Advanced Sudoku: X-Wing, Swordfish, Chains

X-wing

Pick a digit, say 4. Find two rows in which 4 is a candidate in exactly two cells each, and those cells sit in the same two columns. The four cells form a rectangle. In each row the 4 must be in one of the two cells, so the two 4s occupy the two columns between them, one each. Therefore 4 can be removed from every other cell in those two columns.

The same pattern works with columns as the base and rows as the cover. X-wings are the most common advanced step on hard days and are worth a dedicated scan: for each digit, look for rows with exactly two candidates.

Swordfish

A swordfish is an X-wing with three rows and three columns. Find three rows in which a digit's candidates all fall within the same three columns (each row needs two or three of them, not necessarily all three). The digit must take one cell in each of those rows, using all three columns once each, so the digit can be removed from the rest of those three columns. Swordfish are rarer and harder to see; look for them only when X-wings fail, and only for digits with few candidates left.

Skyscraper

A near miss of an X-wing. Two rows each have a digit in exactly two cells, sharing one column but not the other (the two "roofs" are in different columns). The two cells in the shared column cannot both be true, so at least one of the roof cells is true. Any cell that sees both roofs (shares a house with each) cannot hold the digit. The skyscraper is worth knowing because it turns many "almost X-wings" into eliminations.

XY-wing

Find a cell with exactly two candidates, X and Y: the pivot. Find two more two-candidate cells that each see the pivot: one holding X and Z, the other holding Y and Z. Whatever the pivot is, one of the wings must be Z: if the pivot is X, the XZ wing is Z; if the pivot is Y, the YZ wing is Z. So Z can be removed from any cell that sees both wings. The pattern is easy to find once pencil marks are complete: look for bivalue cells and follow their shared candidates.

Unique rectangle

Every proper sudoku has one solution, and that fact can be used. If four cells forming a rectangle across exactly two boxes all hold the candidates 1 and 2, and nothing else, the puzzle would have two solutions (the 1s and 2s could swap). So a proper puzzle cannot end up there. When three corners of such a rectangle hold only 1 and 2, and the fourth holds 1, 2 and 3, the fourth corner must be the 3. The Daily Sudoku always has a unique solution, so this reasoning is safe here.

Simple colouring

Pick a digit. Wherever a house has exactly two candidates for it, the two cells are linked: one is true and the other false. Colour one cell A and the other B, and follow the links across the grid, alternating colours. Two outcomes give eliminations:

• If two cells of the same colour share a house, that colour is false everywhere, and the other colour is true everywhere.
• Any uncoloured cell that sees both an A cell and a B cell cannot hold the digit, because one of the two is certainly true.

Colouring is the gentlest form of chaining and solves many hard grids on its own.

Forcing chains, briefly

When nothing else works, take a bivalue cell and follow the consequences of each of its two values a few steps. If both paths place the same digit in some cell, that placement is true regardless; if both paths remove the same candidate, it is gone. This is still logic, not guessing, because you are proving something about every case. Keep chains short, and prefer the named patterns above, which are chains that other solvers have already proved for you.

A scanning order for hard days

1. Complete the pencil marks fully and check them.
2. Singles, pairs, triples, pointing and claiming, until nothing moves.
3. X-wings for each digit, then skyscrapers.
4. XY-wings using the bivalue cells.
5. Unique rectangles wherever a 2 × 2 of identical pairs appears.
6. Simple colouring on the digit with the fewest candidates.
7. Swordfish, then short forcing chains.

After any elimination, return to step 2. The hard part of advanced sudoku is not understanding a pattern; it is remembering to look for it.

Common mistakes

• Testing patterns on incomplete pencil marks. Every technique here depends on knowing all the candidates in a house.
• Confusing X-wing bases and covers. The rows (or columns) with exactly two candidates are the base; eliminations happen in the covering columns (or rows), never in the base.
• Using uniqueness on a puzzle that might have several solutions. Safe on the Daily Sudoku; not safe on a hand-made grid from an unknown source.
• Chasing swordfish before X-wings. Swordfish are rare; X-wings and skyscrapers do most of the work.

Glossary

• Bivalue cell: a cell with exactly two candidates.
• Base and cover: in a fish pattern, the houses that confine the digit (base) and the houses it is removed from (cover).
• Fish: the family of X-wing (2 × 2), swordfish (3 × 3) and larger patterns.
• Sees: two cells see each other when they share a row, column or box.
• Strong link: two cells that are the only candidates for a digit in a house (one must be true).