Chains, Colouring & Uniqueness
Simple Coloring
Simple Coloring works on one digit. Wherever a row, column or box leaves that digit exactly two places, one of them must hold it; chain such pairs together, colour the cells alternately A and B, and either every A cell or every B cell holds the digit. So any other cell that sees both colours loses the digit — and if two cells of one colour share a unit, that colour is false and the other is true.
What you need to see on the grid
Try Simple Coloring one digit at a time, once singles, subsets, fish and wings have run dry and the digit is down to two places in several rows, columns or boxes. Four terms carry the whole method:
- Strong link (conjugate pair)
- A row, column or box in which the digit has exactly two places. One of the two cells must hold it and only one can, so if one is not the digit, the other is.
- Chain and colours
- Strong links that share a cell join into a chain. Colour one cell A and switch colour across every link: within one chain, all the A cells hold the digit or all the B cells do. Colours of two separate chains say nothing about each other.
- Colour trap
- A cell outside the chain that sees one cell of each colour. Whichever colour is true, the digit sits in a cell it sees, so the cell loses the digit — without knowing which colour is true.
- Colour wrap
- Two cells of the same colour in one row, column or box. That colour would put the digit there twice, so it is false: the digit leaves all its cells, and every cell of the other colour holds it.
How to apply it, step by step
One position runs through the first four steps: the digit 3, late in an expert puzzle from the app, at the exact point where its hint needs Simple Coloring; step 5 shows the second rule on another real puzzle. Every diagram shows the whole grid. The cells a step needs are filled in; the others are left blank and hatched, so you know you can ignore them. A candidate is missing from a drawn cell because its row, column or box already holds that digit, or because an earlier move removed it: before the colouring, a Naked Pair {2,9} in column 9 took 2 and 9 out of R2C9 and 2 out of R9C9, and a Naked Triple {1,3,5} in box 9 took 5 out of R7C8 and 1 and 3 out of R9C8. Colour A is drawn as a dark disc, colour B as a light one, and each coloured cell carries its letter.
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Step 1Pick one digit and find its strong links
Choose a digit and look for the rows, columns and boxes where it has exactly two places left. Each one is a strong link: one of its two cells must hold the digit, and only one can.
Here the digit is 3. Row 2 leaves it only R2C6 and R2C9; column 9 only R2C9 and R9C9; box 3 only R2C9 and R1C8; box 9 only R9C9 and R8C8; row 8 only R8C8 and R8C5; column 5 only R8C5 and R3C5. Check each unit in full — a unit with a third place would not be a strong link. Box 9 counts only because the Naked Triple has already taken the 3 out of R9C8.
Rows 2 and 8, columns 5 and 9, boxes 3 and 9, in full. Each of the six units has exactly two places for 3. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 4 5 R2 6 4 8 5 7 R3 6 4 8 R4 2 6 R5 8 4 R6 6 R7 7 4 R8 9 2 6 4 8 7 R9 9 8 - Digit already on the grid
- Strong link: exactly two places for 3
- A place for 3 in a strong link
- Candidate 3, the digit studied
- Not needed for this step
Filled in: rows 2 and 8; columns 5 and 9; boxes 3 and 9. The other 41 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 2Link the pairs into a chain and colour it
Strong links that share a cell join into one chain. Start anywhere — say R2C6 — and give it colour A. Its partner across row 2, R2C9, takes colour B. Keep going and switch colour at every link: R9C9 is A (column 9) and so is R1C8 (box 3); R8C8 is B (box 9), R8C5 is A (row 8) and R3C5 is B (column 5).
The colours carry the logic. If R2C6 is 3, then R2C9 is not, so R9C9 and R1C8 are, so R8C8 is not, and so on down the chain. If R2C6 is not 3, every step flips. Either all the A cells are 3 or all the B cells are: exactly one colour is true, even though we do not know yet which one.
Technique: Simple Coloring
The same six units. A: R2C6, R9C9, R1C8, R8C5. B: R2C9, R8C8, R3C5 — the colour changes across every link. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 4 5 R2 6 4 8 5 7 R3 6 4 8 R4 2 6 R5 8 4 R6 6 R7 7 4 R8 9 2 6 4 8 7 R9 9 8 - Digit already on the grid
- Strong link: exactly two places for 3
- Candidate coloured A
- Candidate coloured B
- Cell of the chain
- Candidate 3, the digit studied
- Not needed for this step
Filled in: rows 2 and 8; columns 5 and 9; boxes 3 and 9. The other 41 cells are left blank: they are not needed for this step. Colour A: R1C8, R2C6, R8C5 and R9C9. Colour B: R2C9, R3C5 and R8C8. Nothing is removed yet: this step only reads the grid.
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Step 3Remove the digit from a cell that sees both colours
Now look outside the chain for a cell that still lists the digit and sees one cell of each colour. R3C4 does: it shares box 2 with R2C6 (A) and row 3 with R3C5 (B). So does R3C6: column 6 with R2C6 and row 3 with R3C5. Whichever colour is true, one of those two cells holds 3 and both R3C4 and R3C6 see it, so neither can be 3. This is the colour trap.
You never had to decide which colour is true — the trap works either way. A cell that sees two cells of the same colour proves nothing.
Technique: Simple Coloring
The coloured chain, plus R3C4 and R3C6: each sees R2C6 (A) and R3C5 (B), so both lose 3. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 4 5 R2 6 4 8 5 7 R3 6 4 8 R4 2 6 R5 8 4 R6 6 R7 7 4 R8 9 2 6 4 8 7 R9 9 8 - Digit already on the grid
- Strong link: exactly two places for 3
- Candidate coloured A
- Candidate coloured B
- Cell of the chain
- Candidate 3, the digit studied
- Candidate removed
- Not needed for this step
Filled in: rows 2 and 8; columns 5 and 9; boxes 3 and 9; cells R3C4 and R3C6. The other 39 cells are left blank: they are not needed for this step. Colour A: R1C8, R2C6, R8C5 and R9C9. Colour B: R2C9, R3C5 and R8C8. Removed: 3 from R3C4 and R3C6.
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Step 4Collect what the elimination unlocks
A colour trap removes candidates rather than placing a digit, so look at the cells it just cleared. R3C4 had two candidates, 2 and 3. With 3 gone, only 2 is left: a Naked Single, R3C4 = 2 — the app's next hint in this puzzle.
Technique: Naked Single
Row 3 after the trap. R3C4 has only 2 left. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 7 5 2 6 4 8 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- The cell that must be 2
- Candidate 2, the digit studied
- Digit placed
- Not needed for this step
Filled in: row 3. The other 72 cells are left blank: they are not needed for this step. Placed: R3C4 = 2.
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Step 5When a colour meets itself, that colour is false
Sometimes the chain proves more. In another expert puzzle from the app, after two Naked Pairs {7,8} — one in row 3, one in box 6 — the digit 7 has eight strong links: row 1, column 8, box 3, box 6, row 3, column 7, box 1 and column 1. They join into one chain: R1C6, R2C1, R3C9, R6C8 and R9C7 are colour A; R1C8, R3C2, R4C1 and R4C7 are colour B.
But R4C1 and R4C7 are both B, and both in row 4. If B were true, row 4 would hold two 7s, which is impossible. So colour B is false: 7 leaves every B cell — R1C8, R3C2, R4C1 and R4C7 — and colour A is true. This is the colour wrap. Look for it first: once a colour wraps, you need no trap for that chain. Here it leaves box 1 one place for 7, R2C1 — the app's next hint in that puzzle. (A Turbot Fish and an Empty Rectangle on 7 are also available on this board; they only clear the 7 from R4C6, which the chain does not use.)
Technique: Simple Coloring
A second real puzzle, digit 7: eight strong links in full and row 4. R4C1 and R4C7, both B, share row 4, so every B cell loses 7. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 4 9 2 6 5 3 R2 5 2 4 9 R3 2 3 6 1 R4 4 3 5 2 R5 5 3 9 R6 9 1 R7 8 4 3 R8 9 6 R9 2 - Digit already on the grid
- Strong link: exactly two places for 7
- Row 4: two B cells meet
- Candidate coloured A
- Candidate coloured B
- Cell of the chain
- Candidate 7, the digit studied
- Candidate removed
- Not needed for this step
Filled in: rows 1, 3 and 4; columns 1, 7 and 8; boxes 1, 3 and 6. The other 31 cells are left blank: they are not needed for this step. Colour A: R1C6, R2C1, R3C9, R6C8 and R9C7. Colour B: R1C8, R3C2, R4C1 and R4C7. Removed: 7 from R1C8, R3C2, R4C1 and R4C7.
Common mistakes
Colouring across a unit with three places
A unit where the digit has three places says only that at most one of them holds it. Two of those cells can both be empty, so a colour carried across that unit means nothing and every conclusion after it falls. Count the places before you draw each link: exactly two. In the example, box 9 becomes a link only after the Naked Triple removes the third 3, from R9C8.
Trapping a cell that sees only one colour
A cell that sees two A cells, and no B cell, keeps its candidate: if A is the false colour, neither of them holds the digit and it may still go there. The same goes for colours from two separate chains — an A in one chain and a B in another are unrelated. The trap needs one cell of each colour of the same chain in sight.
Filling in a colour that is not proven
Colouring alone never says which colour is true; it only says the two are opposites. Placing the digit in all the A cells because the chain looks right is a guess. Place digits only after a wrap has proved a colour false, or after a trap leaves a single somewhere.
Learn the move, not just the digit
In a real game, when Simple Coloring is what opens the next cell, the hint in the app names it and walks through these same steps on your board — the strong links, the two colours and the cell they clear.