Sudoku Rules: How to Play, Strategies and Variations

What is Sudoku?

Sudoku is a logic puzzle played on a 9×9 grid. The grid is split into nine rows, nine columns and nine 3×3 boxes (sometimes called blocks or regions), marked off by thicker lines. A few cells start with digits already filled in. These are the givens, and your job is to fill every empty cell so that each row, column and box contains the digits 1 to 9 exactly once.

Although it uses numbers, Sudoku involves no arithmetic. The digits could just as easily be nine colors or nine letters: all that matters is that each symbol appears once in every unit. What makes the puzzle work is that a well-made Sudoku has exactly one solution, and every cell can be reached by logic alone. You never need to guess.

That combination of tiny rulebook and deep logic is why Sudoku became a fixture of newspapers, puzzle books and phone apps. Beginners can finish an easy grid in ten minutes, while the hardest published puzzles demand techniques with names like X-Wing and Swordfish.

A short history

The mathematical idea behind Sudoku is old. A completed grid is a special kind of Latin square, an arrangement in which every symbol appears once in each row and column, which the Swiss mathematician Leonhard Euler studied in the 18th century. Sudoku simply adds the extra condition on the 3×3 boxes.

The puzzle itself was first published in 1979 in Dell Pencil Puzzles and Word Games, a magazine from the American publisher Dell, under the name Number Place. It is generally credited to Howard Garns, a retired architect from Indiana. In 1984 the Japanese publisher Nikoli introduced it in Japan with the name Sūji wa dokushin ni kagiru, roughly "the digits must be single", which was soon shortened to Sudoku. Nikoli also popularized the house style that most setters still follow: a symmetrical pattern of givens and a single logical solution.

The worldwide craze started in Britain. Wayne Gould, a New Zealander living in Hong Kong, wrote a computer program to generate puzzles and persuaded The Times of London to print one in November 2004. Rival British papers soon followed, and during 2005 Sudoku had spread across the world's newspapers. The first World Sudoku Championship was held in Lucca, Italy, in 2006, and the event is now run by the World Puzzle Federation. The Wikipedia article on Sudoku covers the full story.

The rules of Sudoku

Sudoku has just three real rules, plus two conventions about how puzzles are built. Everything else in this guide follows from them:

1. 1Every row holds 1 to 9Each of the nine horizontal rows must contain every digit from 1 to 9 exactly once. No digit may repeat within a row.
2. 2Every column holds 1 to 9Each of the nine vertical columns must also contain every digit from 1 to 9 exactly once.
3. 3Every box holds 1 to 9Each of the nine 3×3 boxes, outlined by the thick lines, must contain every digit from 1 to 9 exactly once.
4. 4Givens never changeThe digits printed at the start are fixed. You fill the empty cells around them, and a correct solution always agrees with every given.
5. 5One solution, no guessingA proper Sudoku has exactly one valid solution, so every digit can be deduced. If you ever feel forced to guess, there is a deduction you have not spotted yet.

Because each row, column and box contains 1 to 9 once, each of them also adds up to 45. Standard Sudoku never needs that fact, but variants such as Killer Sudoku rely on it heavily.

A worked example

Here is a classic 30-given puzzle and its unique solution. The givens are in dark ink and the digits you would add are in blue. Look at the top-left box: it already contains 5, 3, 6, 9 and 8, so only 1, 2, 4 and 7 are left for its four empty cells. The top row already has 5, 3 and 7, which rules out 7 in the first row of that box, and a quick check of the columns narrows each remaining cell further.

Check any unit in the solution and you will find all nine digits with no repeats. That is the whole game: repeat small eliminations like the one above until the grid is full.

How to play: a step-by-step method

There is no single correct order, but this routine takes you through almost any easy or medium puzzle without stalling:

1. Survey the grid. Before writing anything, look for the rows, columns and boxes with the most givens. The fuller a unit is, the fewer options its empty cells have.
2. Cross-hatch one digit at a time. Pick a digit that appears often, say 8, and look at a box that lacks it. Trace the rows and columns of the existing 8s through that box. If only one cell in the box is left uncovered, the 8 goes there.
3. Work through all nine digits. Repeat the cross-hatching for 1 through 9. Each placement creates new lines to trace, so a single pass often unlocks several more cells.
4. Complete nearly finished units. Any row, column or box with only one or two empty cells can usually be finished by checking which digits are missing.
5. Switch to pencil marks when you stall. Write the possible candidates small in the corner of each empty cell. With candidates on paper, patterns such as pairs and hidden singles become visible.
6. Check as you go. Before inking a digit, confirm that it does not already appear in the cell's row, column or box. One early mistake can quietly spoil the rest of the grid.

Strategies strong solvers use

Once cross-hatching feels natural, these techniques handle everything up to the hard puzzles in most newspapers. They build on each other, so learn them in order.

Expert grids add chains and coloring techniques, but the vast majority of published puzzles fall to singles, pairs and pointing. Master those before reaching for anything fancier.

Popular variations

The three core rules are easy to extend, and setters have produced hundreds of variants. These are the ones you are most likely to meet:

• Killer Sudoku. Givens are replaced by dotted cages with sum totals, mixing Sudoku logic with a little arithmetic. See our Killer Sudoku rules.
• Thermometer Sudoku. Thermometer shapes are drawn on the grid, and digits must rise from the bulb to the tip. Our guide to Thermo Sudoku explains the extra logic.
• Sudoku X (Diagonal Sudoku). The two long diagonals must also contain 1 to 9 once each.
• Mini Sudoku. Smaller 4×4 or 6×6 grids with 2×2 or 2×3 boxes. Ideal for children or a two-minute warm-up.
• Jigsaw Sudoku. The 3×3 boxes are replaced by irregular nine-cell shapes, which breaks some familiar box patterns.
• Samurai Sudoku. Five 9×9 grids overlap at their corner boxes, so deductions in one grid feed the others.

Common mistakes

The most common error is checking only two of the three units. A digit might be missing from its row and column but already present in its box. Train yourself to scan all three every time before you write.

The second is solving hard puzzles entirely in your head. Easy grids can be done without notes, but on medium puzzles and above, pencil marks turn invisible patterns into obvious ones. Keep them tidy and erase candidates as soon as a digit is placed.

Third, solvers often develop tunnel vision, staring at one stubborn box for minutes. When progress stops, do a full sweep of all nine digits across the whole grid. The breakthrough is usually somewhere else.

Finally, resist guessing. Trial and error can finish a grid, but a wrong guess can take minutes to reveal itself, and you learn nothing from it. If you are stuck, the next step is almost always a hidden single or a pair you have not noticed.

What makes a Sudoku hard?

It is tempting to judge a puzzle by its number of givens, and easy puzzles often do have more of them. But difficulty really depends on which techniques the puzzle requires. A grid with 26 givens that yields to hidden singles is easier than one with 30 givens that needs an X-Wing. Most publishers grade puzzles by running them through a solver that tries techniques from simplest to hardest.

There is a floor, though. In 2012 a team led by mathematician Gary McGuire showed through an exhaustive computer search that no standard Sudoku with fewer than 17 givens can have a unique solution. Puzzles with exactly 17 do exist, and many thousands have been found.

The number of possible completed grids is enormous as well: 6,670,903,752,021,072,936,960 (roughly 6.7 sextillion), as calculated by Bertram Felgenhauer and Frazer Jarvis in 2005. You will not run out of puzzles. If you enjoy the digit-placing side of Sudoku, try Kakuro or KenKen next, which add sums to the mix, or Futoshiki, which uses greater-than signs.

Frequently asked questions