Hackenbush

👥 2 players 📍 Indoor📍 Anywhere ⚡ Calm 🧩 Moderate ⏱ 10-30 minutes 🎂 Ages 6+

Quick Pitch

Hackenbush is a mathematical strategy game where players take turns removing edges (called "hacks") from a graph.

Hook

Draw a picture with lines — a stick figure, a tree, a house — anchored to a "ground" line at the bottom. Each player can erase one line segment per turn, but any parts of the picture no longer connected to the ground fall away automatically. The player who erases the last segment wins. Hackenbush looks like a drawing game but is actually one of the deepest combinatorial strategy games ever analyzed mathematically — invented by a game theorist to illustrate ideas about surreal numbers.

Equipment Needed

  • Sheet of paper
  • Pencil or pen (ideally two colors, e.g., red and blue)
  • Ruler (helpful for drawing straight edges)
  • Eraser (optional)

Setup

  1. Draw Graph: Create a graph with edges and vertices

    • Vertices: labeled dots or circles
    • Edges: lines (each marked with player's color)
    • Ground vertices: designated as connected to "ground" (draw line from ground symbol to graph)

  2. Color Edges: Each player's edges are one color

    • Red edges: Player 1's edges
    • Blue edges: Player 2's edges
    • Uncolored/neutral edges: Either player can remove them

  3. Designate Ground: Choose vertex (or vertices) as "grounded"

    • Edges must remain connected to ground vertex through chain of edges
    • Edges disconnected from ground fall and are removed

Example Starting Position:

    Ground
      |
      A --- B(R)
     /|\
    / | \
  C(B) D(R) E(B)
   |   |
   F   G

(R) = Red edges, (B) = Blue edges

  1. Decide Turn Order: Player 1 (Red) goes first, or decide by agreement

Rules

Objective

Remove the last edge, leaving opponent unable to move.

Gameplay

  1. Players alternate turns

  2. On each turn, a player removes one edge of their color

    • Can only remove edges of your assigned color
    • Cannot remove opponent's edges directly

  3. After edge removal:

    • Check if any remaining edges are disconnected from ground
    • Any edges not connected to ground "fall" and are removed
    • All edges connected to fallen edges also fall

  4. Game continues until one player cannot move (no remaining edges of their color connected to ground)

Cascade Example

Starting configuration:

Ground
  |
  A(R)---B(B)
   |      |
  C(R)  D(B)

Edges: A-Ground(R), A-B(B), B-Ground(implied), B-D(B), A-C(R)

Player 1 (Red) removes edge A-C:

Ground
  |
  A(R)---B(B)
         |
       D(B)

C falls (disconnected). Now if Player 2 removes B-D:

Ground
  |
  A(R)---B(B)

D falls, chain is broken, B becomes "floating" (disconnected from ground).

Continue until no legal moves remain.

Expert Player

Tips

Key Strategic Concepts:

  1. Ground Connection: Understand which edges remain connected to ground

    • Removing edge can cascade and remove many others
    • Plan cascades carefully

  2. Edge Value:

    • Edges supporting many others are valuable
    • Removing support edge causes cascade (can be good or bad)

  3. Critical Edges: Some edges are critical to graph structure

    • Removing them causes major cascades
    • Sometimes beneficial, sometimes detrimental

  4. Blocking Edges: Your color edges can block opponent from accessing some edges

    • Strategic placement creates locks

Playing Strategy:

  1. Analyze Dependencies:

    • Which edges would fall if certain edges removed?
    • Plan cascades to remove opponent's edges

  2. Control Cascades:

    • Use cascades to your advantage
    • Remove edges strategically to trigger helpful falls

  3. Preserve Complexity:

    • Keep game complex to maintain options
    • Simplify when you have advantage

  4. Endgame:

    • Count remaining edges
    • Plan final moves to ensure you make last move

Blue-Red Hackenbush Specifics:

When both players have different colored edges:

  • You can only remove your color
  • Opponent's edges may fall due to cascade
  • Strategic placement matters more

Example strategy:

  • Remove edges to force cascading removals of opponent's edges
  • Use your edges as strategic "anchors"
  • Force opponent into positions with no good moves

Variations

  • One-Color (Green) Hackenbush: All edges same color, all edges can be removed by either player (simpler, more symmetric)
  • Blue-Red Hackenbush: Two colors, each player removes only their color (more complex, partisan)
  • Hackenbush on Grass: Vertices on "ground level" have alternate removal rules
  • Multiple Components: Separate connected components in the same game
  • Directed Hackenbush: Edges have direction, affecting cascade rules
  • Weighted Edges: Different edges worth different point values
Learn More — History & Origins

History & Origins

Hackenbush was invented by Elwyn Berlekamp around 1974 and became famous through the landmark game theory book Winning Ways for Your Mathematical Plays, co-written by Berlekamp, John Conway, and Richard Guy (first published 1982). Conway had been developing a revolutionary mathematical framework called surreal numbers — a vast generalization of ordinary numbers — and Hackenbush turned out to be one of the cleanest game-theoretic examples of how surreal numbers arise naturally from game positions.

The game became a standard teaching example in combinatorial game theory because Blue-Red Hackenbush positions (where each player can only remove edges of their own color) correspond directly to surreal number values, making the game a concrete illustration of abstract mathematics. Conway's broader work on combinatorial games, inspired in part by Hackenbush, eventually led to his book On Numbers and Games (1976), which is now a foundational text in the field.

Cultural Context

Hackenbush occupies an unusual position: it is simultaneously one of the most mathematically rich games ever analyzed and one of the most visually intuitive to explain. Showing someone a stick figure drawn above a ground line and explaining "each player erases one line, anything that falls off the ground is gone" takes about twenty seconds — the rules are genuinely that simple. But the depth beneath those rules is extraordinary, connecting to a mathematical structure that extends the real numbers into something far larger.

For this reason, Hackenbush is a favorite game in mathematics education, particularly in courses on combinatorial game theory, recreational mathematics, and the history of mathematical ideas. The game demonstrates that profound mathematical structure can emerge from the simplest imaginable rule systems.

See Also

Mathematical Analysis

Grundy Numbers:

  • Each position has a Grundy number (nimber)
  • XOR of all component Grundy numbers determines winner
  • Complex calculation for large graphs

Key Theorem (Colon Principle):

  • For two-component game, overall Grundy number is XOR of component Grundy numbers
  • Allows analysis of complex graphs by analyzing components

Surreal Numbers:

  • Hackenbush positions can be analyzed using surreal numbers
  • Mathematical framework beyond simple win/lose

Small Graph Analysis

Simple Path (A-B-C grounded):

Ground-A-B-C

Three edges. First player removes one, second removes one, first removes last. First player wins.

Simple Tree:

      A
     /|\
    B C D

Structure and edge colors determine winner.

Cycle:

A-B-C-A

Different properties than trees due to multiple paths to ground.