I was teaching myself how to play a hex board game by reading some books a couple days ago. I learned how to do $2$ x $2$ and $3$ x $3$ hex games by starting at the principal diagonal.
I wanted to know what the winning strategy would be for player one (white) on a $4$ x $4$ Hex game starting from the principal diagonal to block the second player's move (black).
Consider a $4$ x $4$ Hex.
Show that White has a winning strategy, starting anywhere on the principal diagonal that is in any of the hexagons $1,6, 11,$ or $16$.
Here is the setup:
Let the Hexagons be represented by numbers such as:
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
I do not know how to draw hexagons here so I replaced them with numbers. This is suppose to be a regular Hex board game. Sorry if this confused anyone.
Let White have the first move (match up and down). Let black have the second move (match left to right). White has to start at 1, 6 ,11, or 16 since it is part of the principal diagonal. Show that White can win starting at this position.
White opens up at 6 (principal diagonal).
If White plays $6$ as his opening move, then Black can force a win as follows.
First, Black plays $10$. After that, Black makes sure to play at least one element in each of the following disjoint pairs: $\{4,8\},\{7,15\},\{9,13\},\{11,14\},\{12,16\}$.
$10$ connects to the left side via $9$ or $13$.
If Black plays 7, then $10$ connects to $7$ which connects to $4$ or $8$ on the right side; if Black plays $15$, then $10$ connects via $11$ or $14$ to $15$, which connects to $12$ or $16$ on the right side.