Sudoku You’ve mastered X-Wings and Skyscrapers — fixed geometric patterns built from exactly two conjugate pairs. But what if the chain of strong links is longer, winding through boxes, rows, and columns? That’s where Simple Colouring comes in.
Also known as Singles Chains or Single Colouring, this technique assigns two alternating colours to cells along a chain of conjugate pairs for one digit. Once every reachable cell is coloured, you apply two powerful rules to eliminate candidates — sometimes clearing half a dozen cells in one move.
In this guide we explain the logic, walk through a real puzzle example with before-and-after diagrams, and compare Simple Colouring to other single-digit techniques.
✅ What is Simple Colouring?
Simple Colouring (Singles Chains) is an advanced single-digit elimination technique. You choose one candidate digit and find all its conjugate pairs — pairs of cells in a row, column, or box where the digit appears in exactly two positions. These pairs are called strong links.
Starting from any conjugate-pair cell, you assign it Colour A (blue). The other cell in the pair gets Colour B (green). If that cell belongs to another conjugate pair, its partner gets Colour A again. You keep alternating until no more connections exist.
Exactly one colour is “true” — every cell of that colour contains the digit. The other colour is “false” — none of its cells do. You don’t know which colour is true, but that binary guarantee is enough to eliminate candidates.
🔗 Conjugate pairs & strong links
Simple Colouring is built on conjugate pairs (strong links):
A conjugate pair for digit d exists in a unit (row, column, or box) when d appears as a candidate in exactly two cells of that unit. One of them must contain d — if one doesn’t, the other does. It’s a guaranteed either/or.
Unlike techniques such as X-Wing or Skyscraper that use a fixed number of conjugate pairs (exactly two), Simple Colouring chains together as many as needed — three, four, five, or more. The chain can turn corners through boxes, jump across rows, and span the entire grid.
🧠 How Simple Colouring works
Once you’ve built a chain and coloured every reachable cell, two elimination rules apply:
Type 1 — Colour Wrap (contradiction)
If two cells of the same colour see each other (share a row, column, or box), that colour is impossible — both can’t hold the digit simultaneously. Eliminate the digit from every cell of that colour.
Type 2 — Colour Trap (intersection)
If an uncoloured cell with the candidate can see at least one cell of Colour A and at least one cell of Colour B, it can never hold the digit. Whichever colour turns out to be true blocks it. Eliminate the digit from that cell.
Type 1: Same colour sees itself → kill that entire colour.
Type 2: Uncoloured cell sees both colours → kill the digit in that cell.
🔎 Step-by-step example
Let’s walk through a real Simple Colouring Type 2 example on digit 6.
Step 1: Find conjugate pairs for digit 6
Scanning the grid for units where 6 appears in exactly two cells:
- Row 4: R4C6 & R4C7 (strong link) ✔
- Row 5: R5C5 & R5C7 (strong link) ✔
- Column 6: R4C6 & R8C6 (strong link) ✔
- Box 5 (rows 4–6, cols 4–6): R4C6 & R5C5 (strong link) ✔
- Box 6 (rows 4–6, cols 7–9): R4C7 & R5C7 (strong link) ✔
Step 2: Build the chain and assign colours
Start with R4C6 and assign it Colour A (blue):
- R4C6 = A (blue)
- R4C7 = B (green) — conjugate of R4C6 in Row 4
- R8C6 = B (green) — conjugate of R4C6 in Column 6
- R5C5 = B (green) — conjugate of R4C6 in Box 5
- R5C7 = A (blue) — conjugate of R4C7 in Box 6
Cross-check: R5C5 (B) and R5C7 (A) are conjugate in Row 5 — different colours ✔. The colouring is consistent.
Step 3: Apply Type 2 — find cells seeing both colours
Column 7 contains both Colour A (R5C7) and Colour B (R4C7). Any uncoloured cell in Column 7 with candidate 6 sees both colours:
- R1C7 — {1,2,5,
6,7,8}: sees R4C7 (B) and R5C7 (A) via Column 7. Eliminate 6. - R3C7 — {1,5,
6,7,8}: sees R4C7 (B) and R5C7 (A) via Column 7. Eliminate 6. - R7C7 — {2,4,5,
6,8}: sees R4C7 (B) and R5C7 (A) via Column 7. Eliminate 6. - R8C7 — {1,2,4,5,
6}: sees R4C7 (B) and R5C7 (A) via Column 7. Eliminate 6.
Four eliminations from one chain!
Step 4: Continue solving
With four fewer candidate 6s, other techniques can progress. Notice that R7C7 now has {2,4,5,8} instead of five candidates — reducing options for further deductions like Naked Pairs or Hidden Pairs.
Build: A chain of conjugate pairs for one digit, alternating two colours.
Type 1: Same colour sees itself → eliminate the digit from all cells of that colour.
Type 2: Uncoloured cell sees both colours → eliminate the digit from that cell.
Result: Fewer candidates, potential cascading simplifications.
🕵️ How to spot Simple Colouring
1. Pick a digit (1–9) and find every row, column, and box where it appears as a candidate in exactly two cells — these are your conjugate pairs.
2. Starting from any conjugate-pair cell, assign Colour A. Assign Colour B to its partner. Follow connections to other conjugate pairs, alternating colours.
3. Check for Type 1: Do any two cells of the same colour share a row, column, or box? If so, that colour is false — eliminate the digit from every cell of that colour.
4. Check for Type 2: Does any uncoloured cell with the candidate see both a Colour A and a Colour B cell? If so, eliminate the digit from that cell.
5. Repeat for other digits.
Start with digits that have 3–5 placed instances. These tend to produce more conjugate pairs (units with exactly two candidates) while still leaving enough open cells for useful eliminations.
🔄 Type 1 vs Type 2
| Feature | Type 1 — Colour Wrap | Type 2 — Colour Trap |
|---|---|---|
| Trigger | Two cells of the same colour see each other | Uncoloured cell sees both colours |
| Logic | Contradiction — that colour can’t be true | Intersection — one colour blocks the cell |
| Elimination scope | All cells of the false colour | Just the uncoloured cell(s) |
| Typical yield | High — often clears many cells at once | Moderate — usually 1–4 cells |
| Frequency | Less common | More common |
In practice, Type 2 is more frequent. Type 1 is rarer but more powerful — when you find it, you can eliminate the digit from every cell of the contradicted colour, often solving multiple cells instantly.
⚠️ Common mistakes to avoid
1. Only use strong links (conjugate pairs)
Every connection in the chain must be a strong link — the digit appears in exactly two cells in that unit. If a unit has three or more candidates, it’s a weak link and cannot be used for Simple Colouring.
2. Keep colours consistent
If a cell is reachable via two different paths and gets conflicting colours, you’ve found a Type 1 contradiction rather than an error. Double-check the chain connections before concluding.
3. Check all three unit types
Strong links can exist in rows, columns, and boxes. A common mistake is only checking rows and columns and missing box-based connections that extend the chain.
4. Uncoloured cells must have the candidate
Before recording a Type 2 elimination, verify the cell actually contains the digit as a candidate. Seeing both colours is necessary but not sufficient if the cell doesn’t have the digit.
📅 When to look for Simple Colouring
- Basic techniques: Naked Singles, Hidden Singles, Full House.
- Intermediate: Naked Pairs, Hidden Pairs, Naked Triples, Pointing Pairs, Box/Line Reduction.
- Advanced (single-digit): X-Wing, Skyscraper, Simple Colouring.
- Advanced (multi-digit): XY-Wing, XYZ-Wing, W-Wing.
- Expert: Swordfish, Jellyfish, Multi-Colouring, ALS chains.
Puzzles requiring Simple Colouring are typically rated Hard to Expert. The technique is at the same tier as X-Wing and Skyscraper but can find eliminations those fixed patterns miss because the chain length is unrestricted. Try our hard puzzles.
🚀 Beyond Simple Colouring
Simple Colouring is the first step into colouring and chain techniques. Once you’re comfortable with it, the next level opens up:
| Technique | What it adds | Complexity |
|---|---|---|
| Simple Colouring | Strong links only, one digit | Advanced |
| Multi-Colouring | Links between separate colour clusters | Expert |
| 3D Medusa | Extends colouring to multiple digits simultaneously | Expert |
| X-Cycles | Adds weak links for alternating inference chains | Expert |
| Forcing Chains | Multi-digit chains exploring “what if” paths | Master |
Simple Colouring also generalises many fixed patterns you already know:
- X-Wing is a chain of exactly two conjugate pairs sharing both endpoints.
- Skyscraper is a chain of two conjugate pairs sharing one endpoint.
- 2-String Kite is a chain of two pairs connected through a box.
🎯 Practice Simple Colouring
- Fill all pencil marks: Simple Colouring requires complete candidate lists for the target digit.
- Scan digit by digit: For each digit, find all conjugate pairs. Build chains from any pair and colour alternately.
- Check both types: After colouring, scan for same-colour conflicts (Type 1) and dual-colour visibility (Type 2).
- Verify with the solver: Use our sudoku solver to confirm your eliminations.
Sudoku Hard
Hard puzzles where Simple Colouring and other advanced techniques are regularly needed.
▶ Play Hard SudokuSkyscraper Guide
Master the closely related Skyscraper technique — a fixed two-pair chain pattern.
▶ Read Skyscraper guideX-Wing Guide
Learn the X-Wing — another single-digit conjugate-pair technique that Simple Colouring generalises.
▶ Read X-Wing guideSudoku Solver
Enter your puzzle and watch the solver identify Simple Colouring and other techniques automatically.
▶ Open solverFrequently asked questions
Simple Colouring (Singles Chains) picks one digit, finds all strong links (conjugate pairs), then colours cells alternately. Type 1 finds same-colour contradictions; Type 2 eliminates from uncoloured cells seeing both colours.
Type 1 (Colour Wrap): two same-colour cells see each other → that entire colour is false. Type 2 (Colour Trap): uncoloured cell sees both colours → eliminate the digit from that cell.
Both are single-digit conjugate-pair techniques. X-Wing uses exactly two pairs in a rectangle. Simple Colouring builds chains of any length through rows, columns, and boxes, finding eliminations fixed patterns would miss.
After exhausting basic and intermediate techniques. Simple Colouring sits at the advanced tier alongside X-Wing and Skyscraper. Try it before moving to multi-digit chain methods like XY-Wing or W-Wing.
Singles Chains is another name for Simple Colouring. “Singles” means one digit; “Chains” means the connected path of conjugate pairs built across the grid.