Coloring

What is Coloring?

Coloring (also called Simple Coloring or 2-String Kite in some contexts) is a powerful chain-based technique that uses alternating assignments of a single candidate digit to derive logical conclusions across the entire board. The foundation of coloring is the concept of a "conjugate pair" — a unit (row, column, or box) where a digit appears in exactly two candidate cells. These two cells form a binary choice: one must have the digit (True) and the other must not (False). By chaining conjugate pairs together, you create a network of cells that alternate between True and False states. Coloring assigns one "color" (say, blue) to one state and another color (green) to the opposite state. Once the chain is built, two types of eliminations become possible: (1) Color Trap — if two cells of the same color see each other (same row, column, or box), that entire color is invalid and all cells of the opposite color receive the digit; (2) Color Wrap — if a cell outside the chain can see one cell of each color, that cell cannot contain the digit regardless of which color state is true, so the digit is eliminated from it. Coloring is the gateway to more advanced chain techniques like Chains and AICs (Alternating Inference Chains).

When to Use Coloring

Use Coloring after X-Wing and Swordfish when the board has strong chains for individual candidates — this means a digit that has many conjugate pairs (units with exactly two candidate cells) across the grid. Coloring is most productive for digits that appear frequently as candidates and form long, interconnected chains. Before applying coloring, ensure the candidate grid is fully updated. Begin by choosing a digit and tracing all its conjugate pairs — then build the largest possible chain before analyzing it for contradictions or eliminations.

How to Apply Coloring — Step by Step

1. 1

   Pick a candidate and identify conjugate pairs

   Choose a digit and find all units (rows, columns, boxes) where it appears in exactly two candidate cells. Each such unit is a conjugate pair. These are the links of your coloring chain. Make a note of all conjugate pairs for the selected digit across the entire board.

2. 2

   Start coloring a chain

   Choose a conjugate pair to start. Assign one cell the color blue (True) and the other green (False). This means: if blue is correct, the blue cell gets the digit; if green is correct, the green cell gets it. Proceed to build the chain by extending from each colored cell to new conjugate pairs.

3. 3

   Extend the chain through conjugate pairs

   From each colored cell, look for other units where the digit appears in exactly two cells and one of those cells is already colored. The uncolored partner cell gets the opposite color. Continue until no more extensions are possible. A single coloring chain may span many cells across the entire board.

4. 4

   Analyze for contradictions (Color Trap)

   Examine the chain for any two cells of the same color that can "see" each other — i.e., they share a row, column, or box. If two blue cells see each other, then the blue assignment is impossible (two cells in the same unit cannot both have the digit). Therefore, all green cells receive the digit, and all blue cells are eliminated.

5. 5

   Eliminate uncolored cells (Color Wrap)

   Look for any cell that is not part of the chain but can see both a blue cell and a green cell. Regardless of which color assignment is correct, this cell will always be "seen" by a cell containing the digit. Therefore, the digit can be eliminated from this cell as a candidate — a Color Wrap elimination.

Related Techniques