Lesson 17 of 17 · Chain technique

Simple colouring: follow one digit through a chain

The guide and all sudoku lessons
The full guide Playing on Puzzled
  1. 1The last empty squareFirst move
  2. 2ScanningCore technique
  3. 3Naked singlesCore technique
  4. 4Pencil marksSolving aid
  5. 5Hidden singlesCore technique
  6. 6Naked pairsPair technique
  7. 7Pointing pairsPair technique
  8. 8Hidden pairsPair technique
  9. 9Naked triplesSubset technique
  10. 10Hidden triplesSubset technique
  11. 11Naked quadsSubset technique
  12. 12Hidden quadsSubset technique
  13. 13X-WingCross-line technique
  14. 14XY-WingWing technique
  15. 15SwordfishCross-line technique
  16. 16JellyfishCross-line technique
  17. 17Simple colouringChain technique

Simple colouring tracks one digit through a connected chain of squares. Start with a row, column or box that has exactly two possible homes for that digit. One of those squares must hold it and the other cannot. This is called a strong link.

Label the two ends A and B. Follow another strong link from either end and alternate the labels as you go. The labels describe two possibilities for the whole connected group: either all A squares hold the tracked digit and all B squares do not, or the other way round. They are not guesses written into the board.

There are two useful rules here. If two squares with the same label share a row, column or box, that entire group is ruled out. This is a colour wrap. If a square outside the chain sees both an A and a B, it loses the digit whichever group is true. This is a colour trap.

Blue A and amber B identify the groups; neither colour means correct. A thin outline picks out the helpers discussed in the current step, and a green outline marks one removal target. All pencil marks in the shaded units come from the displayed board. The two examples, exercise and counterexample use separate seeded puzzles.

Example 1: A target that sees both groups

A: blue · B: amber. Same letter, same side of the chain.

Step 1 of 6

Track 3 in column 4. Its only homes are row 4, column 4 (A) and row 8, column 4 (B). Exactly one must be 3.

Follow 3 down column 4, through the bottom-middle box, and along row 9. Column 4 links row 4, column 4 (A) to row 8, column 4 (B). The box links that B to row 9, column 6 (A); row 9 links that A to row 9, column 8 (B). Each link uses the only two homes for 3 in its full unit.

The target at row 4, column 8 sees A at row 4, column 4 along its row, and B at row 9, column 8 down its column. If A is true, the row blocks its 3. If B is true, the column blocks it. Remove 3 and keep 8 and 9; this does not tell you which group is true.

Example 2: A clash rules out the whole group

A: blue · B: amber. Same letter, same side of the chain.

Step 1 of 6

Start with 5 in row 1. Columns 1 and 6 are its only homes. Label column 1 A and column 6 B.

Track 5. Row 1 links A at column 1 to B at column 6. Column 1 links that A to B at row 4, column 1; row 4 then links that B to A at column 4. Column 4 extends the chain to B at row 2, column 4. The middle-left box also links row 4, column 1 (B) to row 6, column 3 (A). Row 6 and the middle-middle box both link into B at row 6, column 6.

Now compare the two B squares at row 1, column 6 and row 2, column 4. They share the top-middle box, so they cannot both be 5. Since every B in this connected chain would be true together, the whole B group is false. Remove 5 from the green target at row 4, column 1, even though it is not one of the two clashing squares. Its remaining 7 is still shown as a pencil mark.

Example 3: Seeing one group is not enough

A: blue · B: amber. Same letter, same side of the chain.

Step 1 of 6

Track 4. Row 3 links A at column 5 to B at column 9; column 9 continues to A at row 6, column 9.

Track 4 through the A and B squares. Each link follows a unit with exactly two homes for 4: row 3, column 7, column 9, the top-right box or the middle-right box. Together they make one connected chain, with no same-label clash.

The green target at row 4, column 4 sees B at row 4, column 7, but no A square in this chain. If B is true, the target cannot be 4. If A is true instead, the B square is not 4 and this chain does not block the target. Keep the 4, along with 5 and 6. In this puzzle's unique solution the target is actually 4.

Try it yourself

Try the reasoning yourself. Choose an answer to see the explanation.

A: blue · B: amber. Same letter, same side of the chain.

Track the digit shared by the A and B squares, following their two-home links. Which candidate can this chain remove from the green target at row 6, column 8? Check which groups the target sees.

Pick one candidate to cross out in the ringed square, not a digit to place.

How to spot it faster

  • Choose one digit after checking simpler moves. Find a unit with exactly two homes for it, label those ends A and B, and follow other two-home links. Branches are fine: keep alternating labels on every link.
  • Scan each group for two members sharing a row, column or box. A clash rules out every member of that group, not just the two squares that clash.
  • If neither group clashes, look outside this chain for a candidate that sees at least one A and at least one B. The two views can use different units, such as a row and a column.

Common mistakes

Counting candidates instead of homes

A strong link needs exactly two homes for the tracked digit in the whole row, column or box. Its squares may each contain three or more pencil marks. Unlike XY-Wing, this is not a pattern made only from two-candidate squares. A third home in the unit breaks that link.

Treating a colour as a placement

A and B are labels, not answers. A colour trap removes a candidate without deciding which group is true. A colour wrap rules out the clashing group. Do not fill every blue square just because blue was the first label you chose.

Questions

Further reading

Try it

Try an expert 9×9 when simpler moves run out. An expert deal is not guaranteed to need colouring; use the separate exercise here to practise the rule directly. Free hints still stop at pairs.

Play solo on an expert 9×9