The Pencil-and-Paper Classic, Drawn for Real
Sprouts is the rare game whose rules fit on a napkin but whose analysis fills research papers. Start with a few dots. Each turn, draw a curve from one dot to another — or from a dot back to itself — and place a new dot on the curve you just drew. Curves may never cross, and no dot may have more than three curve-ends touching it. When a player has no legal line left to draw, the game is over: under normal rules the player who drew the last line wins, and under misère rules that player loses. This page plays it the way it was meant to be played — you actually draw the curves, with a finger, stylus, or mouse — while the rules enforce themselves: crossings are rejected, each dot’s remaining lives are shown as pips, and the game announces the winner the moment no legal move remains anywhere on the board.
Invented at Cambridge, Analyzed Ever Since
Sprouts was invented in 1967 at the University of Cambridge by John Horton Conway — of Game of Life fame — and Michael S. Paterson, and it spread through mathematics departments long before it reached puzzle books, helped by Martin Gardner’s Scientific American column. The charm is that a doodling game turns out to be pure topology: because lines can’t cross, every move carves the paper into regions, and dots stranded in different regions can never be connected again. Playing well means thinking about walls, not just dots.
Why a Game Can’t Outlive Its Lives: the 3n−1 Theorem
Give every dot three lives — one for each line it can still accept. A game with n dots starts with 3n lives. Every move spends two lives (one at each end of the new curve, or two on the same dot for a loop) and creates one new dot that arrives with two of its three lives already used. Net effect: every single move burns exactly one life from the board. The game must stop before the lives hit zero, so no game with n dots can last longer than 3n−1 moves — and a matching argument about the dots that survive shows it can’t end before 2n moves either. The “lines drawn” counter under the board is this theorem made visible: a 3-dot game promises you between 6 and 8 lines, every time. The table below is computed from the same counting functions the game itself runs on.
Who Wins, and Why It Took a Computer to Find Out
Sprouts is deceptively hard to analyze — the number of possible positions explodes with every curve. In 1991, David Applegate, Guy Jacobson and Daniel Sleator brute-forced the game at Bell Labs and found the pattern: with perfect play, the first player wins exactly when the starting dots leave a remainder of 3, 4, or 5 when divided by 6. Julien Lemoine and Simon Viennot later verified the pattern holds for dozens more starting sizes, and no exception has ever been found. If you enjoy games where the winning strategy is genuinely knowable, our Nim game plays the same role with an unbeatable computer opponent — while Hex sits at the opposite extreme: proven to favour the first player, yet nobody knows how.
How to Actually Win at Sprouts
Beginners watch dots; winners watch regions. Every curve you draw splits the paper, and a dot with lives left is only useful to whoever can still reach it. The classic technique is to wall off a private corner: enclose a live dot so only moves inside that pocket remain, then count whether the moves left in each region add up in your favour — sprouts is ultimately a parity battle over who runs out of moves first. Loops are your walls; spend them deliberately. And mind the endgame: a dot with one life can still loop onto itself, which beginners forget and experts bank as a spare move. For a gentler introduction to move-counting instincts, the four in a row game trains the same look-ahead muscle, and the rest of the browser games collection has something for every recess.