Uniform Distribution On Pachinko

Pachinko is a japanese mechanical gambling game similar to pinball. Recently, Akitaya et al. proposed several mathematical models of Pachinko. A number of pins are spiked in a field. A ball drops from the top-side end of the playfield, and falls down. In the 50-50 model, if the ball hits a pin, it moves to the left or right of the pin with equal probability. An arrangement of pins generates a distribution of the drop probability over all columns. We consider the problem of generating uniform distributions. Akitaya et al. show that (1/2)-uniform distribution is possible for a ∈ {0, 1, 2, 3, 4} and conjectured that it is possible for any positive integer a. In this paper, we show that the conjecture is true by a constructive way. 2012 ACM Subject Classification Mathematics of computing → Combinatoric problems