3D Medusa
3D Medusa is a sudoku technique for multi-digit colouring: you paint linked candidates in two colours and let the contradictions finish the job. Simple Colouring follows one digit through the grid; Medusa follows every digit at once, which is why it breaks positions single-digit colouring cannot touch.
What the pattern is
A Medusa cluster is a network of candidates, not cells. Each node is one digit in one cell, and two nodes are joined when exactly one of them can be true:
- Bi-location link: a digit has exactly two places left in a row, column, or box. One of those two is that digit.
- Bi-value link: a cell has exactly two candidates left. One of those two is the answer.
Pick any starting candidate, give it colour A, give its partner colour B, and keep alternating along every link you can reach. When the cluster stops growing you have two colours, and exactly one of them is entirely true. The other is entirely false.
| Simple Colouring | 3D Medusa | |
|---|---|---|
| Digits per cluster | one | any number |
| Links used | bi-location only | bi-location and bi-value |
| Nodes | cells | candidates |
| Elimination rules | 2 | 6 |
When to look for it
Look for Medusa when singles, pairs and wings are exhausted and the grid is full of bi-value cells. That is the fuel: the more cells with exactly two candidates, the further a cluster spreads. Two or three connected bi-value cells plus one conjugate pair is already enough to start.
How to apply it, step by step
- Choose a seed: a bi-value cell or a digit with two places in a unit.
- Colour one candidate A, its partner B.
- Follow every bi-value and bi-location link out of every coloured node, alternating colours. Stop when nothing new can be added.
- Test the cluster against the six rules below.
- Apply what you find. If a whole colour falls, the other colour becomes a set of placements at once.
The six rules, one line each:
- Rule 1: two candidates of the same colour in one cell: that colour is false.
- Rule 2: the same digit twice in one colour inside one unit: that colour is false.
- Rule 3: an uncoloured candidate in a cell that already holds both colours: remove it.
- Rule 4: an uncoloured candidate that sees colour A and colour B of its own digit: remove it.
- Rule 5: an uncoloured candidate seeing one colour in a unit while the other colour sits in its own cell: remove it.
- Rule 6: a cell whose every candidate sees the same colour: that colour is false.
A worked example
This position ends with rule 6. Box 1 holds five bi-value cells: A1 , A2 , B2 , B1 and C1 . Watch the colours spread from A1, where 3 is colour A and 9 is colour B.
The chain runs through the box 1 bivalue cells, then through the 9s (column 1 to H1), the 4s (column 2 to E2, across E2 and E3 in row E, column 3 to H3), the 3s and the 6s. In total 27 candidates get a colour. The cell H8 is not coloured and holds 2, 4 and 9. It sees the colour A 2 in C8 through column 8, and the colour A 4 in H3 and 9 in H1 through row H. If colour A were true, H8 would have no candidate left, so colour A is false and colour B is true.
All 16 colour A candidates are removed: 2 from C8 and D6, 3 from A1, C6 and D5, 4 from B1, E2 and H3, 6 from B7, C1 and J8, 8 from A5 and B2, 9 from A2, E3 and H1. Ten of the colour B cells are left with a single candidate. Notice that the chain crossed digits 2, 3, 4, 6, 8 and 9: a single-digit method would never have connected A1 to H8.
The other rules follow the same logic and are not shown on this board. Rule 4, for example, removes a candidate that sees both colours of its own digit: one of the two is true, so the candidate cannot be.
Common mistakes
- Colouring a link that is not a link. A digit with three places in a unit gives you nothing, and neither does a cell with three candidates.
- Mixing two separate clusters. Colours from different clusters have no relationship; keep them visually distinct.
- Applying rule 4 to a candidate that only sees one colour. It must see both, and both must be the same digit as the candidate you are removing.
- Treating a colour as "probably true". Until a rule fires, both colours are live.
- Forgetting that a fallen colour hands you placements, not just eliminations. Place them all before you re-scan.
Where you meet it
3D Medusa is a diabolical technique. It first becomes necessary in Expert puzzles, where chains and multi-digit logic replace pattern spotting. Below that level the same eliminations are almost always available from a cheaper rule, so try singles, subsets and wings first.