What is coloring in sudoku?

Coloring, also called simple colouring, is a single-digit technique based on conjugate pairs, which are units where a candidate appears in exactly two cells. You alternately mark those two cells with opposite colors and chain the coloring through linked conjugate pairs, so that within one chain all cells of one color are true and all of the other color are false. Analyzing the resulting color pattern reveals contradictions or safe eliminations.

What is a conjugate pair in sudoku coloring?

A conjugate pair is a unit (a row, column, or box) in which a specific candidate digit occurs in only two cells, making the two cells strongly linked. Exactly one of the two must be that digit, so they are given opposite colors during coloring. These strong links are the building blocks that let a coloring chain propagate across the grid.

How does coloring let you eliminate candidates?

Coloring gives two main eliminations. First, if two cells of the same color share a unit, that color cannot be true (two of the same digit in one unit is impossible), so every cell of that color is eliminated and the opposite color is confirmed. Second, if an uncolored cell holding the candidate can see both a cell of one color and a cell of the other, then whichever color is true it will be attacked, so the candidate can be removed from that uncolored cell.

When should you use coloring instead of an X-Wing or chain?

Reach for coloring on a single candidate that forms several conjugate pairs across the grid, especially when no fish or wing is available for that digit. It often finds eliminations that an X-Wing or Swordfish cannot, because it follows a flexible chain of strong links rather than a rigid rectangular shape. If simple coloring on two colors is not enough, more advanced methods like multi-coloring or 3D Medusa extend the same conjugate-pair idea.

Coloring: Sudoku Technique with Examples

Coloring tracks a single digit through a chain of cells where it has only two homes per unit. Two colours mark the two possible truth values, and once the chain is laid down, contradictions and eliminations follow mechanically.

How it works

Pick a digit. Find two cells in the same unit (row, column or box) where the digit has exactly two candidate cells. Those cells are strongly linked. one of them holds the digit, the other does not. Colour one white, the other black.

Now look at each coloured cell and find its strong link in another unit. The new cell takes the opposite colour of its partner. Continue extending the chain. White and black represent the two possible truth values: whichever colour is correct, every cell of that colour holds the digit, and every cell of the other colour does not.

Two payoffs follow. First, any cell outside the chain that sees both a white and a black cell cannot hold the digit. Either colour is correct, so either way the cell is a peer of a placed digit. Second, if two same-colour cells share a unit, that colour cannot be the correct one (it would place the digit twice). Every cell of that colour is eliminated.

4 alternates between two colors. This cell sees both → erase 4.

When to look for it

Late in expert and evil puzzles, when one digit dominates the remaining candidates. The longer the chain, the more eliminations you tend to get.

A good signal: a digit appears in seven or eight units with exactly two candidate cells each. Those bivalue positions chain together easily.

Step-by-step example

Pick a digit. Find every unit where it has exactly two candidate cells. Those are the strong links.
Start at any strong-linked pair. Colour one cell white, the partner black.
From each coloured cell, find other strong links it belongs to. The new cell takes the opposite colour.
Continue until the chain stops growing.
Look for two outcomes. an external cell that sees both colours (erase the digit from it) or two same-colour cells in one unit (erase the digit from every cell of that colour).

Tips for spotting it

Pick the digit with the most strong links remaining. Long chains produce more eliminations.
Coloring is most effective after pointing pairs and X-wings have done their work.
Multi-coloring (three or more colours) handles branching chains. Same idea, more bookkeeping.
Use small marks or two pencil colours so the chain is visible at a glance.

Common mistakes

Linking across cells that share a unit but the digit has three or more candidates there. That is a weak link, not a strong one.
Eliminating a cell that only sees one colour. The cell must see both colours of the chain.
Missing the same-colour contradiction. When it appears, every cell of that colour can be erased.
Trying to colour two digits at once. Each chain tracks one digit only.

Practise it

Run coloring on expert sudoku whenever XY-wing stalls. Pick the most-placed digit, mark its strong links and chain them with two colours. Once the routine is familiar, it becomes the technique that unlocks most evil puzzles before chains and uniqueness ever come into play.

Coloring