AoPS - Problem

Source: XII International Festival of Young Mathematicians Sozopol 2023, Theme for 10-12 grade

Tags: combinatorics

GeorgeRP

23.09.2024 14:41

Alex and Bobby take turns playing the following game on an initially white row of $5000$ cells, with Alex starting first. On her turn, Alex must color two adjacent white cells black. On his turn, Bobby must color either one or three consecutive white cells black. No player can make a move after which there will be a white cell with no adjacent white cell. The game ends when one player cannot make a move (in which case that player loses), or when the entire row is colored black (in which case Alex wins). Who has a winning strategy?

so technically If the entire row is colored black, Alex wins and if a player cannot make a move, that player loses so the ideal goals of the players must be:- Alex’s goal is to make the row black , which ensures her win. Bobby’s goal is to prevent Alex from completing the entire row, thus forcing Alex into a position where she cannot move. although i think that booby have a winning stratergy . If Bobby plays smart, he can stop Alex from winning by keeping the game in a way that makes it hard for Alex to play. Because of this, Bobby probably has the best chance to win since he can control the game and limit what Alex can do.