MathDB

The game of Hive is played on a regular hexagonal grid (as shown in the figure) by 3 players. The grid consists of

k

layers (where

k

is a natural number) surrounding a regular hexagon, with each layer constructed around the previous layer. The figure below shows a grid with 2 layers.
The players, Ali, Shayan, and Sajad, take turns playing the game. In each turn, a player places a tile, similar to the one shown in the figure, on the empty cells of the grid (rotation of the tile is also allowed). The first player who is unable to place a tile on the grid loses the game. Prove that two players can collaborate in such a way that the third player always loses.
Proposed by Pouria Mahmoudkhan Shirazi.

Problem Statement