Sudoku You’ve learned Simple Colouring — picking one digit and colouring its conjugate pairs to find eliminations. But what if the chain could jump across digit boundaries? That’s exactly what 3D Medusa does.
By adding bivalue cells (cells with exactly two candidates) as bridges between digits, 3D Medusa builds a single alternating-colour network that spans multiple digits. The result is one of the most powerful elimination techniques in Sudoku, capable of clearing many candidates in a single step.
In this guide we explain the logic behind 3D Medusa, walk through a detailed example with before-and-after diagrams showing 11 eliminations and 5 naked singles, and show you exactly how to find this pattern in your own puzzles.
✅ What is 3D Medusa?
3D Medusa (sometimes called Medusa Colouring or Multi-Colouring) is an expert-level technique that extends Simple Colouring by allowing the colour chain to cross digit boundaries through bivalue cells.
The name “3D” refers to the three dimensions the technique operates in: rows, columns, and digits. While Simple Colouring works in 2D (rows and columns for a single digit), 3D Medusa adds the digit dimension, letting the chain jump from one digit to another within a bivalue cell.
Like Simple Colouring, exactly one colour is “true” and the other is “false.” But now each node in the chain is a specific candidate in a specific cell (e.g. “digit 5 in R1C2”), not just a cell. A single cell can have one candidate coloured A and another coloured B.
🔗 The two types of link
3D Medusa builds its chain from two types of strong link:
Identical to Simple Colouring. A digit d appears as a candidate in exactly two cells of a row, column, or box. If one cell has d coloured A, the other must be coloured B. This link stays within one digit.
A cell contains exactly two candidates. If one candidate is coloured A, the other must be coloured B — the cell must hold one of them, so they are opposite colours. This link crosses between two different digits.
By alternating between these two link types, the chain naturally weaves through multiple digits. A conjugate pair fixes two cells for one digit; a bivalue cell then pivots to a different digit in the same cell; the next conjugate pair extends the chain for that new digit; and so on.
🧠 The elimination rules
Once the chain is fully coloured, you check for contradictions and eliminations. The most commonly used rules are:
Rule 2 — Colour twice in a house
If the same colour appears twice for the same digit in a single row, column, or box, that colour is false. Eliminate the digit from every cell of that colour.
Rule 4 — Seeing both colours (most common)
If an uncoloured candidate can see cells of both Colour A and Colour B for the same digit, it can be eliminated. One colour must be true, so the digit is already placed somewhere visible to the candidate.
Rule 3 — Uncoloured candidate in a chain cell
If a chain cell has an uncoloured candidate that sees the same colour of that digit elsewhere, the uncoloured candidate can be eliminated. Whether that colour is true (the seen cell holds the digit) or false (the other colour in this cell is true), the uncoloured candidate is blocked.
Rule 4 is by far the most common. In our example below, all 11 eliminations come from Rule 4. Focus on mastering this rule first — scan for uncoloured candidates that can see both colours of the same digit.
🔎 Step-by-step example
Let’s walk through a real 3D Medusa application that eliminates 11 candidates across two digits and creates 5 naked singles.
Step 1: Find conjugate pairs and bivalue cells
Scanning the grid, we find these strong links:
- Digit 5, Row 1: only R1C2 and R1C6 contain 5 → conjugate pair ✔
- Digit 5, Column 2: only R1C2 and R3C2 contain 5 → conjugate pair ✔
- Digit 7, Row 3: only R3C2 and R3C6 contain 7 → conjugate pair ✔
- Digit 7, Box 2 (rows 1–3, cols 4–6): only R1C6 and R3C6 contain 7 → conjugate pair ✔
And these bivalue cells:
- R1C2 = {5,7} — bivalue ✔
- R3C2 = {5,7} — bivalue ✔
- R3C6 = {5,7} — bivalue ✔
R1C6 = {5,7,9} is not bivalue (three candidates), so it doesn’t provide a bivalue link. But it still receives colours through conjugate pairs.
Step 2: Build the chain and assign colours
Start colouring from R1C2, digit 5 = Colour A (blue):
- R1C2(5) = A (our starting node).
- R1C2(7) = B — bivalue link: R1C2 has only {5,7}, so the other candidate gets the opposite colour.
- R3C2(5) = B — conjugate pair for 5 in Column 2: R1C2(5)=A forces R3C2(5)=B.
- R3C2(7) = A — bivalue link: R3C2={5,7}, so 7 gets the opposite of 5.
- R1C6(5) = B — conjugate pair for 5 in Row 1: R1C2(5)=A forces R1C6(5)=B.
- R3C6(7) = B — conjugate pair for 7 in Row 3: R3C2(7)=A forces R3C6(7)=B.
- R3C6(5) = A — bivalue link: R3C6={5,7}, so 5 gets the opposite of 7.
- R1C6(7) = A — conjugate pair for 7 in Box 2: R3C6(7)=B forces R1C6(7)=A.
Cross-check: no same-colour conflict in any house — the colouring is consistent ✔. The chain forms a rectangle across four cells, two digits, and eight coloured nodes.
The chain crosses from digit 5 to digit 7 through bivalue cells (R1C2, R3C2, R3C6). This is what makes 3D Medusa more powerful than Simple Colouring — a single chain colours candidates of both digits.
Step 3: Apply Rule 4 — find candidates seeing both colours
For each digit in the chain, we look for uncoloured candidates that can see both a Colour A cell and a Colour B cell of that same digit.
Digit 5 eliminations:
Colour A has digit 5 at R1C2 and R3C6. Colour B has digit 5 at R1C6 and R3C2.
- R2C4 = {2,
5}: sees R3C6(A) and R1C6(B) through Box 2. Eliminate 5 → naked single {2}. - R2C6 = {
5,9}: sees R3C6(A) and R1C6(B) through Column 6. Eliminate 5 → naked single {9}. - R3C9 = {1,2,
5}: sees R3C2(B) and R3C6(A) through Row 3. Eliminate 5 → {1,2}. - R8C6 = {1,4,
5,7}: sees R3C6(A) and R1C6(B) through Column 6. Eliminate 5 → {1,4,7}.
Digit 7 eliminations:
Colour A has digit 7 at R3C2 and R1C6. Colour B has digit 7 at R1C2 and R3C6.
- R1C1 = {1,2,
7}: sees R1C2(B) and R1C6(A) through Row 1. Eliminate 7 → {1,2}. - R1C3 = {2,
7}: sees R1C2(B) and R1C6(A) through Row 1. Eliminate 7 → naked single {2}. - R5C2 = {
7,8}: sees R1C2(B) and R3C2(A) through Column 2. Eliminate 7 → naked single {8}. - R7C2 = {4,
7,9}: sees R1C2(B) and R3C2(A) through Column 2. Eliminate 7 → {4,9}. - R8C2 = {4,
7,8}: sees R1C2(B) and R3C2(A) through Column 2. Eliminate 7 → {4,8}. - R8C6 = {1,4,
7}: sees R1C6(A) and R3C6(B) through Column 6. Eliminate 7 (also lost 5 above). - R9C6 = {4,
7}: sees R1C6(A) and R3C6(B) through Column 6. Eliminate 7 → naked single {4}.
11 eliminations total, 5 naked singles — from a single application of 3D Medusa!
Step 4: Result
After removing 11 candidates, five cells become naked singles: R1C3=2, R2C4=2, R2C6=9, R5C2=8, R9C6=4. These solved cells trigger further simplifications across the grid.
A single Simple Colouring chain for digit 5 or digit 7 alone would not have produced these results — neither digit had enough conjugate pairs on its own to build a useful chain. 3D Medusa connected them through bivalue cells, creating a four-cell chain that eliminates candidates of both digits simultaneously.
🔄 3D Medusa vs. Simple Colouring
| Feature | Simple Colouring | 3D Medusa |
|---|---|---|
| Digits | One digit at a time | Multiple digits in one chain |
| Links | Conjugate pairs only | Conjugate pairs + bivalue cells |
| Coloured nodes | Cells (one digit per cell) | Specific candidates (digit+cell pairs) |
| Chain length | Often 3–6 cells | Often 4–10+ nodes across multiple digits |
| Elimination scope | One digit | Multiple digits at once |
| Difficulty | Advanced | Expert |
| Typical yield | 1–4 eliminations | 3–15+ eliminations |
Think of Simple Colouring as the “2D” version (rows and columns for one digit) and 3D Medusa as the full “3D” upgrade (rows, columns, and the digit dimension). Every Simple Colouring chain is a valid 3D Medusa chain — but 3D Medusa can find eliminations that Simple Colouring never could.
🕵️ How to spot 3D Medusa
1. Look for bivalue cells (cells with exactly two candidates). These are your digit bridges.
2. For each candidate in a bivalue cell, check if it forms a conjugate pair in any row, column, or box (appears in exactly two cells of that unit).
3. Start colouring: assign one candidate in a bivalue cell Colour A. The other candidate gets Colour B (bivalue link). Follow conjugate pairs to extend the chain, alternating colours.
4. Keep extending through both conjugate pairs and bivalue cells until no more connections exist.
5. Check for eliminations using the rules above, especially Rule 4 (uncoloured candidate sees both colours of the same digit).
Start with bivalue cells that share a conjugate pair for one of their digits. The bivalue cell provides the “digit jump,” and the conjugate pair extends the chain. Clusters of bivalue cells near each other often produce the best chains.
⚠️ Common mistakes to avoid
1. Using weak links
Every connection must be a strong link. For conjugate pairs, the digit must appear in exactly two cells of the unit. For bivalue links, the cell must have exactly two candidates. Three or more candidates do not form a bivalue link.
2. Confusing cell colouring with candidate colouring
In Simple Colouring, you colour entire cells. In 3D Medusa, you colour individual candidates. A cell with {5,7} might have 5=A and 7=B — both colours in one cell. Keep track of which colour belongs to which digit.
3. Missing bivalue links
A bivalue cell provides a link between its two candidates. If you only follow conjugate pairs and forget to check bivalue cells, you miss the connections that make 3D Medusa powerful.
4. Applying Rule 4 across different digits
Rule 4 requires the candidate to see both colours of the same digit. An uncoloured 5 that sees Colour A of digit 7 and Colour B of digit 5 does not qualify — both seen colours must belong to the digit being eliminated.
5. Incomplete pencil marks
3D Medusa depends on accurate candidate lists. A missing or incorrect pencil mark can cause you to miss a conjugate pair or bivalue cell, or worse, lead to incorrect eliminations.
📅 When to look for 3D Medusa
- Basic: 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: 3D Medusa, Swordfish, Jellyfish, Unique Rectangles, ALS-XZ.
Puzzles requiring 3D Medusa are rated Expert or Extreme. The technique sits above Simple Colouring and is one of the most powerful colouring methods. Try our hard puzzles for practice.
🚀 Beyond 3D Medusa
| Technique | What it adds | Complexity |
|---|---|---|
| Simple Colouring | Strong links only, one digit | Advanced |
| 3D Medusa | Adds bivalue-cell links across digits | Expert |
| X-Cycles | Adds weak links for alternating inference chains | Expert |
| ALS-XZ | Two Almost Locked Sets linked by RCC | Expert |
| Forcing Chains | Multi-path “what if” chains | Master |
| Forcing Nets | Branching inference networks | Master |
3D Medusa is the natural next step after Simple Colouring. It is closely related to alternating inference chains (AICs) and subsumes many simpler colouring techniques. Once comfortable with 3D Medusa, advanced solvers often move to full AIC or forcing-chain methods.
🎯 Practice 3D Medusa
Hard Sudoku
Challenging puzzles where 3D Medusa and other expert techniques are regularly needed.
▶ Play Hard SudokuSimple Colouring Guide
Master the single-digit colouring foundation before tackling 3D Medusa.
▶ Read Simple Colouring GuideSudoku Solver
Enter your puzzle and watch the solver identify techniques automatically.
▶ Open SolverFrequently asked questions
3D Medusa extends Simple Colouring across multiple digits by chaining through bivalue cells and conjugate pairs. The “3D” refers to rows, columns, and the digit dimension. It assigns alternating colours to specific candidates and eliminates candidates that see both colours of the same digit.
Simple Colouring works with one digit at a time using conjugate pairs. 3D Medusa adds bivalue-cell links, letting the chain cross digit boundaries. This makes it far more powerful — a single chain can eliminate candidates of multiple digits.
The most common rules: Rule 2 — same colour twice in a house means that colour is false. Rule 3 — uncoloured candidate in a chain cell seeing the same colour elsewhere is eliminated. Rule 4 — uncoloured candidate seeing both colours of the same digit is eliminated. Rule 4 is the most frequently used.
After exhausting simpler techniques including Simple Colouring. 3D Medusa is an expert-level strategy best suited for hard and extreme puzzles with multiple bivalue cells and conjugate pairs that can connect into multi-digit chains.
A bivalue cell has exactly two remaining candidates. In 3D Medusa it acts as a bridge between digits: if one candidate is coloured A, the other must be B. This is how the chain crosses from one digit to another.