Scatter Three Random Points — Is the Triangle Obtuse?
Lewis Carroll posed this one while lying awake at night: scatter three points completely at random on an infinite flat plane. What is the chance that the triangle they form is obtuse (has one angle greater than 90 degrees), rather than acute?
Reveal the answer
It's a trick question: on a truly infinite plane there is no way to choose points 'uniformly at random,' so the probability isn't actually well defined — a lesser-known cousin of Bertrand's paradox for random chords. Carroll's own worked solution, which smuggled in a particular distribution, concluded obtuse triangles were overwhelmingly likely, but the answer changes completely depending on the convention chosen. From Pillow Problems (1893), Carroll's collection of 72 puzzles solved lying awake at night.
— Lewis Carroll (Charles Lutwidge Dodgson), Pillow Problems — Curiosa Mathematica, Part II: Pillow Problems Thought Out During Wakeful Hours (1893)
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