Question 23
Anne, Ellie and Milo play a game on a hexagonal board tiled with hexagons, $n \gt 2$ hexagons on each side. The game begins with a token on a tile in one of corner of the board. Ellie and Milo are on the same team, playing against Anne, and they win if the token lands on center of the board. Anne, Ellie and Milo take turns moving the token, Anne begins, then Ellie and then Milo.
The turns proceed as follows:
* Anne has to move the token to an adjacent hexagon, in any direction.
* Ellie has to move the token by two hexagons in any of the 6 possible directions.
* Milo has a choice of passing the turn, or moving the token by three hexagons in any of the 6 possible directions.
Find all $n \gt 2$ for which Ellie and Milo have a winning strategy.
Figure 1: A board with 5 hexagons on each side.
The turns proceed as follows:
* Anne has to move the token to an adjacent hexagon, in any direction.
* Ellie has to move the token by two hexagons in any of the 6 possible directions.
* Milo has a choice of passing the turn, or moving the token by three hexagons in any of the 6 possible directions.
Find all $n \gt 2$ for which Ellie and Milo have a winning strategy.
Figure 1: A board with 5 hexagons on each side.