Snip

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

Quick Pitch

Snip is a combinatorial graph game where players take turns removing edges (connections) from a graph.

Hook

Draw a web of dots connected by lines — that's your graph. On each turn, you snip one line away. The catch? Cutting certain lines causes a whole chunk of the web to fall off, wiping out extra moves at once. Think carefully before you snip, because the player who makes the last legal cut wins.

Equipment Needed

Sheet of paper
Pencil or pen
Ruler (helpful for drawing clean graphs)
Eraser (optional)

Setup

Draw Graph: Create connected graph structure
- Vertices: labeled circles or dots
- Edges: lines connecting vertices
- Designate "root" vertex (often top or center)
Simple Starting Graphs:
- Path Graph: Linear chain of vertices (1-2-3-4-5)
- Star Graph: Central vertex connected to satellites
- Tree Graph: Branching structure
- Cycle Graph: Vertices in circular arrangement

Example Starting Graph (Tree):

Designate Root: Choose one vertex as the "root" (typically vertex 1)
- Removing edge that disconnects vertex from root removes that branch
Decide Turn Order: Player 1 goes first

Rules

Objective

Remove the last legal edge, leaving opponent unable to move.

Gameplay

Players alternate turns
On each turn, a player removes one edge from the graph
When an edge is removed:
- If it disconnects any vertices from the root vertex, those disconnected vertices are removed along with all their incident edges
- Subgraph becomes smaller
Player unable to remove any edge (no edges remain, or no legal moves) loses

Example Game

Starting tree graph:

      1 (root)
     / \
    2   3
   /
  4

Edges: 1-2, 1-3, 2-4

Player 1: Removes edge 1-2
  Result: Vertices 2 and 4 disconnected from root and removed
  Remaining:
      1
      |
      3

Player 2: Removes edge 1-3
  Result: Vertex 3 disconnected and removed
  Remaining: Only vertex 1

No edges remain. Player 1 cannot move. Player 2 wins.

Key Mechanic: Cascading Removal

When you remove an edge:

Check if any vertices become disconnected from root
Remove those vertices AND all edges touching them
Result is a smaller graph

This creates cascade effects where removing one edge can eliminate multiple vertices.

Expert Player

Tips

Graph Structure Understanding:

Root Control: Root vertex is permanent; removing edges threatens non-root vertices
Bridge Edges: Edges whose removal disconnects vertices from root
- Removing bridge removes entire branch
- Choose carefully when removing bridges
Leaf Vertices: Vertices connected by single edge (leaves)
- Removing edge to leaf removes only that leaf
- Safe moves that don't cause cascades
Branching Depth: How far vertices are from root
- Removing edge to branch removes entire sub-tree
- Plan to control which branches survive

Strategic Principles:

Preserve Complexity: Keep graph complex to maximize opponent's options
- But don't give opponent good moves
- Balance between simplification and constraint
Cascade Efficiency: Use removal edges that disconnect large subtrees
- Removes many vertices at once
- Reduces available edges for opponent
Control Symmetry: Break symmetry to limit opponent
- Asymmetric graphs favor the player creating them
- Remove edges to create unbalanced positions
Endgame Awareness:
- Count remaining edges
- If odd number remain, first player should win
- If even number, second player should win
- (Assuming both play optimally)

Example Strategy on Star Graph:

Starting star (root in center with 5 satellites):

    S1
     \
  S2--R--S3
     / \
    S4 S5

Player 1 should remove edge to one satellite (say S1), disconnecting it:

    S2
     \
      R--S3
     / \
    S4 S5

Now 4 edges remain, and Player 2 is forced to remove another. Continue removing satellites one by one. With careful counting, you can force opponent into final move.

Variations

Directed Graphs: Edges have direction; disconnection based on directed paths
Weighted Graphs: Different edges worth different points or have different removal effects
Multi-Root: Multiple designated root vertices
Cycle Graphs: Graphs with cycles (not just trees) create more complexity
Different Removal Rules: Remove edge and all vertices adjacent to that edge

Learn More — History & Origins

History & Origins

Snip is a lesser-known combinatorial game related to the broader family of graph games (Col, Hackenbush, Nim variants). It appears in academic papers on impartial games and graph games but lacks widespread recreational adoption. The game demonstrates edge-removal mechanics and graph connectivity principles. Like many mathematical games, Snip is studied by combinatorial game theorists but remains specialized to academic and serious game-player communities.

Cultural Context

Snip appears in academic game theory literature as a variant of graph games. The game demonstrates practical applications of graph theory to strategic games.

The game appeals to mathematicians, computer scientists, and graph theorists. It shows how abstract graph structures translate into interesting games and strategic challenges. Understanding optimal strategy requires thinking about graph connectivity and cascading effects.

Mathematical Notes

Graph Theory Concepts:

Trees: Acyclic connected graphs (simple Snip games)
Bridges: Edges whose removal disconnects the graph
Components: Disconnected subgraphs
Degree: Number of edges at a vertex

Game Theory:

Snip is an impartial game
Positions can be analyzed using Sprague-Grundy theory
Some positions are determined winning/losing with perfect play
Parity of edges often determines outcome

Small Graph Analysis

Path Graph (2 edges):

A - B - C

First player removes one edge, second removes the other
First player wins (makes last move)

Star Graph (4 satellites, 4 edges):

  S1
   \
S2-R-S3
   /
  S4
(S5 is 5th edge option)

4 edges remaining
Second player wins if both play perfectly
(Even number of edges generally favors second player)

Binary Tree:

Depth and breadth affect gameplay
Careful analysis needed for each structure