Proven Sudoku

Techniques / Single-digit patterns

Single-digit patterns: skyscrapers, kites and colouring

One digit, chained through the houses where it has only two homes.

8 techniques in this family, introduced at tier 4, 5 with a worked example from a real board. How to read a sudoku grid explains the notation used below.

What these have in common

Like fish, these look at one digit alone. Unlike fish, they work by following a chain of either-or facts rather than by counting rows against columns.

Why the logic holds

If a digit has exactly two possible cells in some house, one of them is true and the other is false. That is a strong link, and it is the only ingredient. Chain two strong links together through a shared house and you can prove that at least one of the two far ends is true, which means any cell seeing both ends cannot hold the digit.

When to reach for it

When fish come up empty. In practice these patterns are more common than Swordfish and easier to see, so many solvers reach for them first.

How the members differ

They are the same chain in different geometry. A Skyscraper is two strong links in parallel lines with one end aligned. A 2-String Kite is two strong links meeting in a box. A Turbot Fish is the general two link case that both are instances of. An Empty Rectangle uses a box where the digit's options form an L shape. Simple Colouring drops the geometry and just paints the alternation across the whole grid, which finds all of them at once; Multi-Colouring and X-Colouring extend that to more than one chain, and 3D Medusa extends it again to alternate between digits inside a cell as well as cells inside a house.

The mistake to avoid

With colouring, calling a contradiction too early. Two cells of the same colour only conflict if they genuinely share a house, and two colours in one cell only conflict if they are the same digit. Check the geometry before you conclude a colour is false.

Start with Skyscraper. Pick a digit.

Every technique in this family