Simple Colouring in Sudoku

Specialist S7 Chains

Simple Colouring looks at one digit at a time. When a row, column or box has exactly two places left for that digit, the two cells are strongly linked: exactly one of them holds it. Join such pairs together and you get a network that can be coloured alternately in two colours, and then exactly one of the colours is the true one. That is enough to remove candidates, even though you do not know which colour is which.

How to spot it

  • Pick one digit and look at nothing else.
  • Find rows, columns and boxes with exactly two cells left where the digit can go. Each such pair is a strong link.
  • Join pairs that share a cell, and they grow into a network.
  • Colour the network alternately: a neighbour takes the opposite colour to the cell you came from. If that works out, you have two colours to reason with.
  • Treat each network separately. Two networks for the same digit share no colour, so their colours can never be compared across them.

Why it works

A strong link means exactly one of the two cells holds the digit - never both, and never neither, because the unit must hold the digit somewhere. Once the colours alternate along the links, the whole network inherits that alternation: every cell of one colour holds the digit, and no cell of the other does.

You do not know which colour is the true one, and you do not need to. Knowing that it is one of them is enough, because you can then rule out whatever would be wrong either way.

Rule 2, the same colour twice in a unit: two cells in one row, column or box cannot hold the same digit. If one colour appears twice in a single unit, that colour cannot be the true one, and the digit leaves every cell of that colour.

Rule 4, both colours from outside: if a cell outside the network sees a cell of one colour and a cell of the other, it loses the digit. One of the two colours is true whichever it is, so the cell sees the digit either way.

Rule 7, one colour empties a unit: suppose one colour were the true one. If the digit then vanishes from every place in some row, column or box, that colour cannot be true, because the unit must hold the digit somewhere. The digit leaves every cell of that colour.

Watch the trap: two cells of the same colour prove nothing. That colour may be the false one, in which case neither of them holds the digit. Only two different colours - or a colour that contradicts itself - license an elimination.

A real example from Sudokuno

The board below is a real Sudokuno puzzle, shown exactly as it stands just before Simple Colouring applies. Everything up to this point was solved with ordinary techniques.
4
5
7
1
6
8
4 9
7 4
6
8
4 1 3
8 9 6 5
9
6 4
7 1
7 6
1
9 4 8
3
1
2 8
6
4 1 7
2
9 8
1

Click away the candidates the technique removes - 1 of them - then press Check. Nothing is saved, and your own game is untouched.

Example: Simple Colouring - Sudokuno - not yet published in the puzzle picker

  • Candidate 3
  • Houses: row 2, column 7 and box 3
  • Colour A: R2C7, R5C4 and R6C9
  • Colour B: R5C7, R6C5 and R8C9
  • Removes 3 from R8C4
  • Step 18 of 58 in the solve

Key: colour A colour B target cells

How the hint helps you

Sudokuno's hint reveals the structure in three steps, without giving away the digit straight away.

  1. Hint 1 Follow one digit through the board. Where a row, column or box has exactly two places left for the digit, those two cells are strongly linked. Sudokuno highlights the whole network already here, in one neutral colour.
  2. Hint 2 Now the network splits into the two colours that alternate along the links, so exactly one colour is the true one. The hint says what to look for: the same colour twice in a unit, a cell outside that sees both colours, or a colour that would empty a unit.
  3. Hint 3 The hint shows which candidates can go, and why. Rules 2 and 7 clear the whole colour that cannot be true; Rule 4 removes the digit from the cell outside the network.

The hint's colours are just the two sides of the network - they are not the same thing as your own colouring tool, and they never touch the colours you painted yourself. Which side is called what does not matter; the alternation between them does the work.

Ready to try it yourself?

You will get a real Sudokuno puzzle at Specialist level S7, where Simple Colouring is needed to finish it.

Practise Simple Colouring

What Sudokuno covers

Sudokuno colours one digit at a time, and only networks built from strong links. Multi-Colouring and 3D Medusa, which link several digits together, are not implemented, and neither are X-Cycles or longer inference chains.

Where to go next

  • X-Wing An X-Wing appears when one candidate has exactly two possible cells in each of two different rows, and both pairs sit in the same two columns. The four cells form a rectangle. The candidate must then occupy two opposite corners, one per row, so it can be removed from every other cell in those two columns.
  • Swordfish A Swordfish is the same idea as an X-Wing with three lines instead of two. One candidate has only two or three possible cells in each of three rows, and together those cells touch exactly three columns. The candidate is then confined to those three columns and can be removed from every other cell in them.

All Sudoku techniques