The Broken Stick Project

TL;DR

This paper discusses the probability that three segments formed by two random points on a line segment can form a triangle, and reports on a high school research project exploring variations of this classical problem.

Contribution

It presents new insights from a high school research project on the broken stick problem and its variations, involving collaborative student efforts over a year.

Findings

Probability that three segments form a triangle is 1/4.

High school students contributed ideas and progress over a year.

The project expanded understanding of the broken stick problem.

Abstract

The broken stick problem is the following classical question. You have a segment $[0,1]$. You choose two points on this segment at random. They divide the segment into three smaller segments. Show that the probability that the three segments form a triangle is $1/4$. The MIT PRIMES program, together with Art of Problem Solving, organized a high school research project where participants worked on several variations of this problem. Participants were generally high school students who posted ideas and progress to the Art of Problem Solving forums over the course of an entire year, under the supervision of PRIMES mentors. This report summarizes the findings of this CrowdMath project.