How to Solve Sudoku: A Practical Guide from Scanning to X-Wing

How to read the grid before you start

A standard Sudoku is a 9 by 9 grid split into nine 3 by 3 boxes. Every row, every column, and every box must end up containing the digits 1 through 9 exactly once. That single rule is the source of every technique below. When a guide says a digit is eliminated from a cell, it always means the digit already appears somewhere in that cell's row, column, or box, or that logic proves it must go elsewhere in one of them.

It helps to have names for positions. In this guide rows are numbered 1 to 9 from top to bottom, columns 1 to 9 from left to right, and a cell is written as r4c7 for row 4, column 7. Boxes are numbered 1 to 9 reading like a book, so box 1 is the top-left box and box 5 is the center.

Before writing anything, spend a few seconds noticing which digits appear most often among the givens. A digit that already appears six or seven times is usually the easiest place to start, because only two or three spots remain for it and scanning will find them quickly.

Scanning: place digits without writing notes

Scanning, sometimes called cross-hatching, means picking one digit and one box, then using the rows and columns that pass through that box to rule out cells. It is the fastest technique and solves a large share of easy puzzles on its own.

Worked example: you want to place a 5 in box 1. You notice a 5 in row 2 (it sits in box 2) and a 5 in row 3 (it sits in box 3). That rules out every cell of box 1 in rows 2 and 3, so the 5 must go in row 1. You also see a 5 in column 1, down in box 7, which rules out r1c1. Only r1c2 and r1c3 remain. If r1c2 is already filled with another digit, the 5 goes in r1c3.

Work one digit through all nine boxes before moving to the next digit. That rhythm keeps you from re-checking the same box over and over, and every placement you make creates new scanning opportunities for other digits.

Pencil marks and naked singles

When scanning stops producing placements, switch to pencil marks, also called notes or candidates. For each empty cell, write the small digits that are still allowed there. It feels slow at first, but it turns a hard puzzle into a set of visible patterns, and every technique from here on depends on accurate notes.

A naked single is a cell with only one candidate left. Worked example: r5c5 sits in a row that already contains 1, 2, and 7, a column that contains 3, 4, and 6, and a box that contains 5 and 8. Combine those and eight digits are taken, so the only possibility for r5c5 is 9.

The most common source of errors is stale notes. Every time you place a digit, immediately erase it from the candidates of every cell in the same row, column, and box. Many players build a habit of doing this before anything else, because one missed erasure can send a whole solve down the wrong path.

Hidden singles: the digit with only one home

A naked single looks at one cell and asks which digits fit. A hidden single flips the question: it looks at one row, column, or box and asks where a particular digit can go. If a digit fits in only one cell of a unit, it belongs there, even if that cell still shows several candidates.

Worked example: in row 4, the candidates for the empty cells are {2,8}, {2,6,8}, {1,8}, {1,2}, and {3,8}. The digit 6 appears in only one of those lists, the cell with {2,6,8}. Row 4 must contain a 6 somewhere, so that cell is 6, and you can erase its other candidates.

Hidden singles are easy to miss because the cell looks crowded. A good routine is to sweep each box digit by digit, asking whether that digit has exactly one possible cell left. Scanning from the previous section is really a hidden single search done without notes.

Pairs: naked and hidden

Pairs are the first technique that removes candidates without placing a digit, and they are what separates easy puzzles from medium ones.

A naked pair is two cells in the same unit that each contain exactly the same two candidates. Worked example: in column 6, r2c6 and r7c6 both show only {3,7}. Between them, those two cells must hold the 3 and the 7 for that column, in some order. So no other cell in column 6 can be 3 or 7, and you can erase both digits from every other cell in the column. If the two cells also share a box, erase them from the box too.

A hidden pair is the reverse. Two digits appear as candidates in only two cells of a unit, even though those cells have extra candidates. Worked example: in box 9, the digits 1 and 4 appear only in r7c8, which shows {1,4,6,9}, and r9c7, which shows {1,2,4}. Those two cells must take the 1 and the 4, so you can strip the extra candidates and reduce both cells to {1,4}.

The same logic extends to triples, three cells sharing three candidates between them. Look for pairs first, since they are far more common and easier to spot.

Pointing pairs and box-line reduction

These two techniques use the overlap between a box and a row or column. They are worth learning together because they are mirror images of each other.

Pointing pair: inside one box, a digit is possible only in cells that lie on the same row or column. Worked example: in box 5, the candidate 2 appears only in r5c4 and r5c6. Box 5 must contain a 2, and it will be on row 5 either way. That means row 5's 2 is inside box 5, so you can erase 2 from every row 5 cell outside the box, such as r5c1 or r5c9.

Box-line reduction, sometimes called claiming, runs the other direction. Inside one row or column, a digit is possible only in cells that sit within a single box. Worked example: in column 3, the candidate 8 appears only in r1c3 and r2c3, both in box 1. Column 3's 8 must be in box 1, so box 1's 8 is in column 3, and you can erase 8 from every other cell of box 1.

An introduction to the X-Wing

The X-Wing is the first pattern most players learn that spans more than one unit. It looks for a single digit forming a rectangle across two rows and two columns.

Worked example: in row 2, the digit 7 is possible only in columns 3 and 8. In row 6, the digit 7 is also possible only in columns 3 and 8. Row 2's 7 must go in c3 or c8, and row 6's 7 must go in the other one, since two 7s cannot share a column. Either way, columns 3 and 8 each get their 7 from rows 2 and 6. So you can erase 7 from every other cell in column 3 and column 8.

The pattern works the same way with columns as the base: if a digit is restricted to the same two rows in two different columns, eliminate it from the rest of those rows. X-Wings are uncommon in easy puzzles, so only go looking for one after singles, pairs, and pointing pairs have all run dry.

A routine that keeps a solve moving

Techniques are only half of solving well. The other half is the order you check them, because the simple ones fire far more often. When you get stuck, step back down the list rather than staring at one corner.

If you are practicing, Lumen Sudoku's Classic mode has eight verified puzzles, each with one solution, plus notes, undo, and hints that explain the reasoning behind a step instead of just filling a cell. Try to name the technique yourself before you ask for a hint, then check whether the explanation matches.

1. Scan each digit through every box.
2. Fill in complete pencil marks once scanning stalls.
3. Check for naked singles, then hidden singles in every row, column, and box.
4. Look for naked and hidden pairs.
5. Check pointing pairs and box-line reductions.
6. Only then search for an X-Wing, and return to singles after every elimination.