### Paper and Pencil Games

# Sprouts

A two player game of drawing lines and dots that has a complex strategy.

Sprouts is a game of drawing lines and dots that is played between two people. It appears simple at first, but the game has a complicated strategy and interesting mathematical properties.

The game is setup by drawing 2 or more dots on a piece of paper. The more dots you start with, the longer the game. The number of moves in a game will be somewhere between 2x and 3x the number of dots that you start with. Two or three dots is a good place to start for new players.

Once the initial dots are placed, players take turns drawing a loop from a dot back to itself, or drawing a line between two different dots. The line can be straight or curvy and can loop around other dots in any way desired. Once the line is drawn, a new dot is added somewhere in the middle of this new line. These additional rules must be obeyed:

- A line cannot touch any other line or third dot.
- A dot cannot have more than 3 lines coming out of it (a loop counts as 2 lines).

The player who makes the last move wins.

A dot that has 3 lines coming out of it is considered dead and can no longer be used. Dots with 0, 1 or 2 lines are alive and can be used, however it is possible for a live dot to become unusable (see green dots in example above). Such dots are called "survivors".

## Strategy

There are 17 possible endings for a game that starts with 2 dots (if you combine topologically equivalent drawings), and the player who goes second can guarantee a win for themselves if they play perfectly. It's possible to learn this with practice. With games starting with 3 or more dots, it is mathematically possible to determine who will win, but in practice, it becomes too complicated for a person to play perfectly.

Winning a game of Sprouts comes down to controlling the number of moves that are left in the game. The first player wants there to be an odd number of moves and the second player wants there to be an even number of moves. For example, a 3 dot game lasts 6, 7 or 8 moves with the 8 move game being rare. Essentially, one player is trying to make the game short and the other is trying to make it last one move longer. In larger games, the same thing seems to be true.

If you are trying to make it a short game, your strategy is to make moves that maximize the number of survivors left at the end. Remember, a survivor is a dot that is still alive (has fewer than 3 lines) but cannot legally connect to any other dot.

If you are trying to make it a long game, your strategy is to do the opposite. You want to make "Pharisees" which is the fancy name given to dead dots that are connected to other dead dots.

Wikipedia

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