Simple Colouring
Two-colour a single digit’s conjugate chain to force eliminations.
When a single digit still fills much of the grid, the simple colouring sudoku technique turns its conjugate links into a two-colour map. You pick one digit, find every unit where it has exactly two possible cells, and shade those paired cells in alternating colours so that one colour must be true and the other false. From that map two eliminations fall out: any cell that sees both colours cannot hold the digit, and any colour that repeats inside a single unit collapses entirely, making the other colour all true.
Worked example
Follow digit 1 through its conjugate pairs. R1C1 and R1C5 are the only 1s in row 1, so shade R1C1 colour A and R1C5 colour B. R1C5 and R5C5 are the only 1s in column 5, forcing R5C5 to A; R5C5 and R5C1 are the only 1s in row 5, forcing R5C1 to B. Now R9C1, which holds {1,9}, sees R1C1 (colour A) and R5C1 (colour B) down column 1. Whichever colour turns out to be true, a 1 already lands in that column, so R9C1 can never be the 1 and its candidate is removed.
When to use it
Reach for simple colouring once the easy singles and pairs are gone and one digit still has many candidate cells. Scan that digit unit by unit for two-spot units — conjugate pairs where it appears in exactly two cells — and chain them together with alternating colours. Then test any cell that sees both colours, or hunt for two same-colour cells sharing a row, column or box.
How to use it
- Build the conjugate chain. Pick a digit and find units where it has exactly two candidate cells (a strong link). Chain those links together.
- Colour the chain in two shades. Alternate two colours along every strong link. One colour is true and the other false — they cannot both be right.
- Apply the two colouring rules. If two cells of the same colour share a unit, that colour is false — clear it everywhere. Otherwise remove the digit from any cell that sees both colours.
Frequently asked questions
What is simple colouring in Sudoku?
Simple colouring is a single-digit technique. You choose one digit, find every unit where it has just two candidate cells (a conjugate pair), and colour those linked cells in two alternating shades so that one colour is true and the other false. On this page, digit 1 chains from R1C1 to R1C5 to R5C5 to R5C1, and that coloured map lets you strip the 1 from R9C1 because it sees both colours down column 1.
How do I spot a simple colouring?
Pick a digit that still has lots of candidates, then look for units where it can go in only two cells — those two-spot units are your conjugate pairs. Link overlapping pairs with alternating colours until the chain stops growing. In the example the 1s in row 1, column 5 and row 5 each form a conjugate pair, so they thread into one A/B chain across R1C1, R1C5, R5C5 and R5C1. Once coloured, check any cell that sees both colours, such as R9C1.
Is simple colouring the same as an X-chain?
They are close cousins: both work a single digit through conjugate links, so an elimination one finds can often be found by the other. An X-chain traces one alternating path between two ends; simple colouring instead paints the entire connected graph two colours at once, which is why a same-colour clash inside a unit can kill a whole colour. In the example, colouring all four 1-cells lets R9C1 lose its 1 precisely because it sees both colour A (R1C1) and colour B (R5C1).