Pachinko
Author(s)
Akitaya, Hugo A.; Demaine, Erik D; Demaine, Martin L; Hesterberg, Adam Classen; Hurtado, Ferran; Ku, Jason S; Lynch, Jayson R.; ... Show more Show less![Thumbnail](/bitstream/handle/1721.1/128438/1601.05706.pdf.jpg?sequence=4&isAllowed=y)
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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(nk2) pins, which is close to the information-theoretic lower bound of Ω(nk).
Date issued
2017-07Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Computer Science and Artificial Intelligence LaboratoryJournal
Computational Geometry
Publisher
Elsevier BV
Citation
Akitaya, Hugo A. et al. "Pachinko." Computational Geometry 68 (March 2018): 226-242 © 2017 Elsevier B.V.
Version: Original manuscript
ISSN
0925-7721