Simple Colouring in Sudoku

Specialist S7 Chains

Simple Colouring follows one digit through a network of strong links. Colour linked cells alternately: either every cell of colour A holds the digit and none of B does, or the reverse. Those two possibilities can rule out candidates.

A candidate is a digit that can still go in an empty cell. Cells see each other when they share a row, column or 3×3 box. A strong link means a digit has exactly two possible cells in one such group: exactly one must contain it.

How to spot it

  • Pick one digit. Find rows, columns and boxes with exactly two possible cells for it. Each such pair is a strong link: exactly one of the two holds the digit.
  • Join links that share a cell to make one connected network.
  • Give the two ends of every strong link opposite colours. Follow only these links when assigning colours.
  • Look for one of the three deductions below. Treat separate networks independently: their colour labels have no relationship.

Why it works

Each strong link has exactly one end containing the digit. Following the links therefore gives two possibilities for the whole network: every A cell holds it and no B cell does, or every B cell holds it and no A cell does.

Same colour twice: if two cells of one colour share a row, column or box, that colour cannot hold the digit. Remove the candidate from every cell of that colour.

Sees both colours: a cell outside the network that sees an A cell and a B cell sees the digit in either possibility. Remove the candidate from that outside cell.

Leaves no place: if making one colour contain the digit would leave some row, column or box with no place for it, that colour is impossible. Remove the candidate from every cell of that colour.

Seeing two cells of the same colour is not enough for the outside-cell deduction. Both could belong to the colour that does not hold the digit.

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 a candidate to cross it out; click it again to restore it. Remove the 1 candidates ruled out by this technique, then press Check. This exercise leaves your game unchanged.

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 The connected network is shown in one neutral colour. Follow the strong links for one digit.
  2. Hint 2 The network is split into two colours. The hint asks you to find the relevant case: the same colour twice, an outside cell seeing both colours, or a colour leaving a group with no place for the digit.
  3. Hint 3 The candidates to remove are marked. A contradictory colour loses the digit throughout; an outside cell seeing both colours loses it only in that outside cell.

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 covers one-digit networks of strong links and the three deductions above. Colours belong to one network only; techniques combining different networks or different digits are outside this form.

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 Work on one digit. Find three rows with two or three possible cells each for it, all within the same three columns. Those columns must contain the digit in the selected rows, so remove it from their other cells. The same pattern works with rows and columns reversed.

All Sudoku techniques