Sudoku Techniques

Also called a sole candidate. After eliminating the digits already present in the same row, column, and box, one cell is left with a single option. That digit must go there. Gently rated puzzles reduce almost entirely to a sequence of naked singles once you pencil in your candidates.

Look at a row, column, or 3×3 box (any one of these is called a "house") and a single candidate digit. If that digit can only be placed in one cell of the house, it belongs there – even if the cell still shows other candidates. Also known as a unique candidate, hidden singles are the most productive basic step because they hide inside cells that look ambiguous at first glance.

Look at a 3×3 box. If every cell that could hold a particular digit sits in the same row (or column), the digit must come from inside the box, so it can be eliminated from every other cell of that row (or column) outside the box. The logic runs outward: box → line. Pointing and claiming are together known as Locked Candidates.

The mirror image of pointing. If every cell in a row (or column) that could hold a digit sits inside the same 3×3 box, the digit must land in one of those cells. Eliminate it from every other cell of that box. The logic runs inward: line → box. Also called box-line reduction (and, with pointing, Locked Candidates).

If two cells in the same house each contain only the same two digits {a, b}, those digits must occupy those two cells in some order. Eliminate both a and b from every other cell in the house. The cells do not need to be adjacent. Naked pairs, triples, and quads are collectively called naked (or disjoint) subsets.

Three cells in a house whose candidates are all subsets of {a, b, c} – each cell can hold any two or all three – together must contain a, b, and c exactly. Eliminate those three digits from the rest of the house. Classic shapes: {ab, ac, bc}, {abc, ab, ac}, {abc, abc, abc}.

Same idea as a triple, one size up. Four cells whose candidates lie entirely within {a, b, c, d} lock those four digits into those four cells. Eliminate a, b, c, d from the remaining cells of the house.

If two digits each appear as candidates in only the same two cells of a house, those cells must hold those two digits – even if extra candidates are written there. Strike every other candidate from both cells. Hidden pairs, triples, and quads are collectively the hidden subsets.

Three digits in a house whose combined candidate positions span exactly three cells must occupy those cells. Remove all other candidates from those three cells. Each digit might only appear in two of the three cells – it is the union that matters.

Four digits whose only positions in a house are the same four cells must fill those cells. Remove every other candidate from those four cells. Quads are rare but the logic is identical to a hidden pair – just larger.

Find two rows where a digit X has only two possible positions, and those positions share the same two columns. The four cells form a rectangle. Whichever diagonal pair holds X, each of the two columns gets exactly one X – so X can be eliminated from all other cells of those columns. The same pattern works with rows and columns swapped. The X-Wing is the smallest "fish"; Swordfish and Jellyfish extend it to three and four lines.

In each of three rows, digit X can only go in cells that together lie in just three columns (a row meets each column once, so that is at most three X-cells per row). Each of those rows still needs an X, and no two of them can sit in the same column, so the three X's land in three different columns – and with only three columns available, they fill all three, one apiece. Those columns are now full of X: it can't appear anywhere else in them, so eliminate X from every other cell of the three columns. The pattern also works with rows and columns swapped. In fish notation this is a 3-fish.

Same counting as a Swordfish, one size up. In each of four rows, digit X can only go in cells that together lie in just four columns (at most four X-cells per row). Each of those rows needs an X, and no two can share a column, so the four X's land in four different columns – and with only four available, they fill all four, one apiece. Those columns are now full of X, so eliminate it from every other cell of the four columns. The same works with rows and columns swapped. Jellyfish – a 4-fish – is the largest fish used here; any 5-row pattern mathematically reduces to a smaller fish.

Begin with an almost-complete X-Wing. For digit X, take two parallel lines (two rows or two columns) where X is confined to two cells each, lining up on the same two crossing lines – except one of the base lines has a third spot for X, the fin, tucked into the same box as one of its corners. The fin breaks the clean sweep, so X can't be cleared from a whole crossing line. One cell is still doomed either way: If X avoids the fin you have a true X-Wing and its eliminations hold; if X sits in the fin, it's trapped inside that box. The cell where a crossing line meets the fin's box loses on both branches, so X comes out of it. Finned fish are among the most common expert patterns.