Pick-up Sticks and the General Fibonacci Numbers

TL;DR

This paper generalizes a probability result involving Fibonacci numbers and random sticks to higher polygons, showing the probability relates to a Fibonacci-type recurrence for any polygon size.

Contribution

It extends previous work by deriving a formula for the probability that no $(k+1)$-gon can be formed from random lengths, involving a $k$-step Fibonacci recurrence.

Findings

Probability that no $(k+1)$-gon forms is a product involving a $k$-step Fibonacci recurrence.

The result generalizes the known case for triangles to higher polygons.

Method closely follows the original proof, adapting it to the generalized setting.

Abstract

In the article by Edward et al. \cite{Sudbury2025}, it was shown that the probability that no three sticks randomly chosen from the unit interval can form a triangle equals the reciprocal of the product of the first $n$ Fibonacci numbers. The authors further suggested a generalization to higher \((k+1)\)-gons \((k\ge 4)\). This note proves that, indeed, for any \(k\ge 2\), the probability that no $k+1$ of $n$ independent uniform $[0,1]$ lengths can form a $(k+1)$-gon is expressed as a product whose factors involve a $k$-step Fibonacci-type recurrence. The method follows closely the original argument of \cite{Sudbury2025}, while making ex