# Hex game winning strategy

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).

• I am not sure how to draw hexagons here so I replaced them with numbers instead. – davis Jun 10 '14 at 6:12
• It's not at all clear that Black's first move must be 15 ... any of 1, 2, 10, 11 would disrupt an immediate win even if White plays 15 second. – Greg Martin Jun 10 '14 at 6:37
• @GregMartin It was just a strategy I came up with. I was not sure if it would work. Is there a strategy for playing $4$ x $4$ hex game starting at the main diagonal. When I was reading, the authors said there was. – davis Jun 10 '14 at 6:48
• I just started learning how to play this game a couple days ago. I was wondering what the winning strategy would be. – davis Jun 10 '14 at 6:49
• What would the winning strategy be a player one (white) to block player two's moves (black)? – davis Jun 10 '14 at 7:44

## 3 Answers

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.

• The first player (White) has to start at the principal diagonal which can be six. I wanted to know what White's strategy would be starting at this position. – davis Jun 10 '14 at 8:47
• According to E. Mendelson "Introducing Game Theory and its Applications," he said that White has the winning strategy and to show what that winning strategy is starting at 1, 6, 11, or 16. The first player (White) has to start at one of these positions first. – davis Jun 10 '14 at 8:58
• Are you sure you're looking at the right diagonal? Are you sure you're numbering the cells the same was Mendelson does? The corner cell $1$ is very unlikely to be a good move, seeing as it has only two neighbors. I can believe that the cells on the shorter diagonal ($4,7,10,13$) are all winning first moves. – bof Jun 10 '14 at 9:07

@bof yes Mendelson did state this problem the same exact way. He said show White has the winnng strategy at 1, 6, 11, or 16. White has to start at one of these positions.

• @bof yes Mendelson did state this problem the same exact way. He said show White has the winning strategy at 1, 6, 11, or 16. White has to start at one of these positions. – davis Jun 10 '14 at 9:15
• @bof sorry I have to comment this way. I was unregistered and lost my account. – davis Jun 10 '14 at 9:17
• @bof What if the diagonal was the other way where 1 connected to 5 and 8 connected to 3 & 4. Can you help me write a winning strategy for this diagonal with the same rule that White has to start at 1, 6, 11, or 16? – davis Jun 10 '14 at 9:24
• Sorry, I need to get some sleep now. Look at the winning strategy I posted in my answer. Then figure out how to adapt it or use the same idea with the hexagons connected the other way. – bof Jun 10 '14 at 9:44

A winning strategy for the first player on a 5 by 5 board is to play in the center. Now consider the horizontal and vertical striped regions separately. If you have a strategy to connect the green edge to the center, taking place only in the horizontally striped squares, with red going first, then you can apply it separately on both sides to win. (If red plays on horizontal, you respond in kind. ) Here are the responses to all red moves, red dots represent a place that red might have moved, and the second green dot is the response. Yellow lines are matching pairs, any time that red places on one side, you place on the other. If red plays on the white (non striped) squares, pretend they played on some other square. For a 4 by 4 board, use this strategy on one side, and a matching pair on the other. 