A game consists of a grid of $4\times 4$ and tiles of two colors (Yellow and White). A player chooses a type of token and gives it to the second player who places it where he wants, then the second player chooses a type of token and gives it to the first who places it where he wants, They continue in this way and the one who manages to form a line with three tiles of the same color wins (horizontal, vertical or diagonal and regardless of whether it is the tile you started with or not). Before starting the game, two yellow and two white pieces are already placed as shows the figure below. Yolanda and Xinia play a game. If Yolanda starts (choosing the token and giving it to Xinia for this to place) indicate if there is a winning strategy for either of the two players and, if any, describe the strategy.
Problem
Source: OLCOMA Costa Rica National Olympiad, Final Round, 2017 3.3
Tags: combinatorics, game, game strategy, winning strategy
11.08.2024 19:26
shortened Yolanda to Y, and Xinia to X I refer to the grid cells with coordinates such with the columns being 1, 2, 3, 4 from left to right and the rows being C, D, E, F from top to bottom I will also note that if there is ever a way to get 3 in a row from both types of counters then whoever puts one down next will get a win no matter what counter the other player gives them. I refer to this as case 1 Y goes first and it is obvious that giving X a 'B' counter gives X a win, so Y will give X an A counter. If X goes anywhere other than the 2 remaining corners then Y has a case 1, so X will place the counter in either C4, or F4, as their is top-left to bottom-right symmetry, WLoG we can assume the A is placed in C4 X then gives Y an A counter for the same reason, and also for the same reason, Y must place the counter in F1. Y can chose to give X a B counter who, where it is placed, will force X to give Y an A counter who will place it in the cell that blocks the 2 in a row created by X Similar things repeat themselves until X can't help it but let Y have a case 11 so Y will win. The second half of this is just a summary for what you would actually have to write