Chains

Simple Coloring

Simple coloring takes a single digit and follows it around the grid with two colors. It looks technical, but it rests on one idea you already trust: a chain of strong links must alternate.

Strong links, the building block

Fix on one number — say the 6. A strong link is any unit where the 6 has exactly two possible cells: whichever one is not the 6, the other must be. Those two cells are opposites, permanently. Coloring is just the act of writing that opposition down: paint one cell of the pair one color and its partner the other, then follow every strong link outward, alternating colors as you go. The result is a web of the digit's candidate cells split into two camps.

What the two colors mean

Because the colors alternate across strong links, one entire color is the true set and the other is the false set — you just do not yet know which is which. That is enough to act on, thanks to two rules. First, if two cells of the same color ever share a unit, that color cannot be the true one (it would place the digit twice in a unit), so every cell of the opposite color is a confirmed placement. Second, and more common, any uncolored cell that can see both colors cannot hold the digit at all.

Why the "sees both colors" rule works

The second rule is the workhorse, and its logic is clean. One of the two colors is true — that is guaranteed. So an uncolored cell that sees a cell of each color is guaranteed to see the digit's true home, whichever color turns out true. A cell that already sees the number cannot also be the number. Therefore the digit is impossible in that cell, and you remove it — without ever needing to learn which color was the real one.

How to do it at the board

Pick a digit that still appears in several strong links, and a full candidate map so you can see them. Start at any strong link, color its two cells oppositely, then walk the chain: each new strong link off a colored cell colors its partner the other shade. When the chain is laid out, check the two rules — same-color clash in a unit, or an outside cell seeing both colors. Either one hands you a result. If neither rule fires, the chain simply has nothing to say about this digit right now, so color a different number rather than forcing it. Coloring rewards breadth over depth — briefly trying several digits usually beats exhausting a single one — so move on the moment a chain falls quiet.

A worked elimination

Say you color the 6's chain and end up with a blue 6-candidate in the top band and a green 6-candidate in the left stack. Now look at a cell in the corner where that row and that column meet, which still lists 6 among its notes. It sees the blue candidate along its row and the green candidate down its column. One of those two is certainly a 6, so this corner cell sees a real 6 no matter how the chain resolves — and therefore cannot be a 6 itself. Off comes the candidate, and the board moves again.

Where coloring sits

Simple coloring is a genuine hard-board tool, a cousin of the X-Wing and the XY-Wing in that it proves a fact and eliminates on it. Reach for it once the simpler techniques are spent. It rewards patience and a tidy map, so build the habit on Medium Sudoku before taking it to Expert Sudoku, where a single chain can be the only way through. The classic board is always at Sudoku AZ.