In a one-dimensional arrangement of n circles, when one specific circle is colored, each of its adjacent circles has a one-half probability of being colored as well. This study seeks to determine the optimal coloring method that minimizes the expected value of the specified colorings for the given arrangement. We investigate three configurations: "linear arrangement," "circular arrangement," and "circular combined linear arrangement" involving n circles and m circles.
By deriving a recursive relationship based on the structure of each configuration, we develop the generating function and characteristic equation, leading to a comprehensive general solution. Employing algebraic operations on complex numbers and geometric properties, we unveil the optimal coloring method for the "linear arrangement," revealing a fascinating consistency between the optimal strategies for both the linear and circular arrangements. Furthermore, we highlight the intriguing phenomenon where the optimal coloring methods for the "circular combined linear arrangement" vary with different values of n and m.