Sim

👥 2 players 📍 Indoor📍 Anywhere ⚡ Calm 🧩 Moderate ⏱ 10-20 minutes 🎂 Ages 6+

Quick Pitch

Sim is a two-player strategy game where you take turns drawing colored lines between six dots — and whoever completes a triangle in their own color loses.

Hook

Draw six dots in a circle and connect every dot to every other dot with colored lines — you use one color, your opponent another. Each turn you draw one line, trying to force your opponent into completing a triangle. Here's the thing: it's mathematically impossible for all 15 lines to be drawn without someone forming a triangle, so somebody is definitely losing. The question is whether it's you.

Equipment Needed

  • Sheet of paper
  • Pencil or pen (two different colors highly recommended)
  • Ruler (helpful but optional for drawing straight lines)

Setup

  1. Draw 6 Vertices: Draw 6 points arranged in a circle (or any pattern with clear spacing)

    • Label them 1, 2, 3, 4, 5, 6

  2. Draw All Edges: Connect every vertex to every other vertex

    • This creates a "complete graph K₆"
    • Total of 15 edges (each of 6 vertices connects to 5 others = 30/2 = 15 edges)

Example starting position:

        1
       / \
      /   \
     2 --- 3
     |\   /|
     | \ / |
     |  X  |
     | / \ |
     |/   \|
     6 --- 4
      \   /
       \ /
        5

(All lines from each vertex to all others would be drawn)

  1. Decide Turn Order: Random or Player 1 goes first

Rules

Objective

Avoid being the player who completes a monochromatic triangle (three vertices connected by three edges of your color). The first player to create such a triangle loses immediately.

Gameplay

  1. Players alternate turns
  2. On each turn, a player draws one undrawn edge between two vertices in their color
  3. After drawing the edge, check if this completes a triangle of that color:
    • A monochromatic triangle is three edges (forming a triangle) all in the same player's color
    • If three vertices form a triangle of all your color, you lose immediately
  4. Play continues with players drawing edges until someone is forced to complete a monochromatic triangle

Losing Condition

A player loses when:

  • They draw an edge that completes a monochromatic triangle of their color
  • By Ramsey theory, someone must lose (it's impossible to color all 15 edges with 2 colors without creating a monochromatic K₃)

Gameplay Example

Turn 1 (Player A): Colors edge 1-2 in Color A
Turn 2 (Player B): Colors edge 3-4 in Color B
Turn 3 (Player A): Colors edge 1-3 in Color A
Turn 4 (Player B): Colors edge 2-5 in Color B
... continues until someone must complete a triangle

Expert Player

Tips

Understanding Ramsey R(3,3):

  • Ramsey number R(3,3) = 6 means: With 6 vertices, when coloring all edges with 2 colors, a monochromatic triangle is inevitable
  • This guarantees the game must end with one player's loss
  • Someone will definitely lose; strategy is forcing opponent to lose first

Forcing Moves:

  • A "forcing move" creates a situation where opponent's response is limited
  • If you create two edges of your color sharing a vertex, opponent often must block before continuing elsewhere
  • Avoid creating your own dangerous patterns while analyzing opponent's position

Symmetry Breaking:

  • Early game is somewhat symmetric; early moves don't usually determine outcome
  • Players try to build toward triangles while avoiding completing them
  • Position becomes critical in last few moves

Prophylactic Blocking:

  • Always monitor if opponent has two edges of their color connected
  • If opponent can easily complete a triangle, block one connection
  • But avoid blocking yourself into a losing position

Parity Consideration:

  • Odd-numbered turns heavily favor one player based on position
  • With optimal play from position X, you can often determine who will lose
  • Counting edges and open connections helps assess game state

Endgame Analysis:

  • Near the end, careful counting determines who must complete a triangle
  • Computer analysis shows many positions have forced wins/losses
  • Advanced players can read deep into remaining possibilities

Variations

  • Different Graph Sizes: Play on K₅ (5 vertices, 10 edges) — first player often has advantage
  • More Colors: Use 3 colors instead of 2 (3-player version or 2-player with 3 colors)
  • Modified Rule: Last player to move WITHOUT creating a triangle wins (opposite rule)
  • Directed Edges: Edges have direction; triangle rule applies to specific directed patterns
  • Weighted Edges: Edges have point values; highest score wins instead of avoiding triangles
Learn More — History & Origins

History & Origins

Sim was invented by Gustavus Simmons in 1969 and first published in the Journal of Recreational Mathematics. It gained a much wider audience through Martin Gardner's legendary "Mathematical Games" column in Scientific American, which introduced millions of general readers to recreational mathematics. The game became a staple example in combinatorial game theory, appearing in texts including Berlekamp, Conway, and Guy's landmark "Winning Ways for your Mathematical Plays."

The reason Sim uses exactly six dots comes directly from Ramsey theory — specifically from the Ramsey number R(3,3) = 6. This theorem, proven in 1928 by Frank P. Ramsey, states that any two-coloring of the edges of a complete graph on six vertices must contain a monochromatic triangle. In other words: six players at a party all either know each other or don't, and you're guaranteed to find three people who all know each other or three who are all strangers. Sim is this theorem turned into a game.

Cultural Context

Sim sits in an unusual position: it's a game that mathematicians invented to illustrate a theorem, but it's genuinely fun to play without knowing anything about the underlying math. The constraint that creates the game — the inevitability of a monochromatic triangle — is invisible to players but felt in every move. Every time you draw a line, you're narrowing the space of what's possible, and the pressure builds as the board fills.

The game has been completely analyzed by computers for many starting positions, and first-player or second-player winning strategies are known. But playing it well without a computer is a different matter — it requires counting paths and visualizing chains of consequences several moves deep. That gap between "provably solvable" and "actually hard for humans" is part of what makes Sim interesting both as a game and as a teaching tool.

See Also