How to Play Sim

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

Sim is a two-player pen-and-paper game played on a diagram of six dots arranged in a hexagon, with every dot connected to every other dot by a line, giving fifteen lines in total. Players alternate colouring one line per turn, each using a different colour, and the sole aim is to avoid forming a triangle whose three corners and three sides are all in your colour.

Its appeal is the gap between how simple it looks and how quickly it becomes genuinely tricky. A game takes only a few minutes, it needs nothing more than paper and two coloured pens, and it travels easily into classrooms, waiting rooms, or any kitchen table. Because it is mathematically proven that a draw is impossible, every game ends with a definite loser.

A little history

Sim was devised by Gustavus Simmons, an American mathematician and cryptographer, and first described in print in 1969 in the Journal of Recreational Mathematics. Its origins are therefore clearly documented as a recreational mathematics invention rather than a folk game. The name is simply a shortening of its inventor's surname. It became a classroom staple because it illustrates Ramsey theory, a branch of combinatorics, in a form accessible to children: the proof that the game can never end in a draw is a direct consequence of Ramsey's theorem R(3,3)=6.

Setting up

1. Draw six dots arranged evenly in a circle, like the vertices of a regular hexagon.
2. Connect every dot to every other dot with a straight line. There are fifteen lines in total.
3. Decide which player uses which colour.
4. Decide who goes first, by any convenient method such as a coin toss.

How to play

1. Players alternate turns throughout the game.
2. On your turn, choose any one uncoloured line and colour it entirely in your colour.
3. You may colour any available line; you are not restricted by your previous moves.
4. Continue alternating until one player completes a triangle.
5. A triangle counts against you only when all three of its corner dots are among the original six, and all three of its sides are coloured in your colour. Internal triangles formed as sub-triangles of larger shapes count equally.
6. The moment a player colours a line and thereby completes such a triangle in their own colour, that player loses immediately. The turn does not continue.
7. If you are uncertain after a move, both players should scan all triangles formed by the newly coloured line before the next turn begins.

Winning and scoring

The player who forces their opponent to complete a triangle in the opponent's own colour wins. Equivalently, the player who completes a triangle in their own colour loses. Because of Ramsey's theorem, it is mathematically proven that by the time all fifteen lines are coloured, at least one triangle of a single colour must exist. A draw is therefore impossible: every game has a loser and a winner. There is no points scoring; the result is simply win or lose.

The fine points

The most common dispute concerns sub-triangles. If three dots among the six happen to form a triangle visually because connecting lines cross, only triangles whose three vertices are genuine members of the original six dots count. Lines crossing in the interior do not create new valid vertices. Check each claimed triangle by confirming its three corners are original hexagon points.

A second issue arises when a single move simultaneously completes two triangles of the same colour. This is still a loss for the player who made the move: completing even one triangle in your colour on your turn ends the game against you. It does not matter how many triangles appear simultaneously. If by some geometry a move would complete a triangle in both colours at once, the player who made the move loses, since it is their line that triggered the completion.

How to win more often

• In your early turns, avoid colouring two sides of any triangle; the opponent can close it and hand you the loss, or force you into a position where you must close it yourself.
• Count how many two-side chains you have already created. Each such chain is a liability, because eventually the third side must be coloured by someone.
• Try to build positions where every available line for your opponent completes one of their own partial triangles. This is the core of strong Sim play.
• The first player has a slight disadvantage by some analyses, because responding rather than initiating gives more information. Do not treat the opening moves as free; every line chosen constrains the whole board.
• When you are in trouble, look for moves that simultaneously threaten two of your opponent's partial triangles. Forcing them to address one liability often means the other gets closed against them.

Variations

Five-point Sim

The diagram is reduced to five dots with ten connecting lines. A draw becomes possible with five points, so this version is better suited to younger players or those who want a shorter, lower-pressure game. The losing condition remains the same: complete a triangle in your own colour and you lose.

Scored Sim

Play a fixed number of rounds, typically five or ten, recording one point for each win. The player with more points at the end of the agreed number of rounds wins the match. This format suits classroom settings and gives both players a reason to continue after an early loss.

Sim with four colours

Three or four players each take a colour and a shared diagram is drawn on a whiteboard or large sheet. Turn order rotates clockwise. The first player to complete a triangle in their own colour is eliminated; remaining players continue on the same diagram until only one remains. This variant works best with four players and is livelier but less mathematically clean than the two-player original.

Reverse Sim

The winning condition is inverted: you try to be the first to complete a triangle in your own colour rather than the last. This turns the strategy almost entirely on its head and is a useful way to refresh the game for players who have become comfortable with standard Sim.

Sim: frequently asked questions