Simple Colouring
Simple Colouring takes one digit, chains together every place where that digit is squeezed into two cells, and paints the chain in two alternating colours. One colour is true and the other is false. You never find out which is which, and you do not need to. US English spells it "Simple Coloring" or "sudoku coloring", and it is the same method.
What the pattern is
A conjugate pair is a unit (one row, column or box) in which a digit has exactly two remaining candidate cells. One of the two must hold the digit, so the pair works like a switch: turn one off and the other turns on.
Conjugate pairs often share a cell. When they do they form a chain. Colour the first cell A, colour its partner B, then keep alternating at every link. Every A cell now carries the same truth value, and every B cell carries the opposite one.
When to look for it
- Pencil marks are complete and singles, pairs and X-Wings have run dry.
- A digit is already placed in five or six boxes, so its leftover candidates are sparse.
- Two conjugate pairs on the same digit share a cell. That is a chain of three cells, and three is enough.
How to apply it, step by step
- Pick one digit and mark every unit where it has exactly two candidates.
- Start at any cell of one such pair and give it colour A.
- Give its partner colour B, then follow every other pair running through those cells, alternating as you go.
- Stop when the chain stops growing. Candidates of that digit that never joined stay uncoloured.
- Test the finished colouring against the two rules below.
| Rule | What you see | What it gives you |
|---|---|---|
| Colour clash | Two cells of the same colour inside one unit | That colour is false, so the other colour is the answer in every cell it covers |
| Colour trap | An uncoloured candidate that sees a cell of each colour | That candidate is impossible and comes off |
The clash rule is the stronger of the two because it places digits outright. If two cells of one colour share a row, column or box, that colour cannot be true, because a unit holds only one copy of the digit. Every cell of that colour loses the digit and every cell of the other colour takes it. The board below shows the trap form instead, because it fires far more often.
A worked example
Only the digit 1 matters here. Box 6, row F and column 2 each leave exactly two places for it: D7 and F8 in box 6, F8 and F2 in row F, F2 and B2 in column 2. Those three conjugate pairs share cells, so they form one chain of four cells.
Colour D7 with A. Its box 6 partner F8 gets B, F2 across row F gets A again, and B2 down column 2 gets B. Colour A is D7 and F2, colour B is F8 and B2. Exactly one colour is true, so one colour holds all the 1s of the chain and the other holds none.
The chain does not touch H7 directly, so this example follows one more implication out of it. If colour B is true, B2 is 1, so B9 in row B is not 1, and column 9 then puts its 1 in H9. Colour B therefore makes H9 a 1, and colour A makes D7 a 1. Either way D7 or H9 is 1.
H7 sees D7 through column 7 and H9 through row H, so 1 comes off H7. Note the last hop: B2 to B9 is a weak link (row B has 1s in B2, B3 and B9), followed by the strong link between B9 and H9 in column 9. That is a small step beyond a pure colour trap, where an uncoloured cell simply sees one cell of each colour.
Common mistakes
- Linking two cells that merely share a unit. A link needs a conjugate pair and nothing weaker.
- Forgetting box links. Most useful chains bend through a box rather than running along rows and columns only.
- Trapping a cell that sees the same colour twice. Two cells of colour A prove nothing; you need one of each colour.
- Treating a clash as the end of the chain. It kills one colour and confirms the other, which is usually several placements at once.
- Reusing colours across digits. Every digit gets its own independent colouring.
Where you meet it
Simple Colouring is the first single-digit chaining technique, and it starts to matter at hard classic Sudoku, where wings and fish no longer finish the grid on their own. Once your chains need links that are weaker than a conjugate pair, the next step up is X-Cycles.