Inside-Outside Coloring: The Advanced Technique

Seeing Beyond the Lines

When basic logic runs dry, pros turn to the most powerful tool in the Slitherlink arsenal: Inside/Outside Coloring (also known as the Jordan Curve Theorem application).

The loop divides the grid into two distinct regions: Inside the loop and Outside the loop.

The Coloring Rules

Pick a cell (usually outside the grid is "Outside"). Now look at its neighbors:

• Cross a Line: If you move from Cell A to Cell B and cross a LINE, the region flips. If A is Outside, B must be Inside. (And vice versa).
• Cross an X (Empty): If you move from Cell A to Cell B and cross an X (no line), the region stays the same. If A is Outside, B must be Outside.

  Cross a LINE → color flips     Cross an X → color stays

  ┌───────┬───────┐              ┌───────┬───────┐
  │  OUT  ║  IN   │              │  OUT  ×  OUT  │
  │       ║       │              │       ×       │
  └───────┴───────┘              └───────┴───────┘
       ═══ (line)                     × (no line)

How to Use This for Deduction

You can propagate these "colors" (Inside/Outside) across the grid even if you don't know where all the lines are.

Example: Suppose you have a "3" next to an Outside cell (border).
Case 1: The shared edge is a Line. Then the cell with "3" is Inside.
Case 2: The shared edge is an X. Then the cell with "3" is Outside.
But wait! A "3" cell must be Inside the loop (usually, depending on topology, but often it forces a region flip). If you find a contradiction where a cell "must be Inside" but your coloring path says "it's Outside," then your initial assumption about the edge was wrong.

  Propagating colors through a 3×3 grid:

  (border = OUT)
  ·═══·───·───·
  ║ IN│OUT│ IN║     Cross ═ → flip
  ·───·═══·───·     Cross ─ → same
  │OUT║ IN║OUT│
  ·───·═══·───·     Each cell is labeled
  │OUT│OUT│OUT│     by following the
  ·───·───·───·     coloring rules.

The Parity Argument

This is a more abstract version of coloring. Pick any row or column. Shoot a ray from the outside edge all the way through.
You must cross the loop an even number of times to end up back Outside.
If you have a row where you've already found 3 vertical lines and there is only 1 uncertain vertical edge left, that edge must be a line (to make the total 4, which is even). If it were an X, you'd have 3 crossings, meaning you end up Inside the loop at the edge of the board—impossible!

  Parity ray across a row:

  OUT │ ║  │  ║  │  ?  │ border
      │ ║  │  ║  │     │
  ────┘ ║  └──║──┘     └────
        ║     ║     ?
  Cross 1  Cross 2   Cross 3?

  Already 2 crossings (even → OUT). If the
  ray must end OUT at the border, the ? edge
  must be × (keeping count even = 2).

When to Use It

Use coloring when:

• You have a sparse grid with few numbers.
• You have disjointed loops and need to know if they can connect.
• You need to check global validity.

This technique separates the masters from the apprentices. It turns a local puzzle into a global topology problem.