How to Play Sprouts

By Jaques of London · Est. 1795 · Updated 2026-08-17

Sprouts is a two-player strategy game played with nothing more than a pen and a piece of paper. Players take turns connecting dots with curved lines, adding a new dot to each line as they go. Because every new dot can only ever take three connections, the board fills up and legal moves run out surprisingly quickly, making each game compact and tense.

Its appeal lies in how much genuine mathematical depth hides behind such simple rules. A game with just three starting dots can keep sharp players thinking hard, and the whole thing fits on the back of an envelope. It is a favourite for long car journeys, waiting rooms, and kitchen tables anywhere a pen is to hand.

A little history

Sprouts was invented on 21 February 1967 by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University. The game spread quickly through mathematical circles after Martin Gardner wrote about it in his Scientific American column in July 1967, introducing it to a much wider audience. Its deeper mathematical properties, including work on the maximum and minimum number of moves possible from a given starting position, have been studied by combinatorial game theorists ever since, and it remains a subject of ongoing mathematical interest.

Setting up

1. Agree on the number of starting dots, typically between 2 and 6 for a satisfying game.
2. One player draws that many dots spread out across the paper, spaced well apart to leave room for lines.
3. Decide who takes the first turn, by any convenient method such as a coin toss.
4. No further setup is needed. The game begins immediately.

How to play

1. On your turn, draw a curve that connects any two live dots, or connects a live dot back to itself in a loop.
2. The curve must be drawn in one continuous stroke and must not cross itself, cross any existing line, or pass through any existing dot.
3. Once you have drawn the curve, place a new dot anywhere along it. This new dot counts as part of the board from now on.
4. A dot is live if it has fewer than three connections. Count carefully: a dot used as both endpoints of a loop gains two connections from that single curve.
5. A dot that has exactly three connections is called dead. It cannot be used as an endpoint or have any new curve pass through it.
6. The new dot you place on each curve starts with two connections already used (one for each end of the curve touching it), so it can only ever accept one more connection before it too becomes dead. If the curve is a loop from a dot back to itself, the new dot placed on it has both its connections used and is immediately dead.
7. Players alternate turns. A player who cannot draw any legal curve on their turn loses.
8. There is no skipping a turn. If a legal move exists, you must make one.

Winning and scoring

The player who makes the last legal move wins. This is the normal play convention. Because moves are finite and the game always ends, there is no draw. With n starting dots the game will last at least 2n moves and at most 3n minus 1 moves, so with three starting dots the game ends in no fewer than 6 and no more than 8 moves. The misere variant, in which the player who makes the last move loses, is sometimes played and should be agreed before the game starts.

The fine points

The most common dispute concerns whether a curve passes through a dot. Any existing dot on the paper is treated as an obstacle: a curve may not pass through it, only terminate at it. If a line is drawn so that it skims ambiguously close to a dot, players should agree before the game starts to redraw the mark clearly. When playing on paper it helps to draw dots as small, distinct circles rather than vague marks, which removes most ambiguity. A curve that starts and ends at the same dot is legal, but the new dot placed on it is immediately dead because the loop accounts for two of that new dot's three permitted connections.

Another point that trips up new players is counting connections on the starting dot of a loop. If dot A draws a loop back to itself, dot A gains two connections from that one move, not one. Keep a tally beside each dot if the game is getting complex. There is no official tournament body for Sprouts, so house rules on disputed marks are standard practice; the key is to agree them before play begins.

How to win more often

• Count connections constantly. Knowing exactly how many connections each dot has tells you which dots are available and helps you plan several moves ahead.
• Aim to create isolated regions, enclosed areas of the paper where your opponent cannot reach with a new curve from outside. If you control who can play inside a region, you control the endgame.
• With an odd number of moves remaining in a region, the player whose turn it is to play into that region will make the last move there and gain the advantage. Try to manipulate regions so the parity favours you.
• Avoid creating new dots in positions that give your opponent easy connections. A dot placed near several live dots hands your opponent options on the next turn.
• In the opening with three or more starting dots, resist the urge to connect the dots that are farthest apart. Short, tight curves give you more control over where regions form.

Variations

Misere Sprouts

The rules are identical to standard Sprouts except that the player who makes the last legal move loses rather than wins. This reversal changes the optimal strategy substantially and is a worthwhile challenge once both players are comfortable with the standard game.

Brussels Sprouts

In this variant, each starting mark is a small cross rather than a dot. Each arm of the cross counts as one free connection, giving each starting mark four free connections. On each turn a player draws a curve between any two free arms and draws a new cross on the curve, adding four more free connections to the board. Unlike standard Sprouts, the total number of moves in any game of Brussels Sprouts is entirely determined by the number of starting crosses, making it of more mathematical than strategic interest.

Scored Sprouts

For a session of several rounds, players keep a running score. The winner of each game scores one point. First to an agreed total, usually five points, wins the session. This format suits players who find a single short game unsatisfying and want a reason to refine their strategy over time.

Large-board Sprouts

Begin with eight or more starting dots on a large sheet of paper. The game becomes considerably longer and more complex, with multiple distinct regions forming and interacting. This version rewards careful positional thinking and is better suited to experienced players who already understand how smaller games resolve.

Sprouts: frequently asked questions