Pick n sticks at random, each with a length between 0 and 1. What is the probability that no three of them can form a triangle?
The answer, proved by Sudbury, Sun, Treeby, and Wang (arXiv 2504.19911, 2025): the probability is the reciprocal of the Fibonacci factorial — the product of the first n Fibonacci numbers.
For three sticks, the probability of no triangle is 1/(1 × 1 × 2) = 1/2. For four sticks, 1/(1 × 1 × 2 × 3) = 1/6. For five, 1/(1 × 1 × 2 × 3 × 5) = 1/30. For six, 1/240. Each additional stick multiplies the denominator by the next Fibonacci number.
The Fibonacci sequence was not expected here. The problem is continuous — stick lengths are real numbers drawn uniformly from an interval. The triangle inequality is a geometric constraint — three lengths form a triangle if and only if the longest is shorter than the sum of the other two. The Fibonacci sequence is a discrete recurrence: each term equals the sum of the two before it. The connection between a continuous probability problem and a discrete combinatorial sequence is not obvious.
The connection becomes visible through the recursion. When a new stick is added to an existing collection, the probability that it creates no new triangle depends on the sorted order of the existing sticks. The longest stick must exceed the sum of the two next-longest for no triangle to exist. This constraint cascades through the sorted sequence in a way that multiplies the probability by exactly 1/F(n), where F(n) is the nth Fibonacci number. The Fibonacci structure is not imposed — it is the natural recursion of the triangle inequality applied iteratively to sorted uniform random variables.
The extension confirms the structural claim. For quadrilaterals — the probability that no four sticks form a quadrilateral — the Fibonacci recurrence generalizes to the Tribonacci recurrence (each term equals the sum of the three preceding terms). For k-gons, the (k-1)-bonacci recurrence governs the answer. The Fibonacci sequence is not a curiosity of triangles. It is the simplest case of a family of recurrences, each matched to a geometric constraint. The polygon constraint and the recurrence are the same mathematical structure viewed from two different directions.