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From Wikipedia, the free encyclopedia

Sprouts is a pencil-and-paper game with interesting mathematical properties. It was first played by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in 1967.

A 2-spot game of Sprouts

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. [1], 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.

Contents

Analysis

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 3nm remaining lives. Each surviving spot has only one life (otherwise there would be another move joining that spot to itself), so there are exactly 3nm survivors. There must be at least one survivor, namely the spot added in the final move. So 3nm ≥ 1; hence a game can last no more than 3n−1 moves.

Live spots (green) and their dead neighbors (black).

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

n+m = 3nm + 2(3nm) + p

since initial spots + moves = total spots at end of game = survivors + neighbors + pharisees. Rearranging gives:

m = 2n + p/4

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. [2] 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).

Who has the win?

By enumerating all possible moves, one can show that the first player when playing with the best possible strategy will always win in normal-play games starting with n = 3, 4, or 5 spots. The second player wins when n = 0, 1, 2, or 6.

At Bell Labs in 1990, David Applegate, Guy Jacobson, and Daniel Sleator used a lot of computer power to push the analysis out to eleven spots in normal play and nine spots in misère play. Josh Purinton and Roman Khorkov have extended this analysis to sixteen spots in misère play.[3] Julien Lemoine and Simon Viennot have calculated normal play outcomes up to thirty-two spots, plus five more games between thirty-four and forty-seven spots.[4] They have also announced a result for the seventeen-spot misère game.[5]

The normal-play results are all consistent with the pattern observed by Applegate et al. up to eleven spots and conjectured to continue indefinitely, that the first player has a winning strategy when the number of spots divided by six leaves a remainder of three, four, or five. The results for misère play do not follow as simple a pattern: up to seventeen spots, the first player wins in misère Sprouts when the remainder (mod 6) is zero, four, or five, except that the first player wins the one-spot game and loses the four-spot game.

A 2-cross game of Brussels Sprouts lasting eight moves

Brussels Sprouts

A variant of the game, called Brussels Sprouts, starts with a number of crosses, i.e. spots with four free ends. Each move involves joining two free ends with a curve (again not crossing any existing line) and then putting a short stroke across the line to create two new free ends.

So each move removes two free ends and introduces two more. Despite this, the game is finite, and indeed the total number of moves is predetermined by the initial number of crosses: the players cannot affect the result by their play. With n initial crosses, the number of moves will be 5n−2, so a game starting with an odd number of crosses will be a first player win, while a game starting with an even number will be a second player win regardless of the moves.

To prove this (assuming that the game ends), let m denote the number of moves and v,e,f denote the number of vertices, edges, and faces of the planar graph obtained at the end of the game, respectively. We have:

  • e = 2m since at each move, the player adds 2 edges.
  • v = n + m since at each move, the player adds one vertex (and the game starts with n vertices).
  • f = 4n since there remains exactly one free end in each face (and the number of free ends does not change during the game).

The Euler characteristic for planar graphs is 2, so 2 = f-e+v = 4n-2m+n+m = 5n-m , hence m = 5n-2.

Other Variants

One can also make the addition of a point (for Sprouts) or a stroke (for Brussels Sprouts) on a newly drawn line optional. Doing so seems to make sprouts very complicated and nearly impossible to analyze. However, the variant of Brussels Sprouts becomes interesting, as the game stops depending on parity. The two games are tentatively titled Weeds for the Sprouts variation, and Brambles for the Brussels Sprouts variation.

References


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