The Rules of Sudoku, Explained Properly

The rules of sudoku are notoriously simple to state and notoriously easy to misremember. People who’ve played for years still sometimes ask whether digits in the diagonal need to be unique (they don’t, in classic sudoku) or whether the puzzle has a unique solution by definition (yes, if it’s well-formed). This page is the reference we wish existed: the rules of classic sudoku stated cleanly, the corner cases addressed, and the rule sets of every popular variant collected in one place.

Classic sudoku, in one sentence

Fill the 9×9 grid so that every row, every column, and every 3×3 box contains the digits 1 through 9 exactly once.

That’s the entire rule set. Anything stricter is a variant. Anything looser is broken.

The three constraints, expanded

The single sentence above hides three structural constraints. Each cell on a sudoku board belongs to one of each:

1. Row constraint. Every row of nine cells must contain each digit 1–9 exactly once. No row may contain a repeated digit, and no row may be missing one.
2. Column constraint. Every column of nine cells must contain each digit 1–9 exactly once. Same logic as rows, rotated 90 degrees.
3. Box constraint. The grid is divided into nine 3×3 boxes (also called regions, blocks, or nonets). Every box must contain each digit 1–9 exactly once.

Each cell sits at the intersection of one row, one column, and one box, so placing a digit always satisfies, or violates, three constraints at the same time. This is why beginners are surprised at how quickly placements cascade once a few cells are filled in.

The givens (clues)

A sudoku puzzle starts with some cells pre-filled. These are called givens, or sometimes clues. The rules say nothing about how many givens a puzzle must have, but a well-formed puzzle has the following property:

A well-formed sudoku has exactly one solution, reachable by logic alone.

Mathematicians have proven that the minimum number of givens for a unique-solution sudoku is 17. Anything below that forces multiple valid completions, which violates the implicit rule of unique solvability. In practice, easy puzzles tend to start with 35–45 givens, hard puzzles with 22–28, and a 17-given sudoku is a curiosity for solvers seeking maximum difficulty.

What sudoku is not

These are misconceptions we hear regularly:

• You don’t add anything. Sudoku has nothing to do with arithmetic. The digits are labels.
• The diagonals don’t need to be unique. In classic sudoku, neither diagonal carries a uniqueness rule. (If they do, you’re playing the “Sudoku X” variant; see below.)
• You shouldn’t need to guess. A well-formed puzzle is solvable by pure logic. Guessing is a sign that the puzzle is broken or that you’ve missed a deduction.
• Not every clue placement is valid. Even with the right number of givens, a puzzle is only legitimate if it has exactly one solution. Two solutions means it’s underspecified; zero solutions means the givens contradict each other.

The rules of popular variants

Variant sudoku adds rules on top of the classic three. The variant rules apply in addition to rows, columns, and boxes; you don’t replace the base rules.

Sudoku X (Diagonal Sudoku)

Identical to classic sudoku, with one extra rule: each of the two main diagonals (top-left to bottom-right, and top-right to bottom-left) must also contain the digits 1–9 exactly once. The diagonals get their own constraint.

Hyper Sudoku (4-square)

Adds four extra 3×3 regions, offset from the standard nine boxes. Each of the four extra regions must also contain 1–9 exactly once. The extra regions overlap with the standard rows, columns, and boxes; they don’t replace them.

Killer Sudoku

Killer sudoku usually starts with no givens at all. Instead, the grid is partitioned into irregular “cages” outlined by dashed lines, each labelled with a sum. The classic three rules still apply, plus:

• The digits in a cage must add up to the labelled sum.
• A cage cannot contain the same digit twice, even if its shape would technically allow it.

Killer sudoku is the variant where arithmetic finally appears, but only as a constraint helper, not as the puzzle’s purpose.

Samurai Sudoku

Five overlapping 9×9 grids arranged in a plus or X shape. The four corner grids each share one 3×3 box with the central grid. Every grid must follow the classic rules, and the shared boxes must satisfy all of the grids they belong to. We have a full guide in Samurai Sudoku.

Mini Sudoku (4×4 and 6×6)

4×4 sudoku uses digits 1–4, with four 2×2 boxes; 6×6 sudoku uses digits 1–6, with six 2×3 boxes. The constraint structure is identical: row, column, and box, each filled with the appropriate range of digits. These are excellent introductions for children, but they exhaust their difficulty curve quickly.

Mega Sudoku (16×16)

Uses digits 0–9 plus letters A–F, with sixteen 4×4 boxes. Same rule structure, much more space, much longer to solve. Most paper sudoku books include a few of these toward the back.

Jigsaw Sudoku (Squiggly)

Replaces the nine 3×3 boxes with nine irregularly-shaped regions of nine cells each. Rows and columns are unchanged; the box constraint applies to whatever shape is drawn on the grid. Often much harder than classic sudoku at the same clue count, because irregular regions disrupt familiar visual scanning.

Anti-Knight Sudoku

Adds the rule that two cells a chess-knight’s move apart cannot contain the same digit. A favourite of competitive solvers, because the knight constraint creates surprisingly long deduction chains.

What makes a puzzle “valid”

A sudoku puzzle is valid if and only if all of the following are true:

• The givens don’t themselves break any rule (no two of the same digit in a row, column, or box).
• The puzzle has at least one solution.
• The puzzle has at most one solution.

The combination of those last two, an exactly one solution rule, is what makes the puzzle a logic puzzle rather than an open-ended exercise. Reputable sudoku publishers verify every puzzle with a solver before printing it. We do the same in Sudoku Samurai: every puzzle is generated and then independently solver-verified to require no guessing.

Difficulty is not about clue count

A common misconception is that “harder” sudoku means “fewer givens.” It doesn’t. Difficulty is determined by which logical techniques are required to solve the puzzle. A puzzle solvable using only singles is rated easy regardless of how many cells start blank; a puzzle requiring X-Wings or Swordfish is rated hard or master regardless of how many givens it starts with.

This is why some sudoku apps publish puzzles with very few starting clues that turn out to be straightforward, and others with many givens that turn out to be brutal. The clue count is only a rough proxy for the techniques required.

Frequently asked rule questions

Can a digit appear more than once on the board?

Yes; in fact, exactly nine times. Each of the nine digits appears in every solved sudoku exactly nine times: once per row, once per column, once per box. The uniqueness rule is within each row, column, and box, not across the whole grid.

Do the colours mean anything?

No. Some boards are coloured to make the 3×3 box boundaries easier to see, and some apps highlight conflicts in red, but colour is never a rule of the puzzle. A black-and-white sudoku with thicker box borders carries the same rules as a colourful one.

Is there a time limit?

Not in the rules. Newspapers and apps may track solving time for comparison, but there is no rule that says a sudoku must be solved within any window. Take an hour, take three. The puzzle waits.

Can a sudoku have no solution?

Strictly, yes. If the givens are placed in a way that contradicts the rules, the puzzle has no valid completion. A reputable publisher will catch and reject these before printing.

New to the puzzle? Start with how to play sudoku for a guided first solve. Once the rules feel automatic, our techniques guide covers every logical method you’ll need from beginner to master difficulty.