## Paper and Pencil Game Puzzle

A forum for good logic/math puzzles.

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ErinPickle
Posts: 9
Joined: Mon Nov 17, 2014 9:56 pm UTC

### Paper and Pencil Game Puzzle

First the rules of the game:

Number of Players: 2

Setup/Board Description: Square Grid (such as graph paper.) The Top and Bottom edges are owned by Player 1, and the Left and Right edges are owned by Player 2. Corners are shared.

Suggested Board Sizes and Play Times:
15 min - 11x11
30 min - 16x16
60 min - 22x22

Overview:
Each player tries to connect their two edges of the board by drawing dots and connecting them together.

Each turn:
* A player draws a dot in ANY empty square on the board.
* Then the player connects it (draws a line) to all of their adjacent dots in the eight adjoining squares, unless it would cross an opponent's connection.

Win condition: A player connects their two edges of the board.

Pie rule (optional, this rule nullifies the 1st player's inherent advantage): After the first move is made, the second player has one of two options:

1) Letting the move stand, in which case the second player remains the second player and moves immediately, or
2) Switching places, in which case the second player becomes the first-moving player, and the "new" second player then makes their "first" move. (I.e., the game proceeds from the opening move already made, but with roles reversed.)

Now the puzzle:

In the following 7x7 n00b game, Red moved at d3, then Blue at b4, then Red at d5. So it's Blue's turn to move. Is there any way:

1a) Blue can stop red from forming a connection between d3 and d5?
1b) If not, where should have Blue moved instead of b4 on their last turn?
2) Can Blue stop Red from winning the game at this point (i.e., has Red already won)?

Sizik
Posts: 1170
Joined: Wed Aug 27, 2008 3:48 am UTC

### Re: Paper and Pencil Game Puzzle

Some questions:
Do the lines connecting dots have to be straight, or are they free-form?
How thick are the lines?
Do they have to go between the centers of the dots, (i.e. perpendicular to the edge of the dot), or can they come out at any angle?
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SirGabriel
Posts: 40
Joined: Wed Jul 16, 2014 11:54 pm UTC

### Re: Paper and Pencil Game Puzzle

Red has already won. And, although I can't prove it, I'm fairly certain that blue could have guaranteed a blue victory if they had chosen d5 instead of b4 as their first move.

ErinPickle
Posts: 9
Joined: Mon Nov 17, 2014 9:56 pm UTC

### Re: Paper and Pencil Game Puzzle

Some questions:
Do the lines connecting dots have to be straight, or are they free-form?

Straight.

How thick are the lines?

As thick as the line your marker, pen, crayon, colored pencil, etc makes.

Do they have to go between the centers of the dots, (i.e. perpendicular to the edge of the dot), or can they come out at any angle?

So to explain it in a bit more detail:
Each turn you draw a dot in an empty square. Then if there is a pre-existing dot of yours in any of 8 squares surrounding it, you draw a line between the dot you just made and that dot. You do this for any and all dots in the 8 squares around your 'new' dot - so you could possibly draw up to 8 straight lines radiating from your new dot on a turn.
However, if there's an opponent's connection already in the way of drawing a connection, then you don't draw it.

SirGabriel: Yes, you are correct I believe. There is also another thing the 2nd player (hint hint) could have done on their first turn that would have helped their chances greatly, especially on a small board

If there's interest, I'll post another problem for this game later tonight or tomorrow.

Wildcard
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### Re: Paper and Pencil Game Puzzle

You would probably greatly enjoy the game "Tak." http://cheapass.com//wp-content/uploads ... -Rules.pdf
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Shufflepants
Posts: 16
Joined: Wed Jul 20, 2016 3:12 pm UTC

### Re: Paper and Pencil Game Puzzle

We can see that there exists a strategy by which the first player can always win by "strategy stealing" argument.

1) There must be a winner when the whole board is filled because a complete wall blocking the other player would be a winning path itself.
2) Having an additional dot on the board can never hurt your position.
3) If there existed a strategy for the second player to win, the first player could move anywhere and then on their second move start using the second player's strategy.

Now, for the specific position you've given:

Code: Select all

`..R.B.....R.`

There's no way blue can disconnect the 2 red dots. If blue plays in any of the spaces marked with X's

Code: Select all

`..R.BXXX..R.`

Red can always play in either of the other two spaces to connect.

So, now red only needs to connect the d5 dot to the top and the d3 dot to the bottom.
By symmetry, we can look at just connecting the d5 dot to the top, and the d3 dot can be connected to the bottom in the same way.

The winning strategy for Red:

Code: Select all

`.12345...678.....R...`

If Blue plays on 3, Red plays on 7, and then on Red's next move, either 2 or 4 will be free to connect to connecting Red to the top.
If Blue plays on 7, Red plays on either 6 or 8, and then on Red's next move if Red played on 6, either 1 or 2 will be free to connect to or if red played on 8, either 4 or 5 will be free to connect to.
If Blue plays on 1,2,4,5,6 or 8 Red can play at 7 and then on the next move, at least one of 2,3, or 4 will be free to connect to.
If Blue doesn't play on a numbered spot, Red can just play as if Blue had played on a numbered spot.
If Blue plays on the other half of the board, Red responds following this same strategy but flipped.

This game is very similar to hex https://en.wikipedia.org/wiki/Hex_(board_game). You might find it more interesting. It is still a theoretical win for the first player, but since the board is slightly less connected, the winning strategies are less obvious and immediate.