The Complete Guide to Sudoku

Sudoku is a game of elimination rather than memorization. Start with the rules and the cells with the fewest possibilities. Add pencil marks when scanning stops making progress, then move from singles to patterns that remove candidates across a box, line, or the whole grid.

Part 1 · How to Play

The classic rule

Sudoku is played on a 9×9 grid. Fill every empty cell so that each row, each column, and each of the nine 3×3 boxes contains the digits 1 to 9 exactly once. A puzzle begins with some digits already given. A properly made puzzle has one completed grid that satisfies the rule.

The digits are labels, not quantities. Classic Sudoku uses no addition, subtraction, or comparison; nine letters or nine colors could represent the same puzzle.

Rule questions that often cause confusion

• A digit cannot appear twice in the same row, column, or 3×3 box. Across a completed grid, each digit appears nine times: once in every row, once in every column, and once in every box.
• The two long diagonals have no special constraint in classic Sudoku. A variant that requires 1 to 9 on the diagonals is usually identified as Sudoku X or Diagonal Sudoku.
• Equal digits cannot touch edge to edge because those cells share a row or column. They may touch at a corner when they are in different 3×3 boxes.

Classic Sudoku and Killer Sudoku

Killer Sudoku keeps every classic rule and adds cages. The digits inside a dashed cage must add up to the small total printed in its corner, without repeating within that cage. Cage totals can supply the information that given digits provide in a classic puzzle, so a Killer puzzle may begin with few or no givens.

Candidates and notes

A candidate, or pencil mark, records that a digit is still possible in a cell. Notes are bookkeeping rather than a rule, and an easy puzzle may be solved without them.

In this game, entering a digit clears that candidate from the notes of the 20 cells that share its row, column, or box. Turn Notes on with the Notes button or the M key to enter pencil marks instead of answers. Auto Notes fills every empty cell with the digits its row, column, and box still allow.

Start writing candidates where a row, column, or box is nearly complete. Short candidate lists make pairs and other patterns easier to see.

Part 2 · Strategies

Start by scanning

Instead of choosing an empty cell and asking what belongs there, choose a digit and ask where it is still allowed to go.

1. Follow a digit that already appears often. Each copy blocks a row, a column, and a box.
2. Look at the row, column, or box closest to completion.
3. Where those two views meet, there may be only one place left for the digit. Place it and scan again.

Naked and hidden singles

Naked Single. Remove every digit already present in a cell’s row, column, and box. If one candidate remains, that cell is forced.

Hidden Single. Choose a digit and inspect one row, column, or box. If only one cell in that unit can take the digit, place it there even if the cell still shows several other candidates.

Singles place a digit directly. The techniques that follow usually remove candidates until another single appears.

Pairs and triples

Naked Pair. If two cells in one unit contain the same two candidates, those two digits must occupy the two cells. Remove both candidates from every other cell in the unit.

Hidden Pair. If two digits can appear only in the same two cells of a unit, remove all other candidates from those cells.

Triple. Three cells whose combined candidates contain only three digits work the same way. The cells do not need identical lists: {2,7}, {7,9}, and {2,9} form a triple because their union is {2,7,9}.

For a self-contained pair example, suppose two cells in column 9 are both {5,8}. The other cells in that column can lose 5 and 8 even though it is not yet known which member of the pair holds which digit.

Locked candidates: where a box meets a line

Pointing. If every possible location for one digit inside a box lies on the same row or column, remove that digit from the rest of that line outside the box. For example, if the candidates for three empty cells in a box are {5,9}, {3,5}, and {3,9}, and the two cells containing 9 lie in the same column, every other cell in that column outside the box can lose 9.

Box–line reduction. Apply the same relationship in reverse. If every place for a digit in one row or column lies inside a single box, remove that digit from the other cells of the box.

X-Wing and Swordfish

X-Wing. Find a digit with exactly two possible cells in each of two rows, with the candidates falling in the same two columns. Whichever arrangement is correct, those two rows occupy the digit in both columns, so remove it from the other cells in those columns. Rows and columns can be swapped in the description.

Swordfish. Extend the same idea to three lines. If every candidate for one digit in three rows lies within the same three columns, remove that digit from the other cells in those columns. Each row does not have to contain exactly two candidates.

XY-Wing

An XY-Wing uses a pivot with candidates {X,Y} and two cells the pivot can see with candidates {X,Z} and {Y,Z}. All three cells have two candidates. If the pivot is X, the {X,Z} wing must be Z; if the pivot is Y, the {Y,Z} wing must be Z. Either way one wing contains Z, so any cell that can see both wings can lose Z.

Part 3 · Frequently Asked Questions

How is difficulty determined?

This game uses the target number of starting clues for its four levels: about 44 for Easy, 34 for Medium, 28 for Hard, and 24 for Expert. Fewer clues generally require a longer chain of elimination, but their positions matter as much as their count. The generator checks that a puzzle has one solution; it does not separately grade which human solving techniques are required, so puzzles in the same level can feel different.

Why should a Sudoku have one solution?

A properly made Sudoku has exactly one solution. A grid with two or more solutions is treated as flawed rather than simply harder because some cells would have no reason to prefer one valid digit over another. This game checks for a single solution while generating each puzzle.

A single solution means luck is not required, but the reasoning may still be long. On a difficult grid, testing an assumption and following it to a contradiction can be a practical shortcut rather than a random choice.

What is the smallest number of clues?

For a classic 9×9 Sudoku with one solution, the known minimum is 17 clues. Gary McGuire, Bastian Tugemann, and Gilles Civario used an exhaustive computer search to establish that no 16-clue puzzle has a unique solution. Their result appeared in Experimental Mathematics; thousands of 17-clue puzzles with one solution are known. Seventeen clues do not by themselves guarantee a unique solution (McGuire, Tugemann, and Civario, “There is no 16-clue Sudoku”).

The number of clues does not determine difficulty by itself. Published puzzles usually use more than 17, and this game’s Expert level targets about 24 givens.

What should I check when the puzzle feels impossible?

On paper, an earlier wrong entry can distort every deduction that follows. In this game, Auto-Check Mistakes is on by default and marks a digit that disagrees with the solution when it is entered. If that setting is off, only duplicate digits are flagged, so review the entries you were least certain about or undo to the last point you trust.

Then work through this order:

1. Exhaust naked and hidden singles.
2. Add or refresh candidates in nearly complete units.
3. Look for naked pairs, hidden pairs, and triples.
4. Compare each box with its rows and columns, then each line with its boxes.
5. Search the whole grid for X-Wing, Swordfish, or XY-Wing only after the local patterns are exhausted.
6. Recheck the area you just changed. A stale pencil mark can make an ordinary position look blocked.