Chains, Colouring & Uniqueness

3D Medusa

3D Medusa is Simple Coloring in three dimensions: it colours candidates — one digit in one cell — rather than cells, and besides conjugate pairs (a digit with exactly two places in a row, column or box) it links the two candidates of every bivalue cell, so one network of colours A and B spreads across several digits. Within the network either every A candidate is true or every B candidate is. If a colour puts two of its candidates in one cell, or the same digit twice in one unit, that colour is false and all its candidates go; otherwise, a candidate that sees its own digit in both colours is removed.

What you need to see on the grid

Reach for 3D Medusa after Simple Coloring, when colouring each digit on its own has found nothing but the grid still holds bivalue cells among digits that have two places left — the bivalue cells are what let a colour change digit. Four terms carry the whole method:

Coloured candidate: What gets a colour is one digit in one cell — the 7 in R1C5, say — not the whole cell. One cell can hold a candidate of colour A, one of colour B and others left uncoloured.
Two kinds of strong link: A conjugate pair: a row, column or box in which a digit has exactly two places, so one of them holds it. A bivalue cell: a cell with exactly two candidates, so one of them is its answer. Either way exactly one end is true, and the two ends take opposite colours. A unit with three places, or a cell with three candidates, links nothing.
Colour wrap: A colour that contradicts itself: two of its candidates in the same cell, which has only one answer, or the same digit twice in one row, column or box. That colour is false — every candidate of it goes, whatever its digit — and the other colour is true.
Colour trap: When neither colour wraps: an uncoloured candidate that sees its own digit in colour A and in colour B. Whichever colour is true, that digit sits in a cell it sees, so it goes. Some references list further Medusa rules; the hint in the app uses only the wrap and the trap.

How to apply it, step by step

One position runs through all five steps, part-way through a hard puzzle. Every diagram shows the whole grid: row 2 and box 2 are drawn in full, plus a few single digits elsewhere that explain missing candidates; the other cells are left blank and hatched, so you know you can ignore them. The pencil marks are complete: a candidate is missing only because its row, column or box already holds that digit, or because an earlier step removed it. Colour A is drawn as a dark disc, colour B as a light one, and each coloured cell carries the letters of its colours.

Step 1Start from a bivalue cell

Pick a cell with exactly two candidates. One of them is its answer and the other is not, so give them opposite colours. R2C4 holds only 2 and 5: colour its 2 A and its 5 B. Which one you call A makes no difference.

Then, for each coloured digit, look for a conjugate pair in the units of that cell — a unit where the digit has exactly two places. 2 has only two places in row 2, R2C4 and R2C8: the 2 in R1C1 keeps it out of R2C1 and R2C2, and the 2 in R4C6 out of R2C6. 5 has only two places in box 2, R2C4 and R1C5. Those two links carry the colours out of R2C4. 2 in box 2 and 5 in row 2 have three places each, so they link nothing.

Row 2 and box 2 in full. R2C4 holds only 2 and 5: 2 is coloured A, 5 is coloured B. 2 has two places in row 2 (R2C4, R2C8); 5 has two in box 2 (R2C4, R1C5).
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1	2	4		3		9			8
R2			1		6		3		4
R3	7								5
R4						2
R5				7
R6					1
R7
R8				8
R9						5

Digit already on the grid
Unit holding a conjugate pair (a digit with two places)
Candidate coloured A
Candidate coloured B
The bivalue cell you start from
Candidates 2 and 5, the digits studied
Not needed for this step

Filled in: row 2; box 2; cells R1C1, R1C2, R1C9, R3C1, R3C9, R4C6, R5C4, R6C5, R8C4 and R9C6. The other 56 cells are left blank: they are not needed for this step. Colour A: R2C4 (2). Colour B: R2C4 (5). Nothing is removed yet: this step only reads the grid.

Step 2Spread the colours through units and cells

Every link flips the colour. Across row 2, the other place for 2, R2C8, takes B. Across box 2, the other place for 5, R1C5, takes A. R1C5 is bivalue itself — 5 and 7 — so its 7 takes the opposite colour, B. That is the move Simple Coloring never makes: the colour has just jumped from digit 5 to digit 7. In box 2, 7 has two places, R1C5 and R2C6, so the 7 in R2C6 is A; R2C6 is bivalue too, 7 and 8, so its 8 is B.

The logic is Simple Coloring's, carried through more digits. If R2C4 is 2, it is not 5, so R1C5 is 5, so it is not 7, so R2C6 is 7, so it is not 8. If R2C4 is 5, every step flips. Either all the A candidates are true or all the B candidates are — we do not yet know which.

The colours so far. A: 2 in R2C4, 5 in R1C5, 7 in R2C6. B: 5 in R2C4, 2 in R2C8, 7 in R1C5, 8 in R2C6. R1C5 and R2C6 each hold one candidate of each colour.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1	2	4		3		9			8
R2			1		6		3		4
R3	7								5
R4						2
R5				7
R6					1
R7
R8				8
R9						5

Digit already on the grid
Unit holding a conjugate pair (a digit with two places)
Candidate coloured A
Candidate coloured B
Cell with a coloured candidate
Not needed for this step

Filled in: row 2; box 2; cells R1C1, R1C2, R1C9, R3C1, R3C9, R4C6, R5C4, R6C5, R8C4 and R9C6. The other 56 cells are left blank: they are not needed for this step. Colour A: R1C5 (5), R2C4 (2) and R2C6 (7). Colour B: R1C5 (7), R2C4 (5), R2C6 (8) and R2C8 (2). Nothing is removed yet: this step only reads the grid.

Step 3Finish the network and find a colour that contradicts itself

Keep going until no link is left. Row 2 holds one more conjugate pair: 7 has only two places in it, R2C6 and R2C8. The 7 in R2C6 is A, so the 7 in R2C8 is B.

Now R2C8 holds two B candidates, its 2 and its 7. A cell has only one answer, so they cannot both be true — yet within one network the candidates of a colour are all true or all false. So colour B is false. This is the colour wrap in its 3D form: Simple Coloring looks at one digit at a time, so it can never see two candidates of one colour in the same cell.

Always check for a wrap first. Had neither colour contradicted itself, you would look for a colour trap instead: an uncoloured candidate that sees its own digit in both colours.

Technique: 3D Medusa

The whole network. Row 2 links the 7 in R2C6 (A) to the 7 in R2C8 (B), so R2C8 now holds 2 and 7, both coloured B: colour B contradicts itself.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1	2	4		3		9			8
R2			1		6		3		4
R3	7								5
R4						2
R5				7
R6					1
R7
R8				8
R9						5

Digit already on the grid
Unit holding a conjugate pair (a digit with two places)
Candidate coloured A
Candidate coloured B
Cell with a coloured candidate
Not needed for this step

Filled in: row 2; box 2; cells R1C1, R1C2, R1C9, R3C1, R3C9, R4C6, R5C4, R6C5, R8C4 and R9C6. The other 56 cells are left blank: they are not needed for this step. Colour A: R1C5 (5), R2C4 (2) and R2C6 (7). Colour B: R1C5 (7), R2C4 (5), R2C6 (8), R2C8 (2) and R2C8 (7). Nothing is removed yet: this step only reads the grid.

Step 4Remove every candidate of the false colour

Colour B is false, so every B candidate goes, whatever its digit and wherever it is: 5 from R2C4, 2 and 7 from R2C8, 7 from R1C5 and 8 from R2C6. Do not stop at the two that clashed in R2C8 — the wrap condemns the whole colour.

Candidates that were never coloured are not part of the proof, and this step leaves them where they are.

Technique: 3D Medusa

Colour B is false: its five candidates are struck out — 5 in R2C4, 2 and 7 in R2C8, 7 in R1C5 and 8 in R2C6.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1	2	4		3		9			8
R2			1		6		3		4
R3	7								5
R4						2
R5				7
R6					1
R7
R8				8
R9						5

Digit already on the grid
Unit holding a conjugate pair (a digit with two places)
Candidate coloured A
Candidate coloured B
Cell with a coloured candidate
Candidate removed
Not needed for this step

Filled in: row 2; box 2; cells R1C1, R1C2, R1C9, R3C1, R3C9, R4C6, R5C4, R6C5, R8C4 and R9C6. The other 56 cells are left blank: they are not needed for this step. Colour A: R1C5 (5), R2C4 (2) and R2C6 (7). Colour B: R1C5 (7), R2C4 (5), R2C6 (8), R2C8 (2) and R2C8 (7). Removed: 2 from R2C8; 5 from R2C4; 7 from R1C5 and R2C8; 8 from R2C6.

Step 5Collect what the wrap unlocks

With colour B gone, every cell of the network is down to one candidate. R2C8 held 2, 7 and 9 and has lost 2 and 7, so it must be 9 — a Naked Single. The other three cells keep only their colour-A candidate, the true colour: R2C4 is 2, R1C5 is 5 and R2C6 is 7.

Four cells settled by one wrap is typical of 3D Medusa: the network reaches several digits, and the false colour takes all of them at once.

Technique: Naked Single

After the wrap: R2C8 has only 9 left, so 9 goes there. R2C4, R1C5 and R2C6 are down to 2, 5 and 7.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1	2	4		3		9			8
R2			1		6		3	9	4
R3	7								5
R4						2
R5				7
R6					1
R7
R8				8
R9						5

Digit already on the grid
The cell left with one candidate
Candidate 9, the digit studied
Digit placed
Not needed for this step

Filled in: row 2; box 2; cells R1C1, R1C2, R1C9, R3C1, R3C9, R4C6, R5C4, R6C5, R8C4 and R9C6. The other 56 cells are left blank: they are not needed for this step. Placed: R2C8 = 9.

Common mistakes

Crossing a link that is not strong

A cell carries the colour from one digit to another only when it has exactly two candidates; with three, two of them can both be false, so opposite colours inside it are a guess. A unit is the same: a digit with three places in it gives no link. Count before every link — exactly two candidates, or exactly two places — or every conclusion after it falls.

Removing only the candidates that clashed

A wrap proves the whole colour false, not just the two candidates that met. Every candidate of that colour goes, in every cell and for every digit the network reached. Stopping at the clash throws most of the move away.

Mixing the colours of two networks

Colours mean something only inside one network, where strong links join every coloured candidate. If you started a second network elsewhere, its A has nothing to do with the first one's A. A wrap needs both clashing candidates in the same network, and a trap needs the digit in both colours of the same network.

Keep going

Simple ColoringChain one digit's strong links and colour them in turn — one colour is true, so a cell seeing both loses the digit.
XY-ChainA run of bivalue cells pins one digit to one end of the chain or the other — every cell that sees both ends loses it.

Learn the move, not just the digit

In a real game, the hint in the app shows the next cell you can fill and explains why the digit must go there. When 3D Medusa cleared the way, the explanation names 3D Medusa as the reason a candidate was ruled out, unless a simpler pattern explains the same removal or another technique of the same level was used first on the way. It does not draw the colours for you: the steps above are how you find them.

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