Sudoku Written by Brian Hamilton and Karan Hamilton.
Brian and Karan are the puzzle lovers and game makers behind Sudoku Online Puzzles. You can learn more on the About page.
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Puzzle 1933, one of our Evil sudoku puzzles, has 27 givens. After 15 hidden singles, five naked singles, three pointing pairs and a pointing triple, 34 squares are empty, and just two of our Sudoku Helper’s eighteen techniques still find a move: an Almost Locked Set and Simple Colouring. Follow the 9s through the houses that leave them only two places, colouring as you go, and six squares split into two teams. R1C1 and R1C6 can see both teams, so neither can be a 9, and R1C1 is left holding a 1.
Simple colouring, spelled simple coloring in the US and also called singles chains, is the simplest of the chain techniques: one digit, strong links only. We ran it over our own puzzles to see how much of its reputation holds up. Our classic levels never need it. About one Evil puzzle in seven does. And the part of the chain that does the work is usually just three links long, not the board-spanning web the name suggests.
Choose Simple Colouring in the Sudoku Trainer for ten chains on real positions. Click every square of the chain, four to ten of them, and press Check. Nine of the ten are colour traps, a square that sees both colours; the other one is a colour wrap.
What is simple colouring in sudoku?
It is a way of following one digit. Find every row, column and box where the digit has exactly two places left. Each of those pairs is a strong link, or conjugate pair: one square of the pair holds the digit and the other doesn’t. Where two pairs share a square they join up, and everything joined that way is a cluster. Colour the squares of a cluster in two colours, alternating along every link, so that the two ends of each link always differ.
Every link has exactly one true end, so the colouring can come out only two ways: every square of one colour holds the digit, and no square of the other does. The colouring never tells you which. Two rules don’t need to know:
- A square that sees both colours loses the digit. It shares a row, column or box with a square of each colour, and whichever colour is true puts the digit beside it.
- A colour that sees itself is false. If two squares of one colour share a house, that colour would put the digit there twice. Every square of that colour loses the digit, and every square of the other colour gets it.
Some guides call the first a colour trap and the second a colour wrap.
A strong link needs exactly two places. With three, “not this one” still leaves two, so a colour can’t be passed through. Colouring never crosses such a house, but it can look across one. In puzzle 1933, row 1 holds four 9s and links nothing, yet R1C1 and R1C6 both see a coloured 9 along it, and that is how they lose theirs.
How do you colour a chain? Puzzle 1933, step by step
Work with the 9s alone. The pictures below show only them, plus every mark in R1C1 and R1C6.
- List the strong links. Seven houses hold exactly two 9s: rows 5, 7 and 8, column 7, and boxes 4, 7 and 9. Row 5 and box 4 name the same two squares, R5C2 and R5C3.
- Start on a busy square. R8C7 sits on three links, in column 7, row 8 and box 9. Colour it green.
- Give its partners the other colour. R1C7 (column 7), R8C2 (row 8) and R7C8 (box 9) are blue.
- Carry on until nothing new is coloured. R8C2’s other link, in box 7, makes R9C1 green, and R7C8’s, along row 7, makes R7C6 green. No more links lead out of these six squares.
- Look for squares that see both colours. R1C1 sees green R9C1 down column 1 and blue R1C7 along row 1. R1C6 sees green R7C6 down column 6 and the same blue R1C7. Neither can be 9.
The pair in R5C2 and R5C3 is a cluster of its own, and alone it removes nothing: a square that sees both is in row 5 or box 4, where those two are the only places for a 9.
Most colouring moves remove one candidate. This one removes two, and one of them matters at once, because R1C1 had only 1 and 9 left. To follow along, open the position in our Sudoku Helper. Its first Next Hint fills in the candidates from the digits; the next four hints are the same pointing moves our solver made (press Apply Hint after each), and the hint after them is this colouring, reported as Rule 2 on 9.
What are the two simple colouring rules?
Every guide has the same two, under different names and numbers. The numbers only matter when you read a solver’s output:
| What you see | What goes | Our helper | SudokuWiki, 3D Medusa |
|---|---|---|---|
| A square sees both colours (a colour trap) | The digit, from that square | Rule 2 | Rule 4 |
| Two squares of one colour share a house (a colour wrap) | The digit, from every square of that colour | Rule 1 | Rule 2 |
A square “sees” another when the two share a row, column or box. SudokuWiki calls them “two colours elsewhere” and “twice in a unit”, and uses its 3D Medusa numbers for them, hence 4 and 2. In July 2026 it added a rule 7, for a colour that would leave some row, column or box with no place for the digit; our helper doesn’t check that one.
Some guides list a third rule, for a house that holds both colours: every other place for the digit in that house goes. That is the first rule again, since every such square sees both colours. And 3D Medusa keeps both rules, as its rules 2 and 4, then adds four more for clusters that cross from one digit to another.
When does a colour contradict itself?
Seldom, on our boards: of the 855 colouring moves our helper made across the 5,000 Evil puzzles, only 11 used the contradiction rule. When it does fire, it decides a whole colour at once. Evil puzzle 366 is 38 moves in, with 25 squares empty. The 4 is already placed three times, and the six missing 4s have thirteen possible places. Twelve of them join into one cluster through fifteen strong links. Colour it, and blue lands twice in row 4, twice in column 8 and twice in box 6.
One clash would be enough. If blue were true, row 4 would hold two 4s, so blue is false: all seven of its 4s go at once. Green is then true everywhere, and R4C8, the one 4 outside the cluster, is left as the only place for a 4 in row 4. One move decides all six missing 4s, and our helper finishes with naked singles alone.
All eleven contradictions we found looked like this one. The clusters were large, 8 to 14 squares; each removed five to eight candidates; and in ten of the eleven, naked singles alone finished the puzzle afterwards. In each, the two clashing squares were six links apart by the shortest route, too far apart to notice without colouring the whole cluster. The helper offers this one straight from the digits: open the position, press Next Hint twice, and choose Simple Colouring from the Available Techniques menu.
How often do you need simple colouring?
The counts here come from our helper playing our own puzzles the way its hints do: the simplest move available, every time, until the grid is full or nothing applies.
On our Easy, Medium, Hard and Expert levels, never. In 200 puzzles from each, colouring was not once the simplest move. Our generator deals those levels only when singles can finish them, with help from locked candidates on Hard and, on Expert, naked pairs. A colouring move was often on the board anyway, at 56 of the 59 Hard positions where singles had run out and at 200 of the 277 Expert ones, but something simpler always came first.
On Evil, about one puzzle in seven needs it: 710 of all 5,000, and 88 of the first 500 (the bank is sorted by score, so that opening stretch is harder than the rest). Mostly once: 585 of the 710 used it a single time. At Evil positions with no single left, a colouring move was there 43.2% of the time, more often than any of the helper’s techniques except the Almost Locked Set. It is needed far less often than that because the helper tries thirteen other techniques first, from pointing pairs to the Skyscraper.
| Rule | Moves | Removed | Single at once |
|---|---|---|---|
| 2: sees both colours | 844 | One candidate in 733 of them, never more than 4 | 90 |
| 1: a colour sees itself | 11 | 5 to 8 | 11 |
Every Simple Colouring move our helper played on the 5,000 Evil puzzles, taking the simplest move each time; rules numbered as the helper numbers them. “Single at once”: the move left a naked or hidden single that did not exist before it. Measured October 2026.
Colouring is sold as a long chain winding across the grid. On our boards the cluster behind a rule 2 move had five squares at the median, and 271 of the 844 had only four. Even in bigger clusters the part that proves the elimination is short: in 747 of the 844 moves, three strong links between a green square and a blue one were enough, as in puzzle 1933. Five were needed 90 times, seven or nine just 7 times.
SudokuWiki’s count over a benchmark set of 50,000 puzzles points the same way: its twice-in-a-unit rule made 47 of 3,003 Simple Colouring moves, 1.6%, against 11 of 855, 1.3%, in ours. We quote our helper’s counts because they are what its hints do. Our 23-technique test solver tries finned fish before any wing, and it needed colouring on only 23 of the first 500 puzzles: at 43 positions where the helper would have coloured, a finned X-Wing got in first, and at 8 more a finned Swordfish.
Is simple colouring an X-chain? X-Wing, Skyscraper and 3D Medusa
Yes. Colouring is the part of the X-chain family that uses strong links only. Written as a chain, R1C1’s elimination in puzzle 1933 reads (9) r1c7=r8c7-r8c2=r9c1: if R1C7 isn’t 9, column 7 puts the 9 in R8C7, row 8 then keeps it out of R8C2, and box 7 puts it in R9C1. One end or the other is a 9, and R1C1 sees both. The row 8 link is strong, but here it does a weak link’s job, which any strong link can. A colour that sees itself is the loop version: the run of links between the two clashing squares, closed by the house they share, is an X-cycle in which two weak links meet, and the square where they meet loses the digit.
X-chains go one step further and use the genuine weak link: a house with three or more places, such as row 1 in puzzle 1933. Colouring can look across that house but never pass a colour through it; an X-chain can. That is also why colouring is not a longer X-Wing or Skyscraper, whatever some guides say:
- An X-Wing joins two strong links through its two cover lines, and a cover line that loses anything holds the digit at least three times. If only one cover line has extras, the other is a strong link and colouring finds the same eliminations; if both do, nothing in the X-Wing links its two base lines, and colouring would need some other route between them.
- A Skyscraper joins two strong links with one weak link. When that link happens to be strong as well, colouring sees the same move; usually it isn’t.
- 3D Medusa takes colouring across digits, adding squares with two candidates as links. Cut down to one digit, its rules 2 and 4 are exactly the two rules above, which our helper numbers 1 and 2.
- Multi-colouring plays two clusters of one digit against each other, through a square of one that sees a square of the other. That is one genuine weak link, so it is an X-chain too. Our helper doesn’t have it.
How do you find a colouring move by hand?
- Make the pencil marks complete. A missing mark turns a house with three places into a fake strong link, and a colour passed through it can strike out the answer. Our pencil marks guide has a routine for that.
- Take one digit and list its two-place houses, each as a pair of squares. For the 9s of puzzle 1933 that is seven houses but six pairs, since row 5 and box 4 share theirs, and the list is the whole map.
- Look for runs of three links. A square that sees both ends of a single link is already a locked-candidates move, so a useful run has at least three. Its two ends are opposite colours; check every square that sees both. This finds most moves: 747 of our 844.
- Then colour the whole cluster and check longer runs, and any house holding two squares of one colour.
- Move to the next digit. A digit with fewer than three strong links won’t give a colouring move that locked candidates would miss.
With your position in the Sudoku Helper, two presses of Next Hint bring up its Available Techniques menu. If Simple Colouring is on it, choose it: the hint names the digit and the rule. For rule 2 it shades every square of the cluster alike and leaves the colouring to you; for rule 1 it shades the colour that survives and marks the other for removal.
Stuck after simple colouring, or can’t find one?
A colouring move rarely opens the board by itself: only 90 of the 844 rule 2 moves made a single straight away. Check the squares that lost the digit first, as R1C1 shows. Then colour the same digit again, because clusters change as marks go. Of the 710 Evil puzzles that needed colouring, 125 needed it more than once, and puzzle 1933 colours its 9s a second time three moves later.
If nothing opens, try an Almost Locked Set. After their first colouring, 443 of those 710 puzzles next needed one, more than every other harder technique put together. On 51 nothing harder ever applied and the helper stalled; in all, it finished 463 of the 710, and on 72 of those colouring was the hardest step it needed.
If you can’t find a colouring, and the helper’s menu doesn’t list one, there isn’t one on those marks. Colour across digits instead with 3D Medusa, or let the chain use weak links: X-chains, then XY-chains. Near the end of a puzzle, check for a BUG; forcing chains are the last resort. The guide to sudoku techniques shows how each of these follows on from colouring.