Sprouts is a pencil-and-paper game with interesting mathematical properties. It was invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in 1967.
The game is played by two players, 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.
In so-called normal play, the player who makes the last move wins. In 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. , p. 21)
The diagram on the right shows a 2-spot game of normal-play Sprouts. After the fourth move, most of the spots are dead–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.
Suppose that a game starts with n spots and lasts for exactly m moves.
Each spot starts with three lives (opportunities to connect a line) and each move reduces the total number of lives in the game by one (two lives are lost at the ends of the line, but the new spot has one life). So at the end of the game there are 3n−m remaining lives. Each surviving spot has only one life (otherwise there would be another move joining that spot to itself), so there are exactly 3n−m survivors. There must be at least one survivor, namely the spot added in the final move. So 3n−m ≥ 1; hence a game can last no more than 3n−1 moves.
At the end of the game each survivor has exactly two dead neighbors, in a technical sense of "neighbor"; see the diagram on the right. No dead spot can be the neighbor of two different survivors, for otherwise there would be a move joining the survivors. All other dead spots (not neighbors of a survivor) are called pharisees (from the Hebrew for "separated ones"). Suppose there are p pharisees. Then
since initial spots + moves = total spots at end of game = survivors + neighbors + pharisees. Rearranging gives:
So a game lasts for at least 2n moves, and the number of pharisees is divisible by 4.
Real games seem to turn into a battle over whether the number of moves will be m or m+1 with other possibilities being quite unlikely.  One player tries to create enclosed regions containing survivors (thus reducing the total number of moves that will be played) and the other tries to create pharisees (thus increasing the number of moves that will be played).
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