Colouring and chains
Where the techniques stop being patterns you recognise and start being arguments you follow. Worth understanding for the shape of the reasoning — though no puzzle on this site requires it.
The idea
When a digit has exactly two possible squares in a unit, those two squares are locked together: one is true and the other is false. That relationship is called a conjugate pair, and it is the link everything here is built from.
Give one square of the pair a colour and the other the opposite colour. Follow the links from unit to unit and the colours spread across the grid in an alternating chain — every square you reach is either "the same as the start" or "the opposite of the start".
What the colours buy you
Two conclusions follow, and they are the whole payoff:
- If two squares of the same colour turn up in one unit, that colour is impossible — it would place the digit twice in the same unit. Every square of the other colour is therefore true.
- If an uncoloured square can see both colours, one of them is true, so that square cannot hold the digit at all. The candidate is eliminated.
Longer chains — the family that includes XY-Wings, W-Wings and the various forcing chains — are all elaborations of this. The links get more varied; the alternating logic does not change.
Do you need this?
For puzzles on this site, no. Every grid generated here is verified solvable with cross-hatching through to the X-Wing, so treat this page as a signpost to what is beyond rather than as homework.
It is still worth reading once. Knowing that a chain exists is what stops you concluding a puzzle needs a guess — and reaching for a guess is how a solved grid becomes an unsolvable one.
Common mistakes
Colouring across a link that is not a conjugate pair. The two squares must be the only two homes for that digit in some shared unit. A digit with three homes gives you no colour information at all.
And colouring more than one digit at a time. Each digit gets its own independent colouring; the chains do not interact, and mixing them produces conclusions that are simply not true.
Common questions
What is colouring in Sudoku?
Colouring tracks a single digit across the grid. Wherever that digit has exactly two possible squares in a unit, the two are opposites: one is true and the other is false. Give them opposite colours, follow the links from unit to unit, and a contradiction anywhere settles the whole chain.
What is a conjugate pair?
Two squares in one unit that are the only two remaining homes for a digit. Exactly one of them holds it, with no third possibility. That strict either/or is the link every colouring chain is built from.
What can I conclude once a chain is coloured?
Two things. If two squares of the same colour turn up in one unit, that colour would place the digit twice, so it is impossible and every square of the other colour is true. Separately, any uncoloured square that can see both colours cannot hold the digit at all.
Do I need colouring to solve the puzzles on this site?
No. No puzzle here requires it. It is worth understanding for the shape of the reasoning, which is the basis of the harder chain techniques, but every grid on this site can be finished with the techniques before it.
Where to go next
That is the last of them. The full list is worth a second pass once you have used a few in anger.
Coming to this cold? The step before it is x-wing, and this technique is usually the one you reach for when that stops opening squares.
All nine techniques: the techniques index. New to the rules? Start here. Ready to use it? Open a board. Run a puzzle site? Embed this board.
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