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arXiv · 1601.05706

Pachinko

Hugo A. Akitaya·Erik D. Demaine·Martin L. Demaine·Adam Hesterberg·Ferran Hurtado·Jason S. Ku·Jayson Lynch
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Abstract

Inspired by the Japanese game Pachinko, we study simple (perfectly "inelastic" collisions) dynamics of a unit ball falling amidst point obstacles (pins) in the plane. A classic example is that a checkerboard grid of pins produces the binomial distribution, but what probability distributions result from different pin placements? In the 50-50 model, where the pins form a subset of this grid, not all probability distributions are possible, but surprisingly the uniform distribution is possible for $\{1,2,4,8,16\}$ possible drop locations. Furthermore, every probability distribution can be approximated arbitrarily closely, and every dyadic probability distribution can be divided by a suitable power of $2$ and then constructed exactly (along with extra "junk" outputs). In a more general model, if a ball hits a pin off center, it falls left or right accordingly. Then we prove a universality result: any distribution of $n$ dyadic probabilities, each specified by $k$ bits, can be constructed using $O(n k^2)$ pins, which is close to the information-theoretic lower bound of $\Omega(n k)$.

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Hugo A. Akitaya, Erik D. Demaine, Martin L. Demaine, Adam Hesterberg, Ferran Hurtado, Jason S. Ku, Jayson Lynch. 2016-01-21. Pachinko. https://arxiv.org/abs/1601.05706

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