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Pick-up Sticks and the Fibonacci Factorial

Aidan Sudbury, Arthur Sun, David Treeby, Edward Wang

TL;DR

This work analyzes a pick-up-stick variant where $n$ independent $[0,1]$ lengths are drawn and asks for the probability that no three can form a triangle. Using order statistics and exponential spacings, the authors derive a Fibonacci-type recursion that collapses to the exact closed form $P_n = 1/(F_1 F_2 ... F_n)$, the reciprocal of the Fibonorial. They extend the method to quadrilaterals, obtaining a Tribonacci-type extension with a correction factor, and discuss generalizations to higher $k$-gons, highlighting a deep link between simple geometric probability and Fibonacci-type recurrences. The results illuminate structural contrasts with the Dirichlet-broken-stick model and invite combinatorial proofs and broader polygon-avoidance analyses.

Abstract

We present a variation of the broken stick problem in which $n$ stick lengths are sampled uniformly at random. We prove that the probability that no three sticks can form a triangle is the reciprocal of the product of the first $n$ Fibonacci numbers. Extensions to quadrilaterals and general $k$-gons are also discussed.

Pick-up Sticks and the Fibonacci Factorial

TL;DR

This work analyzes a pick-up-stick variant where independent lengths are drawn and asks for the probability that no three can form a triangle. Using order statistics and exponential spacings, the authors derive a Fibonacci-type recursion that collapses to the exact closed form , the reciprocal of the Fibonorial. They extend the method to quadrilaterals, obtaining a Tribonacci-type extension with a correction factor, and discuss generalizations to higher -gons, highlighting a deep link between simple geometric probability and Fibonacci-type recurrences. The results illuminate structural contrasts with the Dirichlet-broken-stick model and invite combinatorial proofs and broader polygon-avoidance analyses.

Abstract

We present a variation of the broken stick problem in which stick lengths are sampled uniformly at random. We prove that the probability that no three sticks can form a triangle is the reciprocal of the product of the first Fibonacci numbers. Extensions to quadrilaterals and general -gons are also discussed.
Paper Structure (5 sections, 2 theorems, 26 equations)

This paper contains 5 sections, 2 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Picking up (not breaking) sticks
  3. Main result
  4. Quadrilaterals and the Tribonacci extension
  5. An open problem

Key Result

Theorem 1

If $n$ real numbers are chosen uniformly from the unit interval $[0, 1]$, the probability that no three of the numbers can form a triangle is where $F_n$ are the Fibonacci numbers defined by $F_1 = 1$, $F_2 = 1$, and $F_{n} = F_{n-1} + F_{n-2}$ for $n > 2$.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof