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### Connection game : Fertility

Posted: Thu Jul 13, 2017 10:05 am UTC
This new game is based on the idea of fertility.
Each player has 3 kind of pieces :
- males (3)
- females (3)
- off-springs (the number will depend on the size of the board)

Each player has pieces of the same color (blue versus red for example)

Hex Board : 11x11

Blue (or red) has then 3 males, 3 females and 55 off-springs

Rules :

The board start empty
Blue start first
On his turn a player drop one piece on an empty hex case : either a male, either female, either an off-spring.
And players alternate
There game ends when all the board is full
Then start the score counting :
The red begins the counting by linking a male to some female and removing all the off-springs connecting the chosen male and the chosen female. He must do it for the 3 males and the 3 females. He must chose the longest path possible but he must keep in mind the order of removing the off-springs. Changing the order could disrupt some connections. This phase is the hardest one. His score will be equal to the removed off-springs.
Blue player has to do the same counting process.
The males and females are not counted.
During the counting process a player can remove only his own color pieces.

The player with the best score wins the game.

The first player has a slight advantage.
Is there a strategy to win for any of the players?

Thank you.

### Re: Connection game : Fertility

Posted: Thu Jul 13, 2017 9:29 pm UTC
So to restate the rules of the game:

1. The game is played between two players (Red and Blue) on a Hex board of a certain size.
2. Each player received a set of player-colored pieces: 3 male, 3 female, and sufficient offspring to fill the board.
3. Starting with Blue, players take turns placing one piece of their choice on the board.
4. Both players must eventually place all 6 of their male and female pieces.
5. When the board is full, the players score, first Red then Blue.
6. Each player scores by tracing 3 paths of adjacent pieces, each one connecting one of their male pieces to one of their female pieces, passing only through the player's own offspring pieces. None of the paths may have any overlap or self-intersections, but apart from that there is no restriction on how convoluted a path can be. The player's score is the number of offspring pieces passed through.
7. Whichever player has the higher score is the winner.

Is this restatement accurate? If not, where does it go awry? Also, what if a player is unable to complete all 3 paths (showing this is possible is trivial)?

In any case, I can give a partial answer to one of your questions right away:

Spoiler:
"Is there a strategy to win for any of the players?"

Answer: Yes, unless the best strategy for each player results in a tie, like in tic-tac-toe. This game is, after all, a two-player sequential perfect information zero-sum game.

### Re: Connection game : Fertility

Posted: Thu Jul 13, 2017 9:39 pm UTC
The statement is correct.
I could not write it with this wonderful clarity.
In the counting process a player must indicate the path from a chosen male (or female) to chosen female (or male) by removing the off-springs. So an off-spring can not be used twice. He belongs to one family not two or more.
Otherwise as long as off-springs are connected a player can choose any path to maximize his score.
Sometimes a player can not be able to connect some of his pairs (male and female)

### Re: Connection game : Fertility

Posted: Fri Jul 14, 2017 12:00 am UTC
I've had another realization. Take a look at this configuration of pieces (M is male, F is female, O is offspring, and x is an opponent's piece of some kind):

Code: Select all

`  M x x x O O xF O x O F x O O x  M x x`

If you are forced to make two paths, one of the offspring has to be excluded. However, if you are allowed to only make one path, you can connect the female on the left to one of the males while going around the entire loop, thus including six offspring instead of just five. Considering that sometimes you won't be able to make all 3 paths anyway, I feel that you should be allowed to make only 1 or 2 paths even if making 3 is possible, because of situations like the one described above.

### Re: Connection game : Fertility

Posted: Fri Jul 14, 2017 4:05 pm UTC
It is up to the player to optimize his score by making the best choice under the constraints of the rules.
A player could use only one pair if he is able to connect all his off-springs. No player is forced to use all his 3 pairs.
Sometimes even the choice of the order of counting is important. Assume that you are able to use all the 3 pairs starting from one pair could disrupt the connection between one or the 2 others.
During the placement phase any choice has huge consequences. If you start by drop a male or female then you are allowing your opponent to break or reduce the connection. But if you start dropping your off-springs first you will be able to connect them.
Anyway there are man choices and many tactics and strategies that I feel that the game will be hard to master.

### Re: Connection game : Fertility

Posted: Sun Jul 16, 2017 3:45 pm UTC
For this game it must be played in a board 2nx2n to avoid giving a slight advantage to the blue player.
As we talk about families and fertility an idea came to my mind : introducing neutral pieces ("orphans"). So any player could make path using them and as possible consequence disrupting the counting process of his opponent.

### Re: Connection game : Fertility

Posted: Wed Jul 19, 2017 2:40 pm UTC
Poker wrote:I've had another realization. Take a look at this configuration of pieces (M is male, F is female, O is offspring, and x is an opponent's piece of some kind):

Code: Select all

`  M x x x O O xF O x O F x O O x  M x x`

If you are forced to make two paths, one of the offspring has to be excluded. However, if you are allowed to only make one path, you can connect the female on the left to one of the males while going around the entire loop, thus including six offspring instead of just five. Considering that sometimes you won't be able to make all 3 paths anyway, I feel that you should be allowed to make only 1 or 2 paths even if making 3 is possible, because of situations like the one described above.

In the given case above one path suffices to score the max : 6 points.
No player is forced to use the 3 paths. He has to maximize his score and that`s it.