Sudoku Solving Techniques

From basic last-cell logic to advanced patterns like X-Wing and Swordfish.

Use notes (candidates) to spot these patterns. Start with the basics and work up.

In a row, column, or 3×3 box, if eight cells are already filled and only one is empty, that empty cell must contain the one digit that is still missing from 1–9. Place that number in the only free cell.

Watch video tutorial (YouTube)

Same idea as last free cell: within a single row, column, or box, only one cell is still empty. The digit that has not yet been placed in that unit must go in that remaining cell. This is the simplest fill-in rule.

Watch video tutorial (YouTube)

For one empty cell, look at its row, column, and box. If eight of the nine digits 1–9 already appear in those units, the only digit left is the “last possible number” for that cell. Also called sole candidate or naked single when you use pencil marks: the cell has only one candidate left.

When you write candidates (notes) in empty cells, a cell that has only one candidate is an obvious single. That digit must go there: no other cell in the same row, column, or box can have that number. Place it and remove that candidate from peers. This is one of the most common techniques.

Watch video tutorial (YouTube)

Two cells in the same row, column, or box that both have exactly the same two candidates (e.g. 4 and 7). Those two digits must occupy those two cells, so you can remove 4 and 7 as candidates from every other cell in that row, column, or box. This often clears the way for singles.

Watch video tutorial (YouTube)

Three cells in the same row, column, or box that together contain only three candidates (e.g. 2, 5, 8), even if each cell doesn’t have all three. Those three digits must go in those three cells, so you can remove 2, 5, and 8 from all other cells in that unit. Extends the idea of naked pairs.

Four cells in the same row, column, or box that together contain only four candidates. Those four digits must occupy those four cells, so you can remove those candidates from every other cell in that unit. Rare but useful in harder puzzles.

Watch video tutorial (YouTube)

For a given digit (e.g. 6), look at one row, column, or box. If that digit can only go in one cell in that unit (all other cells in the unit already have 6 or can’t have 6), that cell is a hidden single for 6. Place 6 there. The cell might still have other candidates until you eliminate them; the key is that 6 has only one place in that unit.

Two digits (e.g. 3 and 9) that in a row, column, or box can only appear in the same two cells. Those two cells must contain 3 and 9, so you can remove all other candidates from those two cells. Helps simplify the grid when cells have many candidates.

Watch video tutorial (YouTube)

Three digits that in a row, column, or box can only appear in the same three cells. Those three cells must contain those three digits, so you can delete any other candidates from those cells. Harder to spot than hidden pairs but very useful in medium and hard puzzles.

Watch video tutorial (YouTube)

Four digits that in a row, column, or box can only appear in the same four cells. Those cells must contain those four digits; remove all other candidates from them. Uncommon but can crack tough spots.

Watch video tutorial (YouTube)

If a digit in a 3×3 box appears as a candidate only in one row or one column of that box, then that digit cannot appear elsewhere in that row or column outside the box. So you can remove that candidate from all other cells in that row or column. The box “points” at the row or column, limiting where the digit can go.

Same idea as pointing pairs: a digit in a box has candidates only in one row or one column within the box. That digit is restricted to that row or column, so remove it as a candidate from the rest of that row or column outside the box. With three cells instead of two, the logic is the same.

Watch video tutorial (YouTube)

The reverse of pointing: if a digit in a row or column can only appear inside one 3×3 box (because the rest of the row or column is blocked), then that digit can be removed as a candidate from all other cells in that box. The row or column “claims” the digit for that box.

For one digit: if it appears as a candidate in exactly two cells in each of two rows, and those four cells form a rectangle (same two columns), then that digit cannot appear anywhere else in those two columns. So you can remove it from all other cells in those columns. Same logic applies if you swap “rows” and “columns.” The pattern looks like an X when you mark the four corners.

Uses three cells and three candidates. One “pivot” cell has two candidates (e.g. AB). Two “wing” cells each share one candidate with the pivot (e.g. AC and BC) and see each other. If a cell sees both wings and can have C, you can remove C from it, because either the pivot is A (so one wing is C) or the pivot is B (so the other wing is C). Named after the Y shape formed by the three cells.

An extension of X-Wing to three rows (or three columns). For one digit: if it is limited to two or three columns in each of three rows, and those columns line up in a specific way, you can remove that digit from other cells in the involved columns. The pattern involves three rows and three columns; more complex to spot than X-Wing but powerful in hard puzzles.

Watch video tutorial (YouTube)

Same as Y-Wing: a three-cell pattern (pivot with two candidates, two wings each sharing a candidate with the pivot). Any cell that sees both wings and could hold the “common” candidate can have that candidate removed. Often listed separately from Y-Wing but the logic is identical.

For one candidate: color the cells that can hold it in two alternating colors (e.g. chain of cells that “see” each other). If two same-colored cells see each other, that color cannot be the true placement, so you can eliminate that candidate from all cells of that color. If a cell that is not in the chain sees two different-colored cells, the candidate can be removed from that cell.

A well-designed Sudoku has exactly one solution. If four cells form a rectangle (same two rows and same two columns) and share two boxes, and two of the cells are filled while the other two have the same two candidates, then choosing the “wrong” option would create a symmetrical, interchangeable pattern and two solutions. So you can often remove one of the candidates from one of the empty cells to force the unique solution. Several variants (Type 1, 2, 3, etc.) refine when and what you can remove.
How to Play Play Sudoku