Computer-aided solution to the $k$-bonacci pick-up sticks problem

TL;DR

This paper provides a comprehensive solution to the probability problem of forming no $k$-gon from randomly chosen sticks, extending previous results for triangles and quadrilaterals to general $k$-gons, and clarifies its relation to $k$-bonacci numbers.

Contribution

It introduces a complete method for calculating the probability of avoiding $k$-gons, generalizing earlier work and linking the problem to $k$-bonacci number products.

Findings

Derived explicit probability formulas for general $k$-gons.

Connected the problem to products of $k$-bonacci numbers.

Extended known results from triangles and quadrilaterals to all $k$-gons.

Abstract

A full solution to the recently proposed problem of determining the probability that no $k$-gon can be built from $n$ independently and uniformly chosen sticks in $[0,1]$ is proposed. This extends the known results for triangles and quadrilaterals to general $k$-gons and offers a clearer interpretation of the connection to products of $k$-bonacci numbers.