Experimental test of fair three-sided coins

Abstract

A simple model for a fair ‘three-sided coin’ is proposed and tested. Describing the coin as a cylinder with a given height and basis radius, this model efficiently characterizes the problem, constraining the size of the coin. A statistical analysis of the data collected from actual realizations of such coins has been performed, supporting the proposed model. Besides studying the case of a fair three-sided coin, this work represents a model for an explicit application of the scientific method, in which all parts (problem characterization, statement of a hypothesis, experiment, analysis, description, conclusions) have clearly directed its development. Thus, it represents an useful illustration of such method for undergraduate students.

Exact edge landing probability for the bouncing coin toss and the three-sided die problem

“…Thus, the predicted critical angle for the fair three-sided die derived from our model is θ f air c 0.693 rad, hence, the ratio of cylinder height vs. diameter η := h/2r 0.831. We note that this value differs from previous approximations such as in [9,10,13], in which it was argued that η 1/ √ 3 0.577. Just as in Simpson's model [5], these approximations disregard the impact of bouncing occurring in real tosses, which is known to largely reduce the probability of edge outcome [19].…”

“…First, their energy must be greater than or equal to E c ; and second, their corresponding energy immediately after the collision has to be lower than E c , following Eq. (10). The previous constrains are summarized by the following inequalities:…”

“…Simpson's model has been extensively rejected for noninelastic tosses, as reported in the experimental works of Buden [6], Singmaster [7] and Heilbronner [8], but it is still adopted in current scientific outreach [9] and in recent studies due to its simplicity [10]. Nevertheless, this model is more often presented in an extended version where not only the geometry but also the angular momentum are taken into account [11][12][13][14], providing a solution for the non-bouncing toss only.…”

Exact face-landing probabilities for bouncing objects: Edge probability in the coin toss and the three-sided die problem

“…, (10) that, interestingly, only depends on the geometry of the coin (i.e., θ c ), and not on the restitution coefficient γ . For the examples of 1 £, 1 €, and a quarter $ coins, the theory predicts an edge outcome probability of 1 over ∼1000, 3000, and 8000 tosses, respectively.…”

“…Simpson's model has been experimentally rejected for more realistic tosses as reported by Buden [6], Singmaster [7], and Heilbronner [8], yet it is still adopted in current scientific outreach [9] and in recent studies due to its simplicity [10]. Extended versions take into account constraints in the angular momentum [11][12][13][14] but still provide a solution for the nonbouncing toss only.…”