Sudoku Rules

Sudoku is a puzzle about placing the digits 1 to 9 in a grid so that nothing repeats where it should not. There is no arithmetic and nothing to guess: every puzzle on this site has exactly one correct solution, and you can always reach it by reasoning alone. This page explains the rules, how to read a puzzle, what the four difficulty levels mean and how to get better.

The goal

A Sudoku puzzle is a square grid of 81 cells, nine rows by nine columns. Some cells already contain a digit. These starting digits are called clues or givens, and they never change. Your job is to fill every empty cell with a digit from 1 to 9 so that the finished grid obeys the three constraints below. When the last cell is filled correctly, the puzzle is solved.

A well-made puzzle has a single solution. That matters, because it means each deduction you make is forced: if a digit cannot go anywhere else, it has to go there. You never need to try a number just to see what happens.

The three constraints

The whole game rests on three rules. A digit may appear only once in each of these groups of nine cells:

Each row contains the digits 1 to 9 exactly once. A row is a horizontal line of nine cells.
Each column contains the digits 1 to 9 exactly once. A column is a vertical line of nine cells.
Each 3×3 block contains the digits 1 to 9 exactly once. The thick lines divide the grid into nine blocks, each three cells wide and three cells high.

The cell marked with a question mark must satisfy three groups at once. Its row, its column and its 3×3 block are shaded: whatever digit goes in the marked cell cannot already appear anywhere in the shaded cells.

Because every cell belongs to one row, one column and one block, the groups overlap constantly. That overlap is what makes solving possible: a digit that is ruled out by a row, a column or a block disappears from the list of candidates for that cell, and when only one candidate is left, you have found the answer.

How to read a puzzle

The puzzle appears as a nine-by-nine board with thick lines around each 3×3 block. The digits already printed are the clues. Start by scanning them. Pick a digit, say 5, and find every place it already appears. Each of those fives rules out its whole row, its whole column and its whole block for another five. Look at a block that has no five yet: if the surviving empty cells are reduced to just one, you have placed a digit.

Many players work with pencil marks, small digits written in a cell to record the candidates that are still possible there. On paper, keep them tiny and in the same position every time so they are easy to compare. Pencil marks are optional for easy puzzles but become the main tool on harder ones, since the techniques described below all work by comparing candidates across cells.

When you press Generate, you get a fresh puzzle at the level you chose. The Show solution button reveals the finished grid, which is handy for checking a stuck position or confirming that your own answer is right.

Easy, Medium, Hard and Expert

The level of a puzzle is not just a count of empty cells. It describes which reasoning techniques you need to finish it. Every puzzle generated here is checked by a solver that applies human techniques one at a time, and a puzzle only gets a level if the techniques it demands match that level.

Easy

An Easy puzzle starts with 36 to 42 clues and can be solved using only singles. A naked single is a cell where every digit but one is ruled out by its row, column and block. A hidden single is a digit that has only one possible home within a row, column or block, even if that cell could technically hold other digits. If you can spot these two patterns, you can finish any Easy puzzle.

Medium

A Medium puzzle has 30 to 35 clues, and singles alone will stall at some point. You will need at least one of the next techniques. Locked candidates: if a digit can only go in one row or column within a block, it can be removed from the rest of that row or column outside the block, and the other way round. Naked pairs and hidden pairs: two cells in a group that share the same two candidates claim those digits for themselves, so you can strike them from the other cells of the group, and a pair of digits confined to two cells lets you clear every other candidate from those two cells.

Hard

A Hard puzzle has 24 to 29 clues and calls for the advanced techniques. Naked triples and hidden triples extend the pair idea to three cells and three digits. An X-Wing appears when a digit can go in exactly two places in each of two rows, and those places line up in the same two columns: the digit must occupy opposite corners of that rectangle, so it can be removed from everywhere else in those two columns. Expect to keep careful pencil marks and to spend several minutes on a single puzzle.

Expert

An Expert puzzle has 22 to 25 clues and needs at least two steps with three techniques that no other level asks for: XYZ-Wing, Unique Rectangle or Swordfish (using the same technique twice counts). An XYZ-Wing extends the simpler XY-Wing idea, in which a pivot cell with two candidates, x and y, sees two wings, one holding x and z, the other y and z. Here the pivot is a cell with exactly three candidates, x, y and z, and it sees two more cells, one holding only x and z and the other only y and z. Whichever digit the pivot takes, one of the three cells ends up as z, so z can be removed from every cell that sees the pivot and both wings. A Unique Rectangle looks at four cells at the corners of a rectangle that spans exactly two blocks. If three of them hold only the same two candidates and the fourth holds those two plus others, the two can be removed from the fourth cell: otherwise the four cells could swap those digits and the puzzle would have two solutions, and every puzzle here has exactly one. A Swordfish stretches the X-Wing idea to three rows and three columns: if a digit is confined to the same three columns within each of three rows, it can be removed from the rest of those three columns, and the same works with rows and columns swapped. Keep careful pencil marks for every empty cell, because all three patterns are found by comparing candidates across several cells.

Tips for beginners

Start with Easy and stay there until singles feel automatic. Speed comes from pattern recognition, not rushing.
Scan by digit instead of by cell. Go through 1 to 9, looking for where each digit can still fit in every block.
Begin with the rows, columns and blocks that are already almost full; they have the fewest options.
Never guess. If you are stuck, add pencil marks for every empty cell and look for a cell with one candidate or a digit with one position.
Check your work early. A single wrong digit can look fine for a long time and then cause a contradiction much later.
Take a break when you are stuck. A short walk often reveals what staring at the grid could not.
Move up one level at a time, and treat a Medium puzzle that stumps you as a chance to learn a new technique.

Printing a puzzle

You can solve on paper as well as on screen. After you generate a puzzle, press the Print button that appears beside Generate. The printed page contains only the board, the level and the site name, drawn in black on white so it uses little ink and stays sharp on any printer. If the solution is showing when you print, it is printed too, so hide it first if you want a clean puzzle and keep the solution for later. Every new puzzle is generated in your browser, which means you can print as many different grids as you like for a class, a puzzle club or a long train journey.