Chains of deductions
N-step reasoning
Hard puzzles need longer arguments: "if cell A is x then cell B cannot be y, so cell C must be y, so…". The techniques at this level are such chains packaged into recognisable patterns: fish (X-Wing, Swordfish), wings (XY-Wing, W-Wing), chains and colouring, plus tricks based on the uniqueness of the solution.
All of them are built from two kinds of link. A strong link: in a unit, digit d has exactly two places — if one is not d, the other certainly is. A weak link: two candidates cannot both be true — if one is true, the other is false. A valid chain always alternates "if not… then…" (strong) and "if… then not…" (weak).
When every pattern has run out, players use forcing chains: try each option of a cell in turn and follow the consequences; if every branch leads to the same conclusion, that conclusion is true. It is still logic, not guessing.
Where are the strong links?
List the units with only two places for a digit, and the cells with only two candidates. They are the raw material of every chain.
If this end is false, what about the other?
Walk the chain from one assumption. If the two ends force one of them to be true, a cell that sees both ends is eliminated.
When to use it: When the 2-step techniques can no longer eliminate anything — usually only in hard and very hard puzzles.
11 techniques
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