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2017 BMT Discrete #7

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March 9, 2024
combinatorics

Problem Statement

A light has been placed on every lattice point (point with integer coordinates) on the (infi nite) 2DD plane. De ne the Chebyshev distance between points (x1,y1)(x_1,y_1) and (x2,y2)(x_2, y_2) to be  max(∣x1−x2∣,∣y1−y2∣)\ max (|x_1 - x_2|, |y_1 -y_2|). Each light is turned on with probability 12d/2\frac{1}{2^{d/2}} , where dd is the Chebyshev distance from that point to the origin. What is expected number of lights that have all their directly adjacent lights turned on? (Adjacent points being points such that ∣x1−x2∣+∣y1−y2∣=1|x_1-x_2|+|y_1- y_2| =1.)
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