3D Medusa

3D Medusa is multi-digit coloring: you paint linked candidates in two colors 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 coloring 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 color A, give its partner color B, and keep alternating along every link you can reach. When the cluster stops growing you have two colors, and exactly one of them is entirely true. The other is entirely false.

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

1. Choose a seed: a bi-value cell or a digit with two places in a unit.
2. Color one candidate A, its partner B.
3. Follow every bi-value and bi-location link out of every colored node, alternating colors. Stop when nothing new can be added.
4. Test the cluster against the six rules below.
5. Apply what you find. If a whole color falls, the other color becomes a set of placements at once.

The six rules, one line each:

• Rule 1 — two candidates of the same color in one cell: that color is false.
• Rule 2 — the same digit twice in one color inside one unit: that color is false.
• Rule 3 — an uncolored candidate in a cell that already holds both colors: remove it.
• Rule 4 — an uncolored candidate that sees color A and color B of its own digit: remove it.
• Rule 5 — an uncolored candidate seeing one color in a unit while the other color sits in its own cell: remove it.
• Rule 6 — a cell whose every candidate sees the same color: that color is false.

A worked example

Two bi-value cells and two conjugate pairs are enough to reach rule 4.

colors B and G

r2c2 {3,8}   3=B   8=G
r2c7 {5,8}   8=B   5=G     (8 has only r2c2 and r2c7 in row 2)
r6c7 {3,5}   5=B   3=G     (5 has only r2c7 and r6c7 in column 7)

r6c2 sees r2c2 (column 2) and r6c7 (row 6)
3 is B in r2c2 and G in r6c7  ->  3 removed from r6c2

One of the two colors is true, so one of those two 3s is true. Either way r6c2 cannot be 3. Notice that the chain crossed digits 3, 8 and 5 to get there — a single-digit method would never have connected r2c2 to r6c7.

Common mistakes

• Coloring 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. Colors from different clusters have no relationship; keep them visually distinct.
• Applying rule 4 to a candidate that only sees one color. It must see both, and both must be the same digit as the candidate you are removing.
• Treating a color as "probably true". Until a rule fires, both colors are live.
• Forgetting that a fallen color 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.