Sim is a classic pencil-and-paper graph game invented by mathematician Gustavus Simmons, recreated here so you can play Sim free online in your browser. Six dots are arranged in a circle, and every pair of dots can be joined by a single line, giving fifteen possible edges in total. You and the AI take turns claiming these edges, each in your own color, until someone is forced into a losing position.
What makes Sim a deep triangle-avoidance game is its inevitability: with six points it is mathematically impossible to color all the edges without creating a monochromatic triangle, so the match can never truly end in a draw. The challenge is to stay out of trouble while steering your opponent toward a triangle of their own color. Every line you draw both builds your network and creates a potential trap, making this graph game a tense exercise in foresight, pattern spotting, and strategy on a tiny but surprisingly rich board.
Sim was invented in 1969 by Gustavus Simmons, an American cryptographer and graph theorist, and it is really a piece of mathematics disguised as a pencil-and-paper game. The six dots and fifteen possible lines form the complete graph K6, and a classic result of Ramsey theory says any two-colouring of K6 must contain a single-coloured triangle. That is why Sim can never end in a draw — someone must eventually complete a triangle.
Computer analysis has proven that the second player wins with perfect play, but the winning strategy is so intricate that no human plays it flawlessly, which keeps the game fair and fascinating in practice. It is a favourite example in university combinatorics courses, and playing a few rounds is still the most enjoyable way to feel Ramsey theory in action.
Avoid completing a triangle made entirely of your own color. The first player forced to connect three dots into a same-color triangle loses, so push the AI into that trap before it traps you.
Click a dot to select it, then click a second dot to draw your line between them. Click the same dot again to cancel a selection. On touchscreens, tap the two dots in turn.
There is no point total — each game is win, lose, or the rare draw. Force the AI to make a triangle to win and bank the result; complete one yourself and you lose. Try Easy, Medium and Hard for tougher AI play.
Avoid building two lines of your color that share a dot, since the third edge becomes a forced losing move. Watch for edges that would complete a triangle for either side, block the AI's safe replies, and on Hard expect the AI to herd you into a corner with few safe moves left.
Why can Sim never end in a draw?
Because of Ramsey theory. The six dots with all fifteen lines form the complete graph K6, and mathematics guarantees any two-colouring of K6 contains a triangle in a single colour — so one player must eventually complete one.
Who invented Sim?
Gustavus Simmons, an American cryptographer and graph theorist, described the game in 1969. It is named after him and is a standard example in combinatorics courses.
Is there a winning strategy in Sim?
Computer analysis proves the second player can always win with perfect play, but the strategy is so complex that no human follows it exactly — in practice, careful defensive play decides most games.
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