Simple Coloring: learn the sudoku technique step by step at Sudokuen

See the technique in practice

Work through the examples step by step. Each step explains what you see in the puzzle and why the conclusion is valid.

Example:

We follow digit 3. In some rows, columns and boxes there are only two cells left that can take 3. Such cells form pairs: if one cell does not get the digit, the other must.
Several of the pairs share cells, so they hang together in a chain. Think of the chain as light switches linked together: flip one switch and all the others flip at the same time.
We alternate between blue and orange as we color the cells in the chain. There are now only two possibilities: either all blue cells contain digit 3, or all orange cells do.
Look at the cross-marked cell in row 3, column 7. It sees both a blue and an orange cell in its own row, column or box.
One color wins regardless, so digit 3 is guaranteed to land in a cell that this one sees. Then the cell itself cannot have 3, and we remove it from the notes.

We follow digit 1. In some rows, columns and boxes there are only two cells left that can take 1. Such cells form pairs: if one cell does not get the digit, the other must.
Several of the pairs share cells, so they hang together in a chain. Think of the chain as light switches linked together: flip one switch and all the others flip at the same time.
We alternate between blue and orange as we color the cells in the chain. There are now only two possibilities: either all blue cells contain digit 1, or all orange cells do.
Look at the cross-marked cell in row 5, column 8. It sees both a blue and an orange cell in its own row, column or box.
One color wins regardless, so digit 1 is guaranteed to land in a cell that this one sees. Then the cell itself cannot have 1, and we remove it from the notes.

We follow digit 7. In some rows, columns and boxes there are only two cells left that can take 7. Such cells form pairs: if one cell does not get the digit, the other must.
Several of the pairs share cells, so they hang together in a chain. Think of the chain as light switches linked together: flip one switch and all the others flip at the same time.
We alternate between blue and orange as we color the cells in the chain. There are now only two possibilities: either all blue cells contain digit 7, or all orange cells do.
Look at the cross-marked cell in row 7, column 5. It sees both a blue and an orange cell in its own row, column or box.
One color wins regardless, so digit 7 is guaranteed to land in a cell that this one sees. Then the cell itself cannot have 7, and we remove it from the notes.

We follow digit 4. In some rows, columns and boxes there are only two cells left that can take 4. Such cells form pairs: if one cell does not get the digit, the other must.
Several of the pairs share cells, so they hang together in a chain. Think of the chain as light switches linked together: flip one switch and all the others flip at the same time.
We alternate between blue and orange as we color the cells in the chain. There are now only two possibilities: either all blue cells contain digit 4, or all orange cells do.
Look at the cross-marked cell in row 5, column 6. It sees both a blue and an orange cell in its own row, column or box.
One color wins regardless, so digit 4 is guaranteed to land in a cell that this one sees. Then the cell itself cannot have 4, and we remove it from the notes.

We follow digit 2. In some rows, columns and boxes there are only two cells left that can take 2. Such cells form pairs: if one cell does not get the digit, the other must.
Several of the pairs share cells, so they hang together in a chain. Think of the chain as light switches linked together: flip one switch and all the others flip at the same time.
We alternate between blue and orange as we color the cells in the chain. There are now only two possibilities: either all blue cells contain digit 2, or all orange cells do.
Look at the cross-marked cells, namely row 1, column 8 and row 3, column 8. Each of them sees both a blue and an orange cell in its own row, column or box.
Whichever color is correct, digit 2 is guaranteed to occupy a cell visible to these cells. They therefore cannot contain 2, so we remove it from their notes.

Simple Coloring is used on one digit at a time. Look for units where the digit has only two possible positions, because the two cells form a strongly linked pair: if one cell is false, the other must be true. When these pairs form a chain, alternate between two colors along it.

All cells with the same color share the same fate: either all are correct, or all are wrong. This gives two conclusions. If two cells of the same color see each other, that color is impossible, and the digit can be removed from all cells of that color. And a cell outside the chain that sees both colors can never take the digit, because one of the colors will be right anyway.

Choose a digit and mark all units where the digit has only two possible placements.
Start in one of the cells and give it a color, give its partner in the same unit the opposite color, and continue along all strong links.
Check if two cells of the same color see each other. If so, remove the digit from all cells of that color.
Otherwise check if any cells outside the chain see both colors, and remove the digit there.

Linking via units with three or more possibilities. The chain requires strong links, that is, units where the digit has only two possible cells.
Mixing colors mid-chain. One color mistake gives false conclusions, so be careful to alternate color with each jump.
Removing from the chain itself without a conflict. Without a color conflict, only cells that see both colors can be removed.

At the Hard level, you must consider several units together: digits that form rectangles across multiple rows, chains of linked cells and boxes that lock each other. These techniques still eliminate only candidates, but those eliminations are exactly what opens up the puzzle.

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Simple Coloring

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