Simple coloring
A technique that two-colours the strong-link graph of a single digit, then eliminates candidates that see both colours — the entry point into chain reasoning.
Simple coloring is the gentlest of the chain techniques. Pick a digit. Look at every place in the grid where that digit has bilocation — exactly two candidate cells in a unit. Those pairs of cells are connected by strong links; they form a graph. Two-colour that graph: A and B alternating along every strong-link edge. The two colours form a "conjugate pair" — every cell coloured A is the digit if every cell coloured B isn't, and vice versa.
The graph itself is a curiosity. The eliminations come from two situations.
Contradiction within a colour
If two cells of the same colour share a unit, the colour is impossible. Both cells the same colour would mean both cells are the digit at the same time — which violates the no-repeats rule. So the other colour must be the answer everywhere, and every cell of the contradicting colour can have the digit eliminated.
A candidate that sees both colours
If a third cell — one not in the coloured graph — sees a cell of colour A and a cell of colour B, the digit can be eliminated from that third cell. Whichever colour wins, one of the two seen cells will hold the digit, and the third cell can't be the same digit as a cell it shares a unit with.
This second elimination is the more common simple-coloring fire. The technique walks the strong-link graph, two-colours it, then scans every other candidate cell for the digit and asks: does this cell see at least one cell of colour A and at least one of colour B? If yes, the digit is eliminated.
The relationship to harder chains
Simple coloring is what every more complex chain technique generalises. Multi-coloring extends it to multiple disjoint chains and looks for cross-chain interactions. AIC and XY-chain extend the alternation to mix weak links into the chain structure. The intuition for why those harder techniques work is most easily learned by drilling simple coloring on a few worked examples first.
See also
- Strong link— A relationship between two cells in a unit where a digit must occupy exactly one of them — the basic primitive that hidden singles, X-wings, and chain reasoning all rest on.
- Bilocation— When a digit has exactly two candidate cells in a unit. The underlying configuration that strong links, X-wings, and chain techniques all reach for as their starting shape.
- Multi-coloring— An extension of simple coloring to two or more disjoint chains on the same digit, finding eliminations that fire when the chains interact at a distance.
- XY-chain— A chain of bivalue cells linked by shared candidates. Eliminates a digit from any cell that sees both endpoints — the workhorse intermediate-to-advanced chain technique.
- Alternating Inference Chain (AIC)— The general-purpose chain technique. Alternates strong and weak links along a sequence of candidates, eliminating a digit from any cell that sees both endpoints' candidates.
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