What you need
A pencil and a sheet of paper. Sprouts was invented in 1967 by the mathematicians John Conway and Michael Paterson, and made famous by Martin Gardner’s Mathematical Games column in Scientific American. It looks like doodling but hides real topology.
Setup
Draw a few spots on the paper, well spaced. Two or three spots make a short game for learning; five or six give a proper contest. Decide who goes first.
How to play
On your turn you do two things:
- Draw a line joining two spots, or a loop from one spot back to itself. The line may be straight or curved.
- Add a new spot somewhere along the line you just drew, not on its ends.
The line must obey three rules:
- It may not cross or touch itself or any other line, and may not pass through a spot.
- No spot may ever have more than three lines attached to it. A loop from a spot to itself counts as two lines at that spot.
- The new spot already has two lines (one each side), so it can take only one more.
A spot with three lines is dead. A spot with fewer than three is alive and can still be used.
Scoring and winning
Players alternate until someone cannot make a legal move. In the normal game, the player who makes the last move wins: the player unable to move loses. There are no draws.
The game always ends. Each spot starts with three “lives”; every move uses up two lives at the ends of the line and creates a new spot with one life left, so the total falls by one each turn. A game with n starting spots lasts at least 2n moves and at most 3n − 1.
Who wins with perfect play depends on the number of spots. Computer analysis supports the Sprouts conjecture: the first player wins when the number of starting spots, divided by 6, leaves a remainder of 3, 4 or 5, and loses otherwise. Two spots is a second-player win; three spots a first-player win.
Common variants
- Misère sprouts: the player who makes the last move loses. It is harder to analyse and played competitively online.
- Brussels sprouts: start with crosses (spots with four free arms) instead of spots. Each move joins two free arms with a line and draws a short stroke across it, creating two new arms. The number of moves is always 5n − 2 for n crosses, so the winner is decided before the game begins: a fun trick to play on someone once.
- Bigger starts: 7 or 8 spots make long, tricky games on a big sheet.
Tips for first games
- Count moves, not spots. Since the game has a fixed range of possible lengths, your goal is to make the total number of moves odd if you started, even if you did not.
- Enclosing a region with a loop shortens the game: spots trapped inside can only connect to each other.
- Keeping all spots in one open group lengthens the game.
- Draw neatly and leave room; most arguments come from lines that seem to touch.
- With two spots, play both sides to see why the first player always loses.
Frequently asked questions
Can a line go from a spot to itself? Yes, as a loop. It counts as two of that spot’s three lines.
Do the new spots count towards the three-line limit? Yes. A new spot starts with two lines, so it can take one more.
Can lines be curved? Yes, any shape, as long as they never cross or touch another line.
Who wins with three spots? The first player, with perfect play.
Can the game go on forever? No. With n starting spots it ends after at most 3n − 1 moves.
