The problem of the broken stick reconsidered

Abstract

originally caUed the "Senate-House Examinations," after the name of the building in which they took place, and later they became known as the "Mathematical Tripos." Afterwards, the best solutions were published. The ensuing volumes can be found in the Rare Book Collections of the British Library in London and the University Library in Cambridge. In the morning session of January 18, 1854, there was posed [13, pp. 49-52] an elementary problem in geometrical probability that was destined to become a classic. The British call it "The Problem of the Broken Rod," whereas Americans refer to it as "The Problem of the Broken Stick." It says, "A rod is marked at random at two points, and then divided into three parts at these points; shew [sic] that the probability of its being possible to form a triangle with the pieces is 1/4." The exact value of the probability is of little interest, except possibly to numerologists. What is interesting is to see how various later authors, apparently unaware of the original formulation of the problem, reinterpreted what it means to break a stick "at random," and developed fresh methods to solve it. Be advised that I am employing the word "random" in the narrow technical sense used in probability and statistics to refer to any chance phenomenon that is governed by the uniform distribution over a suitable sample space. This usage was recommended by de Finetti [1, p. 152] and [2, p. 62], who was writing on the very topic of random division. However, even with such a restriction, there is more than one way to interpret what is meant by the term "random," depending on the identity of the sample space, just as in Bertrand's Paradox [3], [11]. I shall examine two of these interpretations in detail. Although the problem originated in England, it found its way to France, possibly with the aid of John Venn, wh o was enrolled as an undergraduate at Gonville and Caius College at the time. Presumably, he had taken the exam and done well on it, for he was awarded the title of "Mathematical Scholar" at his college later in the year. The first journal publication was in the founding volume of the Bull. Soc. Math. de France in 1875, written by t~mile Lemoine [6]. Lemoine formulated a discrete version of the problem by considering the rod as a measuring stick divided into equally spaced intervals and allowing the breaks to occur only at their endpoints. This gives rise to a finite number of outcomes, and he treated them in the traditional way, interpreting the word "random" as meaning that all trisections of the rod at a given scale are equally likely. He made tables and calculated the ratio of the number of favorable cases to the number of possible ones. Then he passed to the limit as the scale decreased to zero. In this way, he found the answer to be 1/4, in agreement with the Cambridge Examiners, whom he does not cite. Subsequently, several French mathematicians, including Lemoine himself [7], showed that the same answer could be obtained by formulating the problem directly in terms of the continuum and solving it by use of geometry. References can be found in [7] and [12]. They took as their sample space an equilateral triangle and interpreted the trilinear coordinates of a sample point as the lengths of the broken pieces, as in Figure 1. Relative area provides a uniform distribution of probability on the space. Thus, for them, random trisection of a rod amounted to choosing a point "at random" in the triangle.