SUDOKU

Very hard Sudoku: from patterns to chains

Very hard puzzles need everything from the hard level plus techniques that link several cells together. The key skill is following how one candidate affects another across the grid.

What changes at this level

Pairs, locked candidates and X-Wings still do most of the work, but at some point none of them applies. The remaining eliminations come from wings, uniqueness arguments and short chains. These look complicated at first, but each one is a short sequence of simple "if this, then that" steps.

Techniques you will need

• XY-Wing (4.2) and XYZ-Wing (4.4): a pivot cell and two pincer cells that share candidates. Whatever the pivot turns out to be, one pincer holds the common digit, so any cell that sees both pincers loses it.
• Unique rectangles (4.5–5.0): a well-made Sudoku has one solution, so four cells forming a rectangle across two boxes cannot end up with the same two digits. That rules out a candidate.
• Jellyfish and quads (5.0–5.4): the four-row and four-cell versions of Swordfish and pairs. They are rare, but worth checking when nothing else works.
• Bivalue universal grave (BUG, 5.6–6.0): if every unsolved cell has two candidates except one cell with three, the extra candidate in that cell must be the answer.
• X-chains and simple coloring (from about 6.5): follow one digit through strong links, places where it has only two positions in a unit. Alternate two colors along the chain; a cell that sees both colors cannot hold that digit.

A routine for very hard puzzles

1. Exhaust singles, locked candidates and pairs first, as on the hard level.
2. Highlight the cells with exactly two candidates. Wings and chains are built from them.
3. For each digit, mark the units where it has only two positions; these strong links are the raw material for chains.
4. Look for an XY-Wing around each two-candidate cell before trying longer chains.
5. Check rectangles of two-candidate cells for uniqueness eliminations.

Practical advice

Keep notes tidy and complete; chains are impossible to see with missing candidates. Work through one digit at a time, and when you find an elimination, go back to the simple techniques: one removal often unlocks a run of singles.