What Is Sim? The Six-Dot Game Where Making a Triangle Loses

Sim is a two-player pencil-and-paper game invented in 1969 by the mathematician Gustavus Simmons. Players take turns drawing coloured lines between six dots, and the first person to complete a triangle entirely in their own colour loses. The rules take seconds to learn, yet the game hides real strategic depth — and a genuine mathematical theorem.

What Is Sim?

Sim is played on six dots — the complete graph mathematicians call K6, which has exactly 15 edges. Two players alternate claiming edges, each in their own colour, and neither may recolour an edge once it is drawn. Because every edge belongs to several potential triangles, a line that looks harmless early can become a trap a few moves later. Despite being more than fifty years old, Sim remains largely unknown outside mathematics and puzzle circles, which is exactly why it deserves a modern home as Triad.

How to Play

1. Start with six dots and no lines.
2. On your turn, draw one line between any two dots that aren’t already connected, using your colour.
3. Players alternate turns, and an edge’s colour can never change.
4. Avoid completing a triangle in your own colour — three dots all joined by your lines.
5. The first player to complete a same-colour triangle loses.

A full game lasts only a few minutes, and at most 15 lines can ever be drawn before the board is full.

Why Sim Is Different

Most games reward you for building something: a row in tic-tac-toe, territory in Go, an attack in chess. Sim inverts that instinct — completing the pattern is the one thing you must avoid. That makes it a misère game, a game where the usual winning move becomes the losing one. If you’ve ever wished tic-tac-toe were deeper, Sim is the six-dot answer: the same one-minute simplicity, with none of the forced draws.

The Math: Why a Draw Is Impossible

Sim can never end in a draw, and that isn’t a quirk of the rules — it’s a theorem. Colour the 15 edges joining six points with two colours, and a single-colour triangle is unavoidable. In Ramsey-theory notation this is written R(3,3) = 6, and six is the smallest number of points for which the triangle is forced. The full argument, including the famous “party problem,” is in Ramsey theory in a game.

Is Sim Solved?

Yes — Sim is a solved game. With perfect play the second player can always force the first player to complete a triangle. That sounds deflating, but humans almost never play perfectly, so the game stays sharp at the board. The details of whether Triad is solved and what perfect play looks like are their own story — including why a strategy humans can actually follow wasn’t found until 2020.

Strategy Basics

Strong play in Sim is about restraint: keep your options open, avoid building two sides of the same triangle, and count how many safe moves you and your opponent have left. The player with more safe moves usually wins. For a full walkthrough, see how to win at Triad.

A Worked Example: Counting Safe Moves

The single most useful habit in Sim is a quick census before every move. An edge is unsafe for you if drawing it would complete a triangle in your own colour — three of your dots already joined into two sides, with that edge as the third. Count those forbidden edges, subtract them from the edges still uncoloured, and what remains is your supply of safe moves. Do the same count for your opponent. Because every turn forces a player to draw something, whoever runs out of safe edges first is compelled to draw a losing line.

Concretely, watch what happens each time you draw a second edge out of the same dot. If you already own the edge from dot A to dot B and you add the edge from A to C, you have built two sides of a potential triangle — and the edge from B to C is now permanently off-limits to you. Every such “cherry” (a pair of your edges sharing a dot) removes one edge from your safe supply. Build cherries faster than your opponent and you will be the one forced into the losing move; build them more slowly and the squeeze falls on them instead.

Common Beginner Mistakes

New players lose Sim in a handful of predictable ways. The first is piling edges onto one dot — a third or fourth line from the same point multiplies your forbidden edges and shrinks your safe supply far faster than spreading your lines around the board. The second is playing too quickly: the trap in Sim is laid several moves before it springs, so the line that loses often looks completely harmless when you draw it. The third is seeing an edge as part of only one triangle when, on K6, every edge belongs to four different potential triangles at once. And the fourth is mirroring your opponent — copying their moves feels safe but does nothing for you, because in this misère, single-board game there is no symmetry that protects the imitator. Slow down, spread out, and keep counting safe moves; that alone beats most opponents.

The History of Sim

Simmons published Sim in the Journal of Recreational Mathematics in 1969, and it became a favourite teaching example among mathematicians without ever becoming a household name. The history of Sim traces how a cryptographer’s puzzle became a quiet classic — and how it lost its name to SIM cards and simulation games along the way. If you arrived looking for the SIM card or The Sims, here’s how Sim the game differs from them.

Keep Exploring

Sim sits in a family of deceptively simple games. See where it fits among games that are easy to learn but hard to master, what it reveals about elegance in game design, and why simple games are often harder than complex ones.