by Jason Cantarella, Tom Needham, Clayton Shonkwiler, and Gavin Stewart
In his Pillow Problem #58 from 1884, Lewis Carroll posed the following problem:
Three Points are taken at random on an infinite Plane. Find the chance of their being the vertices of an obtuse-angled Triangle.
In fact, the same question had been asked at least 20 years earlier by Wesley Woolhouse in the 1861 Lady's and Gentleman's Diary:
Many answers have been proposed over the years; interestingly, all suggest that the majority of triangles should be obtuse.
In this paper, we discuss a particularly natural way of choosing triangles at random, answer Carroll's question (about 83.8% of triangles are obtuse), and then provide answers to some related questions, such as J.J. Sylvester's famous Four Point Problem about the probability that a random quadrilateral is reflex.
Debates over these types of questions in the 1860s helped spur the development of geometric probability as a field; it's pretty exciting to be able to introduce some new ideas to this old but fruitful subject.
