21 techniques, ordered by depth of reasoning
Work through them top to bottom. Each technique has an example from a real puzzle that walks through every step of the argument, until a digit is placed or a candidate is eliminated.
See it, fill it
1-step reasoning
Always the first thing to look for, and the thing to return to after every elimination at the higher levels.
Full House
Last Digit
A row, column or box has only one empty cell left: fill in the missing digit.
Difficulty
Hidden Single in a Box
Cross-hatching
Pick a digit and a box, cast rays from the copies of that digit, and find the only cell still able to take it.
Difficulty
Hidden Single in a Row or Column
Hidden Single (line)
In a row or column, a digit has exactly one cell left where it can go.
Difficulty
Naked Single
Sole Candidate
A cell sees 8 different digits in its row, column and box, so only one possibility is left.
Difficulty
Eliminate first, then fill
2-step reasoning
When you have exhausted hidden and naked singles and the board still won't move.
Locked Candidates: Pointing
Pointing Pair/Triple · Locked Candidates Type 1
In a box, digit d lies on a single row (or column), so d is removed from the rest of that row (column).
Difficulty
Locked Candidates: Claiming
Box/Line Reduction · Locked Candidates Type 2
In a row (column), digit d lies inside a single box, so d is removed from the other cells of that box.
Difficulty
Naked Pair
Naked Pair · Locked Pair
Two cells in a unit have the same two candidates only, so those digits are removed from the rest of the unit.
Difficulty
Hidden Pair
Two digits can only go in the same two cells of a unit, so every other candidate is cleared from those cells.
Difficulty
Naked Triple
Three cells in a unit share just three candidates between them, so those digits are removed from the rest of the unit.
Difficulty
Hidden Triple
Three digits appear in only the same three cells of a unit, so every other candidate is cleared from those cells.
Difficulty
Chains of deductions
N-step reasoning
When the 2-step techniques can no longer eliminate anything — usually only in hard and very hard puzzles.
X-Wing
X-Wing · 2×2 fish
Digit d in two rows lies only in the same two columns, so d is removed from the other cells of those columns.
Difficulty
Skyscraper
Two parallel strong links share a base, so a cell that sees both tops cannot be d.
Difficulty
2-String Kite
A strong link in a row and one in a column, tied together in a box.
Difficulty
XY-Wing
XY-Wing · Y-Wing
A pivot {x, y} sees two pincers {x, z} and {y, z}, so a cell that sees both pincers cannot be z.
Difficulty
XYZ-Wing
Like an XY-Wing, but the pivot has three candidates {x, y, z}; z is removed only from cells that see all three cells.
Difficulty
W-Wing
Two far-apart cells with the same pair {x, y}, joined by a strong link on x, so a cell that sees both cannot be y.
Difficulty
Swordfish
Swordfish · 3×3 fish
The X-Wing extended to three rows and three columns.
Difficulty
Simple Coloring
Simple Colors · Single's Chains
Join a digit's strong links into a chain and colour it in two alternating colours: exactly one colour is the answer.
Difficulty
Unique Rectangle
Unique Rectangle Type 1
Avoid the pattern that would give a puzzle two solutions: the fourth corner cannot be left with only {x, y}.
Difficulty
BUG+1
Bivalue Universal Grave + 1
Every empty cell has two candidates except one with three: that cell takes the digit appearing three times in its units.
Difficulty
XY-Chain
A chain of linked bivalue cells; if both ends contain z, a cell that sees both ends cannot be z.
Difficulty