# Broken bricks and the pick-up sticks problem

## TL;DR

This paper explores generalizations of the broken stick problem, introducing discrete brick analogues and sequential pick-up versions, providing combinatorial insights and framing these problems within a broader, largely open family of questions.

## Contribution

It introduces new discrete and sequential variants of the broken stick problem and connects them within a broader context of related open questions.

## Key findings

- Limit behavior of the broken brick problem analyzed
- Combinatorial proof of the broken stick problem provided
- Broader family of related problems identified and discussed

## Abstract

We generalize the well-known broken stick problem in several ways, including a discrete "brick" analogue and a sequential "pick-up sticks/bricks" version. The limit behavior of the broken brick problem gives a combinatorial proof of the broken stick problem. The pick-up version gives a variation on those scenarios, and we conclude by showing a greater context---namely, that the broken stick/brick problem and the pick-up sticks/bricks problem are two extremes in a family of interesting, and largely open, questions.