## A virtual trisectrix in space?

Orbital resonances are simple numerical relationships between the periods of nearby planets or satellites. A salient example in the solar system is given by three of Jupiter's moons: during every single revolution by Ganymede, Europa makes two and Io four (nice animation here). Recently predicted and even more recently observed is a retrograde 1:1 resonance. This occurs with Jupiter and the asteroid 2015 BZ509, as reported by Paul Wiegert, Martin Connors and Christian Veillet in Nature, March 30, 2017. The planet and the asteroid have the same period, but go around the Sun in opposite directions.

The path of BZ, as seen from Jupiter (the analogue of the epicyclic path of Mars as seen from Earth) is characterized as a "trisectrix" in the News & Views commentary in that same issue of Nature. (Those authors, Helena Morais and Fathi Namouni, were the first to work out the possibility of this type of resonance, back in 2013). Strictly speaking there is only one trisectrix, the planar curve with polar equation $r=1+2\cos\theta$, related in fact to angle trisection (nice explanation at 2000clicks.com). The virtual path of 2015 BZ509 is not even planar.

That was the title for a report by Christopher Mele in the New York Times (May 15, 2017; as the Times put it elsewhere, "By Counting His Chicks, Texas Teenager Tops Pecking Order in Math Contest"). "A 13-year-old boy from Texas won a national math competition on Monday with an answer rooted in probabilities -- and a dash of farming. The boy, Luke Robitaille, took less than a second to buzz in at the Raytheon Mathcounts National Competition with the correct answer." Mele gives us the chicks problem and a couple more from past competitions. (Answers below).

• In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of unpecked chicks?
• The smallest integer of a set of consecutive integers is $-32$. If the sum of these integers is 67, how many integers are in the set?
• A bag of coins contains only pennies, nickels and dimes with at least five of each. How many different combined values are possible if five coins are selected at random?

"'You get to think about things and move logically toward solving problems,' said Luke, who came in second place at last year's competition."

The Raytheon competition, Mele tells us, is "a way to promote skills in science, technology, engineering and mathematics -- known as the STEM fields." He reminds us of the current national lack of STEM professionals and how, according to the President's Council of Advisors on Science and Technology, to maintain our "supremacy in science and technology" we will have to "increas[e] the number of students earning STEM degrees by about 34 percent annually."

Mele also spoke with Lou DiGioia, "executive director of Mathcounts, [who] said the contestants' achievements were the results of endless hours of practice and coaching and not necessarily innate math abilities. 'These are not natural prodigies,' he said. 'Nobody watches a basketball game and says, "Oh, LeBron James was born that way.'" (Answers: 25, 67, 21).

Tony Phillips
Stony Brook University
tony at math.sunysb.edu