Editing Sprouts (game) (section)

==Rules==
[[File:Sprouts-2spot-game.png|right|300px|thumb|A 2-spot game of Sprouts. The game ends when the first player is unable to draw a connecting line between the only two free points, marked in green.]]

The game is played by two players,<ref>{{cite journal |last1=Lam |first1=T. K. |title=Connected Sprouts |journal=The American Mathematical Monthly |date=10 April 2018 |volume=104 |issue=2 |pages=116–119 |doi=10.1080/00029890.1997.11990609}}</ref> starting with a few spots drawn on a sheet of paper. Players take turns, where each turn consists of drawing a line between two spots (or from a spot to itself) and adding a new spot somewhere along the line. The players are constrained by the following rules:
* The line may be straight or curved, but must not touch or cross itself or any other line.
* The new spot cannot be placed on top of one of the endpoints of the new line. Thus the new spot splits the line into two shorter lines.
* No spot may have more than three lines attached to it. For the purposes of this rule, a line from the spot to itself counts as two attached lines and new spots are counted as having two lines already attached to them.
* You cannot touch a dot twice with one line then connect it to another.

In so-called ''normal play'', the player who makes the last move wins. In ''[[misère|misère play]]'', the player who makes the last move loses. Misère Sprouts is perhaps the only misère combinatorial game that is played competitively in an organized forum.<ref>{{cite arXiv |last=Plambeck |first=Thane E. |eprint=math/0603027v1 |title=Advances in losing |date=2006 |page=21}}</ref>

The diagram on the right shows a 2-spot game of normal-play Sprouts. After the fourth move, most of the spots are ''dead''&ndash;they have three lines attached to them, so they cannot be used as endpoints for a new line. There are two spots (shown in green) that are still ''alive'', having fewer than three lines attached. However, it is impossible to make another move, because a line from a live spot to itself would make four attachments, and a line from one live spot to the other would cross lines. Therefore, no fifth move is possible, and the first player loses. Live spots at the end of the game are called ''survivors'' and play a key role in the analysis of Sprouts.