Simple coloring

By Sudoku Online Editorial Team · Published October 23, 2026

Exactly one colour of the 1 chain is true. r1c7, r5c9 sees both colours, so 1 is removed there.

Simple coloring follows one digit through a chain of strong links, places where the digit has exactly two possible cells in a group, and colors the cells alternately. Exactly one color is true. Two cells of one color that see each other make that color false, and any cell that sees both colors cannot hold the digit.

On this site simple coloring first appears in Expert puzzles.

Strong links and chains

Choose a digit. In some row, column or box, that digit may have exactly two possible cells. That is a strong link: one of the two cells is the digit and the other is not, and there is no third option. If you know one end is false, the other end is true, and the reverse. Now chain these links together. If cell B is the other end of one link from A and also an end of a link to C, then A and C always agree with each other and disagree with B.

Coloring writes this down. Mark A with one color, B with the other, C with the first color again, and so on across the whole chain. At the end, every cell has a color, and exactly one of the two colors is the set of true cells for that digit. The other color is the set of false cells.

The diagram above is a real position from this site's Expert pool, drawn by the technique engine. The caption reads: exactly one color of the chain for the digit 1 is true, and the cells named in the caption can see both colors, so 1 is removed from them. Notation is row first, then column, counted from the top left.

The two rules

Color trap (seeing both colors). If an uncolored cell sees at least one cell of each color, then whichever color is true, the cell is looking at a cell holding the digit. It cannot hold the digit itself, so remove the digit from it.
Color wrap (a color sees itself). If two cells of the same color see each other, they cannot both be the digit, so that color cannot be the true one. The other color is true: remove the digit from every cell of the false color, and the cells of the true color can be filled in.

This site's engine implements both. It requires the chain to hold at least four cells, since a shorter chain offers too little to work with, and it looks for the wrap first because it is the stronger result.

How to color by hand

Pick a digit that appears as a candidate in many cells and has several two-cell groups.
Find all groups where the digit has exactly two candidate cells and draw the links.
Start at any linked cell and color it, for example blue. Color its partner red, then the partner's other link blue, and so on until the chain stops.
Check each color for a wrap: do two blue cells (or two red cells) share a group?
Check every uncolored cell that still has the digit as a candidate: does it see both a blue and a red cell?

If you do this on paper, use two highlighter colors, or letters A and B. In the game, the notes mode lets you highlight the cells with the digit selected, which helps you see where the strong links are.

Why it works

Each strong link is an either-or. Chained together, the either-ors force the cells of one color to be all true or all false together. For the trap rule, whichever color is true, the cell in question sees a true cell, and a cell cannot hold the same digit as one it sees. For the wrap rule, two same-colored cells that see each other cannot both be true, but they must be both true if their color is the true one, so their color is false. No guessing is involved: each conclusion holds in every case.

Common mistakes

Linking through a group with three candidates. A strong link needs exactly two candidate cells in the group. With three, the either-or is lost.
Mixing digits. A chain is for one digit only. Do not link cells through a different digit.
Coloring after a placement without refreshing. A placed digit or an elimination can break a link or create a new one. Rebuild the chain after changes.
Eliminating from a colored cell by the trap rule. The trap rule removes the digit from uncolored cells only.
Ignoring the wrap. The wrap is rarer but stronger. Check it first.

Where the site's grading needs it

Simple coloring is a fourth-tier technique, with Swordfish and XY-Wing. In the committed frequency data over 1,000 stored classic puzzles per tier, simple coloring was used in 0 puzzles at Easy, Medium and Hard, and in 570 Expert and 337 Evil puzzles. Each puzzle is solved cheapest technique first and a technique counts if used at least once. See how difficulty is graded and the frequency guide.

Why this is more general than a fish

The X-Wing is a coloring chain in disguise: two rows each give a strong link between two columns, and the four cells form a loop of alternating colors. Coloring handles chains that do not close into a loop and that wander across rows, columns and boxes in any order. It is the most flexible tool in the fourth tier, at the cost of needing more bookkeeping.

In the variants

Coloring needs only groups of cells that must contain each digit once, so it works wherever a digit has groups to link. On the diagonal board a diagonal is another group in which strong links can occur, and on the hyper board so is each window. More groups mean more strong links and longer chains.

Check your understanding

A chain has cells A, B, C, D, colored blue, red, blue, red. A and C, both blue, are in the same row. What does that tell you?
An uncolored cell sees B (red) and C (blue). What can you do?
Why must a strong link have exactly two candidate cells?

The answers: two blue cells that see each other cannot both be the digit, so blue is the false color; remove the digit from every blue cell and treat the red cells as true; remove the digit from that cell, because it sees a true cell whichever color wins; and because with three candidates the group could place the digit in a third cell, so one end being false would not force the other true.

Spotting chains faster

Start with the digit that is closest to finished. A digit that is already placed seven or eight times has very few candidate cells left, so most of its groups have exactly two, and the chain practically draws itself. A digit with few placements has long candidate lists and few strong links, which makes coloring unproductive. The same point applies to fish: pick the digit with the shortest lists first.

Practice

Play an Expert puzzle on the main board. When simpler techniques stall, pick a digit with plenty of candidate cells and look for rows, columns and boxes where it has just two. The level-two hint names the technique when coloring is the cheapest step.

Previous: XY-Wing. Next: unique rectangles. Back to the technique list.