Sudoku & variants — core techniques

Standard Sudoku places 1–9 once each in every row, column and 3×3 box. Killer keeps those houses and adds irregular cages with sum targets; digits usually cannot repeat inside a cage. Classic sites sit under Sudoku; Killer lives with variants and arithmetic lists.

Jigsaw (irregular) Sudoku replaces 3×3 boxes with nine-cell polyomino regions. Line–region intersections matter more; solvers often use leftover-set comparisons (sometimes called the Law of Leftovers) between complete lines and partial jigsaw zones.

Thermometer shapes force a strictly increasing sequence from bulb to tip. Candidates that break the increase can be dropped early; the rest of the grid is still standard Sudoku houses.

A circled bulb digit equals the sum of the digits on its arrow. Intersecting arrows share algebra with ordinary row/column/box constraints — treat them as extra sum equations, not decoration.

Both main diagonals become extra houses that must also contain 1–9 once each. Candidate work on the diagonals is the same skill as on a row.

For standard 9×9 Sudoku, published results show no uniquely solvable puzzle with fewer than 17 clues, and uniquely solvable 17-clue grids exist. That floor is about uniqueness of a finished grid — not “difficulty” and not a rule your daily app must follow.

If digit d is limited to the same two columns in two rows (or the transpose), those four cells form an X-Wing: d can be eliminated from the rest of those two columns (or rows). It is a fish of size 2.

A naked pair is two cells in one house that contain only the same two candidates — those two digits leave every other cell in the house. A hidden pair is two digits that appear only in the same two cells of a house — other candidates can be stripped from those two cells.

Under a unique-solution assumption, four cells that would allow a deadly swap of two digits can force eliminations. If a rectangle of two digits spans two boxes in two rows and two columns, extra candidates on the rectangle are often the only way to keep uniqueness — use that to drop candidates carefully.

If digit d in a box appears only in one row (or column), clear d from that row outside the box — a pointing pair. In Killer, any full house sums to 45; comparing cage sums inside a house reveals leftovers for unsummed cells.

Skyscraper, wings, Empty Rectangle, Sue de Coq, Medusa coloring, Exocet and related patterns are collected in the Advanced Sudoku patterns guide. Open Solvers & tools when you want a named step explained on a live grid.