Cheat sheet
One rule per technique. Each thumbnail shades the cells that form the pattern in that technique's example.
1-step reasoningSee it, fill it
Full HouseLast Digit
A unit already holds 8 digits → its last empty cell takes the 9th.
Hidden Single in a BoxCross-hatching
In a box, digit d has only one cell no ray reaches → that cell is d.
Hidden Single in a Row or ColumnHidden Single (line)
In a row (column), digit d has only one valid cell → that cell is d.
Naked SingleSole Candidate
A cell's row + column + box contain 8 different digits → the cell is the 9th.
2-step reasoningEliminate first, then fill
Locked Candidates: PointingPointing Pair/Triple
Box B has d only on row R → remove d from row R outside box B.
Locked Candidates: ClaimingBox/Line Reduction
Row R has d only inside box B → remove d from box B outside row R.
Naked PairNaked Pair
Two cells in a unit hold only {x, y} → remove x and y from the other cells of the unit.
Hidden Pair
In a unit, x and y appear only in cells A and B → A and B keep only {x, y}.
Naked Triple
Three cells in a unit whose candidates together are {x, y, z} → remove x, y, z from the rest of the unit.
Hidden Triple
In a unit, x, y, z lie only in 3 cells → those 3 cells keep only {x, y, z}.
N-step reasoningChains of deductions
X-WingX-Wing
Two rows have d only in the same two columns → remove d from those columns in every other row (rows ↔ columns also works).
Skyscraper
Two rows each with only 2 places for d, one end of each in the same column → remove d from cells that see both other ends.
2-String Kite
A row and a column each with only 2 places for d, one end of each in the same box → remove d from cells that see both other ends.
XY-WingXY-Wing
Pivot {x, y}, pincers {x, z} and {y, z} → remove z from cells that see both pincers.
XYZ-Wing
Pivot {x, y, z}, pincers {x, z} and {y, z} → remove z from cells that see the pivot and both pincers.
W-Wing
Two {x, y} cells + a strong link on x joining them → remove y from cells that see both {x, y} cells.
SwordfishSwordfish
Three rows have d only within the same three columns → remove d from those columns in every other row.
Simple ColoringSimple Colors
A cell that sees both colours → remove d. Two cells of the same colour that see each other → that colour is false.
Unique RectangleUnique Rectangle Type 1
3 corners of a rectangle (2 rows, 2 columns, 2 boxes) hold only {x, y} → remove x and y from the fourth corner.
BUG+1Bivalue Universal Grave + 1
All cells have 2 candidates, one cell has 3 → it takes the digit that appears 3 times in its row, column and box.
XY-Chain
A chain of bivalue cells whose two ends both contain z → remove z from cells that see both ends.
Three things to remember
- Always try the simplest technique first. After every elimination, look for singles again.
- Strong link: “if not A, then B”. Weak link: “if A, then not B”. Every chain alternates the two.
- The most common conclusion: a cell that sees both ends of an “at least one is d” pair cannot be d.