Abstract
Pachinko is a Japanese mechanical gambling game similar to pinball. Recently, several mathematical models of Pachinko have been proposed. A number of pins are spiked in a field. A ball drops from the top of the playfield and the ball falls down. In the 50-50 model, if the ball hits a pin, it moves to the left or right passage of the pin with an equal probability. An arrangement of pins generates a distribution of the drop probability for all of the columns. This problem was considered by generating uniform distributions. Previous studies have demonstrated that the (1/2a)-uniform distribution is possible for a∈{0,1,2,3,4} and is conjectured so that it is possible for any positive integer a. This study describes the constructive proof for this conjecture. This study also formalizes a natural decision problem yielded by this model while investigating its computational complexity. More precisely, given any drop-probability distribution A and any partial drop-probability distribution B, this study uses non-deterministic polynomial-time (NP) hardness to determine if there exists a pin arrangement that transforms A into B.
Highlights
1.1 BackgroundPachinko is a japanese mechanical gambling game similar to pinball (Figure 1)
Pachinko is analyzed in the context of discrete mathematics
The origin of mathematical Pachinko is the book written by Akiyama in 2008 [3], and recently, Akitaya et al study an idealized geometry of a simple form of Pachinko [2]
Summary
Pachinko is a japanese mechanical gambling game similar to pinball (Figure 1). The machine stands up vertically, and the player shoots a metal ball into the playfield. Many pins are spiked in the playfield, and the ball drops from the top of the field. If it goes into a pocket in the field, the player gets some reward. The 50-50 model consists of three factors, field, pins, and a ball. A row is a horizontal line where lattice points exist, and a column is a vertical line where lattice points exist. Since we consider the triangle lattice, intersection points of rows and columns do not necessarily have a lattice point (see Figure 2). The ball drops from the center of the top end and falls down vertically. Arrangement under the 50-50 model, we can calculate the probability that the ball drops to each column. We can define its inverse problem of “deciding whether there exists a pin arrangement generating a given distribution or not”
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