The Broken Stick Problem

The Broken Stick Problem: History, Proofs and Applications Margaret McGuire May 1, 2017

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Introduction

Suppose two points are randomly chosen on a stick of unit length, and that the stick is broken at these points. What is the probability that the resulting segments can be used to form a triangle?

This problem, commonly called the Broken Stick Problem, has been of interest to mathematicians for decades and continues to be a fruitful area of research [3]. Many different types of proof techniques can be used to present the solution, and the problem translates into probability distributions that can model real life phenomena. In this paper, we present a brief history of the Broken Stick Problem and then describe three ways to solve the problem. From there, we discuss the potential uses of the problem as a model for understanding animal demographics in ecology.

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History of the Broken Stick Problem

We begin with a brief history of the Broken Stick Problem. The problem first appeared in an 1854 examination at Cambridge University [6]. This exam had to be completed by students who sought a degree in mathematics. The solutions preferred by the faculty who designed the exam were compiled and published and a proof similar to theirs will be presented in this paper. It is suggested by Goodman [2] that John Venn (who would have taken the 1

1854 exam at Cambridge) brought the problem to France, where Emil´e Lemoine discovered it and wrote the first published paper about the problem in 1857. The problem circulated in France as it was studied by other french mathematicians such as Henri Poincar´e in his book Calcul des Probabilities [5], and Paul L´evy [2]. The problem gained attention in the middle twentieth century as a probabilistic model for predicting real world phenomena; this work was pioneered by MacArthur [4] and continues to be studied by others. Later we will discuss MacArthur’s initial work in this area and reflect on the work of modern authors who find MacArthur’s model to be weak in comparison to other models. With this historical and mathematical context of the Broken Stick Problem in mind, we turn to three proofs of the problem. These include a selection of proofs from across history, from 1854 to modern day.

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Proofs of The Broken Stick Problem

In this section we present three different proofs for the original Broken Stick Problem. Each sub-section discusses the primary mathematical tool used to complete the proof and then provides a proof using that tool.

3.1

Complementary Probability

The first proof of the Broken Stick Problem we present uses the concept of complementary probability, a similar technique to that used by Cambridge mathematicians in 1854. In the Broken Stick Problem, we seek the probability of a triangle being formed, but it is easier to calculate the probability a triangle is not formed. To find the probability that a triangle is formed, we subtract the probability of a triangle not being formed from 1. The following proof of the Broken Stick Problem utilizes this reasoning.

Before we calculate the complementary probability for this problem, we will discuss a Lemma that we will come back to for nearly every proof of the broken stick problem: Lemma. The first part of any proof of the Broken Stick Problem is the same: we must first 2

show that the stick pieces only form a triangle when all of the stick lengths are less than 1/2. We will show this with a direct proof. We know the triangle inequality states that, for triangle side lengths a, b and c, a + b > c, b + c > a, and a + c > b.We also know that a + b + c = 1 since we are breaking a unit stick into three pieces. This implies a + b = c − 1.

By substitution a + b > c becomes 1 â&#x2C6;&#x2019; c < c and therefore 1/2 < c. If we apply the same reasoning to the two other inequalities we see that a < 1/2 and b < 1/2. Thus the stick pieces only form a triangle when each is less than 1/2 in length.

With this lemma in mind, we proceed to prove the Broken Stick Problem using complementary probability.

Proof. We have a unit stick that is broken randomly at two points. Let the two points be x and y, and without loss of generality let x < y. We seek the probability that the three sticks resulting from this breakage can be used to form a triangle. By the above lemma we know the stick pieces only form a triangle when each is less than 1/2 in length. We proceed using complementary probability to show that the chance of forming a triangle with the sticks is 1/4. We will approach this complementary counting with casework. There are two cases to consider.

Case 1: Points x and y land on the same side of the midpoint. This happens with probability 1/2 and no triangle is formed.

Case 2: Points x and y are on opposite sides of the midpoint. Imagine the unit stick is now divided into four equal lengths labeled 1 to 4 from left to right as in Figure 1 (below).

With 1/2 probability we have that x is in region 1. With probability 1/2 we know y is in region 4. It follows that with probability (1/2)(1/2) = 1/4 no triangle is formed. The results of Case 1 and 2 imply that a triangle is formed with probability 1 â&#x2C6;&#x2019; ((1/4) + (1/2)) = (1/4).

Figure 1 summarizes the scenarios where triangles are not formed in each of the two cases discussed.

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Figure 1: Cases demonstrating possible locations of x and y that prevent the formation of a triangle.

In the next section we will investigate a proof technique first used in publication by Henri Poincar´e that utilizes geometric reasoning drawing upon Viviani’s Theorem.

3.2

Viviani’s Theorem

The second proof of the Broken Stick Problem we present relies heavily on concepts from geometry and on Viviani’s theorem in particular. To directly calculate the probability of forming a triangle with three broken sticks we can use geometric reasoning to define a region of all possible results and a sub-region of success (in this case, success means we form a triangle). We can then divide the area of the success region by the area of the possible results region. Viviani’s theorem is at the heart of the geometric reasoning that allows us to define these regions and complete the proof.

Proof. We seek to show by direct proof that the probability of forming a triangle with the broken sticks is 1/4. Viviani’s theorem states that for a point P inside an equilateral triangle ∆ABC, the sum of the perpendiculars pi from P to the sides of the triangle is equal to the apothem. Let the apothem of ∆ABC be 1. Then the broken sticks that may form a triangle are equal to the perpendiculars pi formed when we pick a point inside ∆ABC. Thus p1 + p2 + p3 = 1. Viviani’s theorem is expressed visually in Figure 2.

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Figure 2: Vivianiâ&#x20AC;&#x2122;s theorem states that the length of the apothem (of length 1 in this case) is equal to the sum of the lengths of the perpendiculars drawn from point P outwards to the sides of the triangle. To connect this idea to the problem at hand, we let p1 , p2 and p3 be the broken sticks that may be put together to form a triangle.

Suppose we take â&#x2C6;&#x2020;ABC and subdivide it into four equilateral triangles. If a point P lands within the central sub-triangle, all of the perpendiculars pi are shorter than 1/2, but if the point P lands outside the central sub-triangle at least one of the segments is greater than 1/2. This is exemplified in Figure 3 below. Since the area of the central triangle is 1/4 of the area of the larger triangle, there is a 1/4 probability that a triangle is formed from the perpendiculars p1 , p2 and p3 that correspond to the segments of the broken stick.

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Figure 3: The small equilateral triangle inside â&#x2C6;&#x2020;ABC containing the red point represents the area where the broken sticks form a triangle. Each set of red and green segments sum to the length of the apothem (blue) according to Vivianiâ&#x20AC;&#x2122;s theorem.

With this proof complete, we move forward in the course of mathematical history to a proof from 2009 found on Mathematics Stack Exchange, an online mathematics forum. It merits mention because it is a prime example of how some proofs of this problem can be unintuitive and surprising at first but are also intriguing and even beautiful.

3.3

Sticks into Circles: An Unusual Geometric Approach

Strange geometric tricks do not always strike us as the first way to solve a problem, but such techniques often present the most interesting solutions. The following is a proof that connects the ends of the unit stick to produce a circle and then devises a creative procedure for cutting the circle into segments.

Proof. Suppose we have a unit stick where we connect the ends to form a circle of perimeter 1. Then to solve the Broken Stick Problem, we must break the circle at three points instead of two. As before, our lemma states that the segments will only form a triangle when each stick 6

has a length less than 1/2. We now present a procedure for forming three broken sticks from the circle. Suppose we randomly choose three pairs of points, the points in each pair being opposite each other on the circle. Then suppose we pick one point in each pair randomly and cut at that point. This is visualized in Figure 4. From inspection we see that there are 8 different sets of three cuts we can make, and 6 of these points lead to non-triangles. Since 2 of the 8 events form triangles, the probability of forming a triangle is 1/4.

Figure 4: Circled points indicate that a cut is made at that point. Six scenarios admit no triangle formation (example shown), but two scenarios produce segments all less than 1/2.

Antipodal points are randomly picked each time, and are not necessarily guaranteed to be evenly spaced.

These three proofs provide a small glimpse into the rich world of proofs that have been described for the Broken Stick Problem. While several other proofs exist we will now move on to our discussion of the Broken Stick Problem as the basis for a probabilistic model of real phenomena.

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The Broken Stick as a Model

There have been many proposed applications of the Broken Stick Problem, but here we focus on two related to ecology. In 1957, MacArthur proposed that the Broken Stick model could be used to fit data from ecological populations in genetic equilibrium [4]. MacArthur

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was studying bird demographics, specifically the interaction between the number of different species and the rarity of those species in an ecosystem. An ecosystem is composed of niches, where niches are the environment type that a single species occupies. These niches are not necessarily geographically distinct but they often contain different types of resources, for example, an insect living on the surface of a pond occupies a different niche than a turtle swimming in the water just below the insect. MacArthur claimed that the environment should be compared to a stick of unit length on which we randomly select n â&#x2C6;&#x2019; 1 points. The stick is broken at these points, and the lengths of the n resulting segments are proportional to the abundances of the n species. MacArthur cites Barton and David who claim that the resulting distribution predicts the abundance of the rth rarest species among n species and P m individuals is (m/n) ri=1 [1/(nâ&#x2C6;&#x2019;i+1)]. Comparing this distribution to other models with other assumptions about nichesâ&#x20AC;&#x2122; overlaps and continuity, MacArthur concludes that niches do not overlap much and are more continuous than discrete.

The Broken Stick Problem can also be applied to a related issue in ecology regarding the number of genera within a biological family1 . The specific example we use here is that used by Beres et al. [1] with the family of Rotifers , which are small aquatic organisms. In the case of rotifers we have that eleven families have only one genus and six families have two genera, whereas on the other side of the spectrum one family has eighteen genera, so that 126 genera are distributed over 34 families. This distribution of genera per family has a hollow curve distribution because it is decreasing but concave up. The task we are faced with is to find a model and associated theory that predicts this distribution.

Beres et al. fit the Rotifer data to several distributions, including a Poisson model, a Broken Stick model, and the Hubbell Unified Neutral model. The Poisson model is an average of the results if one were to distribute 126 genera at random over the 34 families of Rotifers but with each family to receive at least one genus. This model does not fit the 1

Organisms are organized into ascending taxonomic categories: species, genus, family, order, class, phylum, kingdom, domain. If a family has many genera then usually there is a greater diversity of species in that family compared to family with only one genus, for example.

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data well as indicated by the difference in shapes of the distributions in Figure 5. We now turn to the Broken Stick model. To model the situation, we take a stick of unit length with 126 subunits and pick 33 random points on it to produce 34 segments. This would occasionally result in families with no genera in them so we actually apply the process to only 92 subunits, and then add one subunit to each of the 34 intervals to ensure no families are empty. Unfortunately, the Broken Stick model does not fit the data well either because it underestimates the frequency of families with a large number of genera and overestimates the frequency of families with smaller numbers of genera (see Figure 6).

Beres et al. attempt to modify the Broken Stick model to improve its fit to the data, but they find that Hubbellâ&#x20AC;&#x2122;s Unified Neutral model (see Figure 7) can provide a much better fit. This model makes assumptions about the real-world that researchers can use to make inferences about the nature of families of organisms. The details of the model and the results we can conclude from it are beyond the scope of this paper, but our purpose in describing this scenario is to acknowledge that the Broken Stick model does not always provide an ideal model.

Figure 5: This figure illustrates data from Rotifer studies compared to predictions based on the Poisson (random) model. Note that for all of the following graphs the histograms show the frequency of having a certain number of genera (number of subunits) in a family. This figure and all subsequent figures are taken from the paper by Beres et al. [1].

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Figure 6: This figure illustrates data from Rotifer studies compared to predictions based on the Broken Stick model. The Broken Stick model provides a better fit than a random distribution, but has some shortcomings.

Figure 7: This figure illustrates data from Rotifer studies compared to predictions based on the Hubbell model. In this case, the Hubbell model represents a better fit than the Broken Stick model.

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Conclusion

In closing, the Broken Stick Problem has a rich history and provides an exciting opportunity to experiment with unusual proof techniques. It may also offer a useful function as a model for real life phenomena, but we have to be careful in what way we apply the Broken Stick model and be cognizant of the implications such a model has for the underlying nature of the data under consideration. The problem continues to be of interest to mathematicians and scientists, especially regarding generalizations of the problem and as a probability distribution model for other real events.

Works Cited [1] Karl A. Beres, Robert L. Wallace, and Hendrik H. Segers. Rotifers and Hubbell’s Unified Neutral Theory of Biodiversity and Biogeography. Natural Resource Modeling, 18:363– 376, 2005.

This paper provides a brief introduction to the the hollow curve distribution and reviews the various models claiming to explain the shape of the curve, one of which is the broken stick model. It ultimately advocates for Hubbell’s unified neutral theory of biodiversity and biogeography. We will use this paper in our critical reflection about whether the Broken Stick model provides the best model for all ecological studies.

[2] Gerald S. Goodman. The Problem of the Broken Stick Reconsidered. https://link. springer.com/article/10.1007/BF02985378. Accessed April 30 2017.

This document contains a useful overview of the history of the Broken Stick Problem from the time of its initial presentation at Cambridge to its investigation by early french mathematicians. It will be used as a reference for the historical summary of the Broken Stick Problem.

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[3] Lingyi Kong, Luvsandondov Lkhamsurren, Abigail Turner, Aananya Uppal, and A. J. Hildebrand. Random Points, Broken Sticks, and Triangles Project Report. http://www. math.illinois.edu/~ajh/ugresearch/brokenstick-spring2013report.pdf.

Ac-

cessed April 30 2017.

This document presents a variety of interesting historical notes about the Broken Stick Problem, including where it was first used and what possible modeling applications it may have. The paper considers two generalized versions of the broken stick problem with n pieces. The document presents solutions to two questions. The first question asks, what is the probability that there exist three of the n pieces that can form a triangle? The second question asks, what is the probability that any triple of pieces chosen from the n pieces can form a triangle? For us this paper functions as a jumping off point for finding good historical and application-based articles.

[4] Robert H. MacArthur. On the Relative Abundance of Bird Species. Proceedings of the National Academy of Sciences of the United States of America, 43:293–295., 1957.

This is the original article in which Robert MacArthur argues for why the broken stick model can predict the behavior of animals in ecological niches. In this case, “the environment is compared with a stick of unit length on which n - 1 points are thrown at random. The stick is broken at these points, and the lengths of the n resulting segments are proportional to the abundances of the n species. This comparison is legitimate, since the mathematical part of the derivation can be used unchanged for spaces of higher dimension.” MacArthur goes on to build a model from this and compares the model to actual data to demonstrate its fidelity. We will use this paper in our discussion of the relevance of broken stick problems to everyday problems.

[5] Henri Poincar´e. Calcul des Probailit´es, pages 123–126. Gauthier-Villars, 1 edition, 1912.

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In this first edition French mathematics textbook, Poincar´e describes the Broken Stick Problem and proposes a solution using what appears to be Viviani’s Theorem. This will be used in the historical section of the paper and relates to the proof that we present using Viviani’s theorem.

[6] Cambridge

University.

Solutions

of

The

in the Senate-House Examination for 1854.

Problems

and

Riders

Proposed

https://archive.org/details/

solutionsofprobl00mackuoft, pages 49-50. Accessed April 30 2017.

This document contains the first known occurrence of the broken stick problem as it appears in the Cambridge University Senate-House examination of 1854.

The senate house examination was an exam taken by students over a course of days resulting in submitted papers with solutions to problems. This exam was a requirement for graduation at Cambridge. Based on the preface of the document, it appears that the problems are largely designed by the faculty of Cambridge, the “moderators and examiners.” This source will be used in our historical discussion of the Broken Stick Problem.

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