by Jonathan Kujawa
While laying in bed on the night of January 20, 1884, Lewis Carroll conjured up the following puzzle:
Three Points are taken at random on an infinite Plane. Find the chance of their being the vertices of an obtuse-angled Triangle.
That is, since any three points on a sheet of paper can be connected to form a triangle, what's the likelihood that one of the angles is more than ninety degrees if you pick those points at random?
If you only know Lewis Carroll from Alice in Wonderland you may be surprised that his thoughts turned to mathematics. In fact, his day job was to be a mathematician at Christ Church college in Oxford under the name Charles Dodgson. In addition to his more famous works of fiction, he was known for writing several mathematical texts. When teaching linear algebra I always take a day to talk about Dodgson Condensation [1].
One of the books he wrote is Curiosa Mathematica, Part II: Pillow Problems Thought Out During Wakeful Hours. It is the compendium of 72 math problems Dodgson pondered and solved while waiting to fall asleep. Helpfully he also gives the date he dreamt up the problem and the solution he devised. Go here if you'd like to take a look at the other 71 problems.
The Obtuse Triangle Problem is No. 58. Before we take a look at his solution we should step back a minute. What does it mean to pick three points at random? Like most politicians' speeches, it sounds good but falls apart under the slightest scrutiny. Are we to pick x and y coordinates for each of these points? Alternatively, we could pick an angle between 0 and 360 degrees and a distance and, starting at the origin, take the point at that angle and distance. Or, since all we care about is the resulting triangle, maybe we should randomly pick an angle between 0 and 180 degrees, pick two side lengths at random, and make the triangle made by drawing two sides of those lengths with that angle between them. I'm sure we could come up with a dozen different ways to randomly pick a triangle.
If a random triangle was a random triangle, and if the world was fair and just, then the odds of an obtuse triangle would be the same regardless of our method. Sadly, the world is neither fair nor just. It will matter how we choose to pick a random triangle [2].
