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Note: This is a question about a strategy puzzle / game, and is not proposing a puzzle itself.

Growing up I often played a pencil and paper game with the following rules:

  • First someone draws some dots on a piece of paper (number is usually small, maybe 6 - 10), and placement doesn't matter as will become clear from the rules of play.

  • Players then take turns drawing a curve starting at a dot and ending at a dot (it can even be the same dot the curve started at) subject to the restrictions

    1. The curve cannot cross over any curve (including itself)

    2. Each dot can have a maximum total of 3 curve end/start points on it

  • After a new curve is drawn another dot is added somewhere on that curve

  • The winner is whoever can make the last valid move.

Due to how one draws the lines, it is possible to separate off points that cannot become "saturated" with three curve ends/starts, hence allowing strategy due to choice of how to have the resulting graph "grow".

Does anyone know the name of this game? And even better, a book analyzing the strategy of this game?

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  • $\begingroup$ Numberphile made a video on this. $\endgroup$
    – Abraham Zhang
    Dec 1, 2014 at 6:19

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This game is called Sprouts. There is a discussion in Winning ways for your mathematical plays, Vol. 3, by Berlekamp, Conway, and Guy, on p. 598.

Sprouts also has a Wikipedia page.

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  • $\begingroup$ It's also mentioned in Sid Sackson's "Gamut of Games" $\endgroup$
    – Almo
    Dec 1, 2014 at 15:54

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