Fibonacci numbers dictate the exact probability that a set of random sticks will fail to connect into a closed polygon.
Randomly broken sticks usually form a closed loop if the longest piece is shorter than the sum of all the others. Many people assume these geometric odds follow standard bell curves or simple linear ratios. A hidden mathematical bridge links this physical problem to p-step Fibonacci sequences where each number is the sum of the previous several terms. The chance of failing to form a polygon of a specific size equals a product of the reciprocals of these specific Fibonacci numbers. This discovery means the growth patterns of sunflower seeds and rabbit populations are baked into the very nature of physical shapes.
Fibonacci numbers and the probability of polygon formation using random length sticks
arXiv · 2604.27573
We present two complementary proofs that, if the lengths of $n$ sticks are sampled at random, then the probability that no $p+1$ sticks can form a $(p+1)$-sided polygon can be expressed as the product of the reciprocals of a series of terms involving the $p$-step Fibonacci numbers. The first proof uses matrix algebra to extend the method previously used by Sudbury et al. to derive expressions for the probabilities of not being able to form triangles and quadrilaterals. The second alternative pro