World

Game

Of

Sprouts

Association

Two players, Left and Right, alternate moves until no more moves are possible. In the normal game, the last person to move is the winner. In misere play, the last person to move is the loser.

The game is played with paper and pen. The starting position is some number of small circles called "spots". A move consists of drawing a new spot g and then drawing two lines, in the loose sense, each terminating at one end at spot g and at the other end at some other spot. (The two lines can go to different spots or the same spot, subject to the following conditions.) The lines drawn cannot touch or cross any line or spot along the way. Also, no more than three lines can terminate at any spot. A spot with three lines attached is said to be "dead", since it cannot facilitate any further action.

If an original spot were a motel, it would have three vacancies. Let's say three "liberties". A move uses up two liberties and puts only one back. The game, therefore, has to wind down. Eventually any live spots remaining are sufficiently depleted and isolated that no more moves are possible and the game is over.

It turns out that live spots at the end of the game are the key to making the game fun to play. These live spots at game's end are called "survivors". Suppose a normal sprouts game begins with an even number of spots. Left (the first player) will win if there are an odd number of survivors. If the game started with an odd number of spots, then Left will win if there are an even number of survivors. A mnemonic is, "Left is the aggressor. He needs to change things."

Here is an amusing, alternative way to teach the rules of sprouts. (1) Explain what constitutes a legal move. (2) Explain what is a survivor. (3) Say something like, "One of us will play for an even number of survivors, and one of us will play for an odd number of survivors."

This alternative formulation is sprouts pure and simple. But a novice taught in this way will play his first game at a much higher level of skill than a novice going by the "real" rules.

Sprouts was invented in 1967 by two extraordinary mathematicians, John H. Conway and Michael S. Paterson. The game was popularized by one of Martin Gardner's inimitable "Mathematical Games" columns in *Scientific American*. Surprisingly, Gardner did not mention anything about survivors, and for years the game was considered by many otherwise knowledgeable people to be almost unplayably complex. It was seen as being completely tactical, with each game being a single, mind-boggling calculation. But with the simple idea of survivors, strategical thinking finds its foothold and positions which Gardner thought difficult become child's play.

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