2561 points on a circle
Source: 2018 Thailand October Camp 1.1
February 18, 2022
combinatorics
Problem Statement
Let given points on a circle be colored either red or green. In each step, all points are recolored simultaneously in the following way: if both direct neighbors of a point have the same color as , then the color of remains unchanged, otherwise obtains the other color. Starting with the initial coloring , we obtain the colorings after several recoloring steps. Determine the smallest number such that, for any initial coloring , we must have .