually, this analysis leads to a table of “Grundy Values,” which are numbers of chips that guarantee that the first/next player to move is the loser if both players use good strategy. The students’ final challenge was to try to invent a strategy for a game starting with 32 chips in the pile!

Next up was Ian Whitehead, Assistant Professor of Mathematics at the prestigious Swarthmore College. Professor Whitehead discussed Apollonian Packings, in which areas are filled in with increasingly smaller circles. The lecture started with a review of the subject, which started over 2000 years ago with Apollonius of Perga and continued through Rene Descartes and into the present. Images of Apollonian packings can be found in many places, including some Shinto shrines in Japan and Kandinsky paintings.

The discussion started with the concept of curvature, which is the reciprocal of the radius of the circle. Smaller circles thus have higher curvatures. Curvatures can also be zero (for a straight line) or negative (for a circle outside of an inner circle that is being discussed). Students were then challenged to figure out patterns in the curvatures of smaller circles in arrangements adjacent to larger circles.

Then our attention turned to tangent circles, starting with groupings of three tangent circles and asking whether a fourth circle tangent to all 3 would be possible. It turns out that two such circles are possible, one “inside” circle and one “outside” circle. That discovery led to a discussion of the “rule of triangles,” which says that the relationship among the curvatures is: Inside tangent circle curvature + Outside tangent circle curvature = 2 X “Triangle”of curvatures, or x + y = 2 X (a + b + c). Students were challenged to calculate the curvatures of various circles in several Apollonian packings when given three of the curvatures as a starting point. Some of them were trickier than you might imagine!

In the afternoon students were treated to a tour of the Princeton University campus. In addition to being one of the most selective institutions of higher learning in the country, it is also blessed with one of the most beautiful campuses in America.