Simple Coloring in Sudoku: Conjugate Pairs and Chains

Simple coloring (or simple colouring, sometimes called single-digit coloring) follows one digit through a chain of cells where that digit must be in one place or the other. You paint the chain in two alternating colors. One color is true and the other is false, and you can often work out which cells are ruled out without knowing which color wins. It is a hard-puzzle technique, and it is the easiest way into chain logic.

Conjugate pairs: the building block

A conjugate pair is a row, column, or box where a digit has exactly two possible cells. If the digit is not in one, it must be in the other. Two cells linked this way always have opposite truth values for that digit: one is the digit, the other is not.

Chains form when conjugate pairs share a cell. If 5 has two spots in row 1 (A and B), and B is also one of only two spots for 5 in column 7 (B and C), then A and C must agree with each other and disagree with B.

How to color a chain

1. Pick a digit that has many conjugate pairs. Digits with only two or three spots left in several groups are good choices.
2. Start at any cell in a conjugate pair and give it color A.
3. Give its partner color B.
4. From each colored cell, follow any other conjugate pair it belongs to and give the new cell the opposite color.
5. Keep going until no conjugate pair extends the chain.

Every cell colored A holds the digit, or every cell colored B does. It cannot be a mix.

The two rules

Rule 1: color wrap (a color contradicts itself)

If two cells of the same color share a row, column, or box, that color would put the digit twice in one group. That color is false. Remove the digit from every cell of that color, and place it in every cell of the other color. This rule is rarer but solves many cells at once.

Rule 2: color trap (a cell sees both colors)

If an uncolored cell with the digit as a candidate sees a cell of color A and a cell of color B, it cannot hold the digit. One of those two colored cells is the digit, and the uncolored cell sees it either way. This is the rule you will use most.

Worked example

Follow digit 5. Three conjugate pairs link up:

• Row 1: 5 can only go in R1C2 or R1C7.
• Column 7: 5 can only go in R1C7 or R6C7.
• Row 6: 5 can only go in R6C7 or R6C3.

Color R1C2 blue. Its partner R1C7 is green. R6C7 pairs with R1C7, so it is blue. R6C3 pairs with R6C7, so it is green. The chain is R1C2 (blue), R1C7 (green), R6C7 (blue), R6C3 (green).

Now look at R3C3, which still has 5 as a candidate. It shares box 1 with R1C2 (blue) and column 3 with R6C3 (green). Whichever color is true, R3C3 sees a 5. Erase 5 from R3C3.

Tips for spotting chains

• Mark conjugate pairs as you go. Snyder notation (writing a digit only when it has two spots in a box) gives you box conjugate pairs for free. Our pencil marks guide explains it.
• Longer chains find more. A chain of two cells is a single conjugate pair, and it finds nothing a pointing pair would not. The payoff grows with every link.
• Check rule 1 first. It takes a few seconds and, when it hits, it places several digits at once.
• Know the relatives. An X-Wing is a four-cell coloring chain in a rectangle. Skyscrapers and 2-String Kites are short chains with a fixed shape. If you understand coloring, those patterns stop being separate things to memorize.

Simple coloring works on one digit at a time. When you need to link different digits, the next step is chains like the XY-Wing (a three-cell chain across digits) and longer XY-chains. For where it sits in the overall order, see our sudoku techniques list.