Hard killer sudoku techniques
Our killer sudoku books start where most puzzle books stop. Every puzzle needs the advanced and expert moves below, usually several of them, and each one is explained the moment it is used in the walkthrough. Here is the whole ladder, with a one-sentence explanation and real puzzles to see it in.
Killer sudoku adds arithmetic to the logic: cages with totals, no starting digits. The cage techniques (combination limits, the 45 rule, innies and outies, cage splitting) come first; once they have placed a few digits, the ordinary sudoku techniques take over.
Expert
Long chains, almost-locked sets, colouring and (at most one per puzzle) a forcing chain. The steps that make the books extreme.
- Remote Pairs
- A chain of cells that all hold the same pair alternates between the two digits. Cells that see both ends of an odd-length link cannot hold either digit. Used in 221 of our killer sudoku walkthroughs, the key move in 121.
- Finned Fish
- An X-Wing or Swordfish with one extra candidate (the fin). Cells that are in the fish pattern and also see the fin lose the digit. Used in 108 of our killer sudoku walkthroughs, the key move in 43.
- Coloring/Multi-coloring
- Following one digit’s strong links, cells get two colours: one colour is true and the other is false. A cell that sees both colours cannot hold the digit. Used in 279 of our killer sudoku walkthroughs, the key move in 149.
- Forcing Chains
- Try each candidate of one cell in turn. Every choice leads to the same result, so that result must be true. Used in 436 of our killer sudoku walkthroughs, the key move in 137.
Advanced
Fish, wings and short chains. In an Extreme book these come up many times in a single puzzle.
- X-Wing/Swordfish
- A digit fits in only two (or three) places in each of two (or three) rows, and those places line up in the same columns (or the other way round). The digit can be removed from the rest of those columns. Used in 14 of our killer sudoku walkthroughs.
- Skyscraper
- Two rows each have a digit in exactly two places, with one end lined up. One of the two free ends must hold the digit, so any cell that sees both free ends cannot. Used in 77 of our killer sudoku walkthroughs.
- 2-String Kite
- A row and a column each have a digit in exactly two places, linked through one box. One of the two far ends holds the digit, so a cell seeing both ends cannot. Used in 88 of our killer sudoku walkthroughs.
- Empty Rectangle
- In one box a digit is confined to an L or cross shape. Combined with a strong link outside the box, it rules the digit out of one cell. Used in 104 of our killer sudoku walkthroughs.
- XY-Wing
- Three two-candidate cells: a pivot {X,Y} sees two wings {X,Z} and {Y,Z}. Whichever digit the pivot takes, one wing becomes Z, so any cell seeing both wings cannot be Z. Used in 55 of our killer sudoku walkthroughs.
- XYZ-Wing
- Like an XY-Wing, but the pivot holds all three digits {X,Y,Z}. A cell that sees the pivot and both wings cannot be Z. Used in 57 of our killer sudoku walkthroughs.
- WXYZ-Wing
- Four cells holding four digits between them, where only one digit is not confined. That digit is removed from cells that see all its copies. Used in 4 of our killer sudoku walkthroughs.
- Sue de Coq
- Cells where a box meets a row or column hold a set of digits that must split between the box and the line. That rules digits out of both. Used in 18 of our killer sudoku walkthroughs.
- Multi-Sector Locking
- A digit the cage must contain can only go in one row, column or box, so it is ruled out of the rest of that row, column or box (or the other way round). Used in 346 of our killer sudoku walkthroughs.
- Advanced Innies (3+ sectors)
- The 45 rule over three or four rows or columns at once fixes a cell or the total of a small group of cells. Used in 345 of our killer sudoku walkthroughs.
Intermediate
Pairs, triples and box-line interactions. Routine in our books; they clear the ground for the hard steps.
- Naked Pairs/Triples
- Two (or three) cells in one row, column or box share the same two (or three) candidates between them. Those digits must go in those cells, so they can be removed from the rest of that row, column or box. Used in 450 of our killer sudoku walkthroughs.
- Hidden Pairs/Triples
- Two (or three) digits can only go in the same two (or three) cells of a row, column or box. Those cells can hold nothing else. Used in 320 of our killer sudoku walkthroughs.
- Naked/Hidden Quads
- The same idea as pairs and triples, with four cells and four digits. Used in 135 of our killer sudoku walkthroughs.
- Pointing Pairs/Triples
- Inside a box, a digit is confined to one row or column. It must go in that box, so it can be removed from the rest of that row or column. Used in 442 of our killer sudoku walkthroughs.
- Box-Line Reduction
- Inside a row or column, a digit is confined to one box. It must go in that line, so it can be removed from the rest of the box. Used in 403 of our killer sudoku walkthroughs.
- Virtual Cages
- The 45 rule can also give the total of a small group of cells (the innies or outies of a row, column or box). That group then works like an extra cage. Used in 450 of our killer sudoku walkthroughs.
- Cage Min/Max Bounds
- The other cells of the cage cannot add up to enough (or are already too much) whichever digits they take, so this candidate cannot make the total. Used in 449 of our killer sudoku walkthroughs.
- Innies & Outies
- The 45 rule over two neighbouring rows, columns or boxes leaves one cell whose value is fixed by the cage totals. Used in 114 of our killer sudoku walkthroughs.
- Forced Cage Splits
- List every way the empty cells of a cage can make its total with different digits and their current candidates. A candidate used in none of them is ruled out. Used in 450 of our killer sudoku walkthroughs.
Basic
The moves every solver makes without thinking. Our books assume them.
- Naked Singles
- A cell with only one possible digit left must take it. Used in 450 of our killer sudoku walkthroughs.
- Hidden Singles
- When a digit fits in only one cell of a row, column or box, it goes there. Used in 450 of our killer sudoku walkthroughs.
- Cage Sum Arithmetic
- When only one cell of a cage is still empty, it must take whatever is left of the cage total. Used in 450 of our killer sudoku walkthroughs.
- Cage Combination Limits
- Only some sets of different digits add up to a cage total (two cells making 3 can only be 1 + 2). Digits that appear in no such set are ruled out of the cage. Used in 450 of our killer sudoku walkthroughs.
- 45-Rule (Box/Row/Column)
- Every row, column and box adds up to 45. Subtract the cages that lie wholly inside it, and the one cell left over must make up the difference. Used in 388 of our killer sudoku walkthroughs.
The books
Step-by-step solutions: Extreme Killer Sudoku 1 · Extreme Killer Sudoku 2 · Extreme Killer Sudoku 3.