Sudoku Rules Explained in Five Minutes

There are three rules in sudoku. You can read them all in one breath, and once you've placed five digits, you'll remember them forever. This post takes five minutes because we'll also cover the small things that trip beginners up, the things rule-summaries always leave out.

Rule one: the row

A row is a horizontal strip of nine cells. There are nine rows stacked from top to bottom of the grid. In every finished row, the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 each appear exactly once.

Concretely: no row may contain two of the same digit. If you've placed a 5 in a row, that's all the 5s the row gets.

Rule two: the column

A column is the same idea, rotated ninety degrees. It runs from the top of the grid to the bottom, nine cells tall. There are nine columns side by side. Each one, when complete, contains the digits 1 through 9, each exactly once.

Place a 7 in column 3? No other cell in column 3 can be a 7.

Rule three: the 3×3 box

The grid is divided into nine smaller squares, three across and three down. Each of those nine boxes is a 3×3 region, nine cells in all. The thicker grid lines mark them out. The box rule says the same thing as the others: digits 1 to 9, each once, in every box.

This is the rule most beginners forget. Rows feel obvious. Columns feel obvious. Boxes need a little practice. But the box rule does more work than the other two combined when you're solving, because a 3×3 box covers fewer cells than a row or column of length nine, so constraints from rows and columns cut into it from multiple angles.

What "valid" actually looks like

Let's make this concrete. Here are the three ways a sudoku can be wrong, and they're the only three:

1. 1
   Row repeat

   The same digit appears twice in any row. Invalid.
2. 2
   Column repeat

   The same digit appears twice in any column. Invalid.
3. 3
   Box repeat

   The same digit appears twice inside any 3×3 box. Invalid.

Notice what is not on this list. A grid with empty cells isn't invalid, it's just unfinished. A grid where one digit appears more times than another across the whole 81 cells isn't invalid either, as long as the row, column and box rules hold. (At the end, every digit appears exactly nine times. Mathematics does the bookkeeping for you.)

Things that are not rules

These come up surprisingly often, so let's clear them out of the way.

• No diagonal rule. In standard sudoku, the two main diagonals don't have to be unique. A diagonal containing 5, 5, 2, 7, 7, 1, 3, 8, 9 is perfectly fine. (Diagonal sudoku, or X-sudoku, is a variant that adds the rule.)
• No arithmetic. The digits never get added or multiplied. Killer sudoku is a variant that adds cage sums, but that is its own puzzle.
• No order required. The digits in a row don't have to be sorted. They just each have to appear once.
• No starting clue minimum on the screen. A well-made puzzle has enough clues for exactly one solution, but the rules themselves say nothing about how many clues you start with.
• No guessing required. Every standard sudoku is solvable purely by logic. If you find yourself wanting to try a digit and see what happens, you've missed a deduction.

"Three rules. No arithmetic. No diagonals. No guessing. That is sudoku in its entirety."

What the clues are for

The clues, the digits the puzzle starts with, exist for one practical reason. The rules alone allow an enormous number of valid completed grids: about 6.67 sextillion, give or take. Without any starting digits, there is no single right answer to chase. The clues anchor the puzzle to one specific solution.

A well-made puzzle has exactly enough clues so that one and only one valid grid can be reached. If you find two different ways to finish a puzzle, then either you've broken a rule along the way, or the puzzle is faulty. (Sudokly puzzles are verified for unique solutions before they are served. So are puzzles from any reputable publisher.)

A quick checklist before you start

Before you make any moves on a new puzzle, run through this:

• ✓I can name the three rules out loud.
• ✓I can spot the nine 3×3 boxes on the grid by following the thicker lines.
• ✓I know I never need to add, multiply, or otherwise do arithmetic.
• ✓I know I never have to guess; every move has a logical reason.
• ✓I know that placing a digit eliminates it from the rest of its row, column and box.

Why the rules are enough

It looks like a small ruleset, and it is, but the constraints are rich enough to generate every difficulty level you can imagine. Three constraints, applied to 81 cells with nine possible digits, produce puzzles that range from "solve in a minute on autopilot" to "spend an hour and use seven different deduction techniques".

That last part is the real magic of sudoku. The rulebook is two sentences long. The skill ceiling is essentially unreachable for most of us. There is always one more technique to learn, and you already know the rules.

What the constraints actually do

Cells

81 in total

Rows

9, each a unique-digits constraint

Columns

9, each a unique-digits constraint

Boxes

9, each a unique-digits constraint

Constraint groups

27 (rows + columns + boxes)

Each cell sits in

Exactly 3 groups (its row, its column, its box)

That last point is what makes sudoku solvable. Every cell is watched by three different groups at once, so the moment you place a digit, you've narrowed down possibilities in twenty other cells without lifting your pen.

You're ready

That is the whole rulebook, with the common confusions sorted. Anything else you read about sudoku, candidates, pencil marks, scanning, naked singles, hidden singles, none of that is a rule. It is technique built on top of these three rules.

If you'd like to put it into practice immediately, an easy puzzle takes about five minutes. If you'd rather build a mental model first, the step-by-step solving guide walks through what to look at, in what order, on a real grid. And if you'd like a printed copy to do over coffee, the printables page has free PDF packs.