Investigating the probability of a cylindrical coin landing on its side

Anton Gaek; Artem Sukhov

Home/Archive/arXiv

arXiv · 2401.03557

Investigating the probability of a cylindrical coin landing on its side

Anton Gaek·Artem Sukhov

View original source ↗Read source document ↗Export metadata ↓

Abstract

The problem of creating a three-sided dice with the probability of it landing on each of its sides being equal to 1/3 has been around for many years. Various approaches have been attempted, but as different authors achieved at different results, no uniform answer has been found. In this paper, the probability of a cylinder-shaped coin landing on its side is investigated, aiming to predict the final position of the coin, based on the starting physical and geometrical parameters of the coin. Pure physical methods and computer modeling have been used to find the parameters of a "fair" coin and to determine which parameters of the system affect the probability of the coin landing on an edge, and in what way. Statistical analysis of the situation has been made, to ensure that the probabilities we are trying to find, correspond to the appropriate physical events. Overall, two main possibilities for modeling the situation are considered: single-axis rotation (flat rotation) and the superposition of rotation against multiple axes (volumetric rotation). For each, a model was built and it was compared with computer modeling and real-life experiments. Dynamical and energetic approaches have been considered, and a model of the coin during its impact with the ground has been created. A robot that is capable of launching cylindrical coins, picking them up, and recording the result of the experiment, controlling starting parameters precisely, and distributing them to our needs, and automatically record the results of the experiment, was created. In conclusion, the parameters of a "fair" coin are provided, which have even probabilities landing on all its sides, as well as the dependencies of the probability of a coin landing on its edge, on different parameters.

Explore related subjects

math.DS

Keep this discovery

BibTeX RIS

Anton Gaek, Artem Sukhov. 2024-01-07. Investigating the probability of a cylindrical coin landing on its side. https://arxiv.org/abs/2401.03557

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗

arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗

arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗