How to Play Sudoku: The Rules and Your First Solve

To play sudoku, fill every empty square so that each row, each column and each box contains every digit exactly once. A classic sudoku is a 9x9 grid split into nine 3x3 boxes, using the digits 1 to 9; some squares are filled in from the start. You never need arithmetic or guessing: you find each digit by asking where it can still go. This guide is for complete beginners and ends with a full grid solved move by move.

The rules of sudoku

Sudoku has one rule, applied three ways. Nikoli, the Japanese publisher that gave the puzzle its name, states it in two lines: place a number from 1 to 9 in each empty cell, and each row, column and 3x3 block contains all the numbers from 1 to 9 (Nikoli).

A few words you will meet in every sudoku guide:

• Square (or cell): one of the 81 small spaces in a 9x9 grid.
• Row: a horizontal line of nine squares. Column: a vertical line of nine squares.
• Box (or block, or region): one of the nine 3x3 areas marked by thicker lines.
• Clues (or givens): the digits printed at the start. You cannot change them.
• Candidates: the digits that could still go in an empty square. Writing them down is called taking notes or pencil marks.

Two things follow from the rule. A digit can never repeat inside a row, column or box. And because each of those holds nine squares and nine digits, every digit must appear in every row, every column and every box. That second fact does most of the work when you solve.

A good sudoku also has exactly one solution. Wikipedia's entry puts it simply: the setter provides a partially completed grid which, for a well-posed puzzle, has a single solution (Wikipedia). That is why logic alone is enough, and it is the subject of why a good sudoku has exactly one solution.

Rows, columns and boxes: the three units

In sudoku, every square belongs to exactly three units: one row, one column and one box. When you place a digit, it rules itself out of every other square in all three. Solvers say those squares "see" each other.

The 6x6 grid plays by exactly the same rules with smaller numbers, which makes it the best place to learn. Puzzle Picnic's Easy and Medium sudokus are 6x6 grids that can be solved with the simple moves below; Hard and Expert are classic 9x9. There is more on the small grid in 6x6 sudoku: rules, boxes and tips.

To talk about squares, this guide uses row and column numbers: R2C5 means row 2 (counting from the top), column 5 (counting from the left).

Three questions that find a digit

Almost every digit in an easy or medium sudoku is found by one of three questions. Ask them in turn and you can finish most easy and medium grids.

1. Which square is the last one empty in its row, column or box? If a unit has only one empty square, the missing digit goes there. Solvers call this a full house (HoDoKu).
2. Where can this digit go in this unit? Pick a box and a digit it still needs. Look along the rows and columns that cross the box: every one that already holds the digit rules out its squares in the box. If only one square is left, the digit goes there. This is a hidden single, and the scanning that finds it is called cross-hatching.
3. Which digit fits this square? Pick an empty square and list the digits already in its row, column and box. If they cover every digit but one, that last digit goes in the square. This is a naked single.

Question 2 is usually the quickest to check by eye, because it works from digits already printed in the grid rather than from lists you have to build. Question 3 takes longer per square, but it finds digits the other two miss. All three are the first rungs of the technique ladder in sudoku strategies, from singles to X-Wing.

Your first solve, move by move

The best way to learn sudoku is to watch a whole grid fall. Here is a 6x6 sudoku with 18 clues. The boxes are 2 rows tall and 3 columns wide: rows 1 and 2 make the top two boxes, rows 3 and 4 the middle two, rows 5 and 6 the bottom two. A computer check confirms it has exactly one solution.

2		1	4	5	6
	4				3
		2
1			6	4	2
		4	3	6	1
		6			4

Moves 1 to 6

Move 1: R1C2 is 3. Row 1 already holds 2, 1, 4, 5 and 6. Its only empty square must take the missing digit, 3. That is a full house.

Move 2: R3C6 is 5. Column 6 reads 6, 3, empty, 2, 1, 4 from top to bottom. The missing digit is 5. Another full house.

Move 3: R2C1 is 6. Look at the top-left box (rows 1 and 2, columns 1 to 3). It now holds 2, 3, 1 and 4, so it still needs 5 and 6, and it has two empty squares: R2C1 and R2C3. Column 3 already has a 6, at the bottom in R6C3, so R2C3 cannot be 6. The 6 must go in R2C1. That is a hidden single.

Move 4: R2C3 is 5. The top-left box now has one empty square, and 5 is the digit it is missing.

Move 5: R6C2 is 1. Turn to the bottom-left box (rows 5 and 6, columns 1 to 3), which holds only 4 and 6. Where can its 1 go? Row 5 already has a 1 (in R5C6), which rules out R5C1 and R5C2. Column 1 already has a 1 (in R4C1), which rules out R6C1. Only R6C2 is left.

Move 6: R3C1 is 4. In the middle-left box (rows 3 and 4, columns 1 to 3), where can the 4 go? Row 4 has a 4 in R4C5, so R4C2 and R4C3 are out. Column 2 has a 4 in R2C2, so R3C2 is out. That leaves R3C1.

2	3	1	4	5	6
6	4	5			3
4		2			5
1			6	4	2
		4	3	6	1
	1	6			4

Moves 7 to 18

Move 7: R5C1 is 5. This one needs the third question. Row 5 holds 4, 3, 6 and 1. Column 1 holds 2, 6, 4 and 1. Together they contain every digit from 1 to 6 except 5, so 5 is the only digit that fits. That is a naked single.

From here each move opens the next. The table lists the rest of the solve; follow it on the grid above.

2	3	1	4	5	6
6	4	5	2	1	3
4	6	2	1	3	5
1	5	3	6	4	2
5	2	4	3	6	1
3	1	6	5	2	4

Notice the rhythm: one hard look at a box or a square, then a run of easy full houses. Most sudokus solve like that, in bursts.

Scanning a 9x9 grid

On a 9x9 grid the same questions work, but there is more to scan, so cross-hatching becomes your main tool. Choose a digit, then visit each box that does not have it yet and rule out squares along the rows and columns that already contain it.

2	×
1	×	×	5	9			6
×	×	×	6			1	5
	5			6	2

The top-left box needs a 5. The 5 in row 2 rules out the box's two empty squares in that row. The 5 in row 3 rules out all three squares of the box's bottom row. The 5 in column 2 (down in row 4) rules out R1C2. One square survives, R1C3, so the 5 goes there.

A routine that keeps you from going in circles:

1. Start with the digit that appears most often in the grid. It has the fewest places left.
2. Visit every box that lacks it and cross-hatch.
3. Place every digit you find, then move on to the next most common digit.
4. After a full pass through 1 to 9, check rows and columns for full houses.

Notes: pencilling in candidates

Notes are small digits you write in a square to remember which candidates are still possible there. On paper they are pencil marks; in an app they are a notes mode. In Puzzle Picnic you tap a square and pick a number, and Notes mode pencils in candidates instead of placing a digit.

You do not need notes for an easy grid like the one above. They start to pay off on 9x9 grids, where a square can have four or five candidates and you cannot hold them all in your head. Two habits matter from the start:

• When you place a digit, erase it from the notes in the same row, column and box.
• Only note what you have checked. A note you are not sure of is worse than no note.

Full notes, the two-square method many fast solvers use, and the corner and centre styles are covered in sudoku notes: when and how to use them.

What to do when you get stuck

When you are stuck on a well-made sudoku, there is a step you have missed: the puzzle never needs a guess. Work through this list before you reach for a hint.

1. Run the three questions again, digit by digit. Most "stuck" moments are a hidden single that slipped past. Go through 1 to 9 in order and cross-hatch every box.
2. Look for nearly full units. A row, column or box with two or three empty squares is easy to check exhaustively.
3. Take notes in the busiest box. Writing the candidates for one box often shows a square with a single candidate, or two squares that share the same pair of digits.
4. Check for an error. If a square has no candidates left, or a digit seems to fit nowhere in a unit, something placed earlier is wrong. Undo back to the last move you are certain of.
5. Learn the next technique. On harder grids the next step may be a pattern such as a pointing pair or a naked pair. Each one you learn opens up harder puzzles.

Never guess and check. In a puzzle with one solution a guess teaches you nothing, and a wrong guess spreads quietly through the grid until it shows up as a contradiction ten moves later.

Frequently asked questions

Do you need to be good at maths to play sudoku?

No. Sudoku uses digits only as symbols; you never add or multiply them. A sudoku with the letters A to I, or nine colours, would be exactly the same puzzle. What it asks for is patience and careful checking.

Do you ever have to guess in sudoku?

Not in a well-made puzzle. A sudoku with exactly one solution can always be solved by logic, though harder grids need techniques beyond the singles shown here. If a puzzle seems to need a guess, either a step has been missed or the puzzle is badly made.

How many clues does a sudoku need?

A classic 9x9 sudoku with a single solution needs at least 17 clues. That was proved by an exhaustive computer search by Gary McGuire, Bastian Tugemann and Gilles Civario (Wikipedia: Mathematics of Sudoku). Most puzzles give more: Puzzle Picnic's 9x9 grids start with 27 to 31 clues at Hard and 23 to 27 at Expert.

How many different sudoku grids are there?

Bertram Felgenhauer and Frazer Jarvis calculated that there are 6,670,903,752,021,072,936,960 completed 9x9 grids, about 6.7 sextillion (Cornell University: counting sudoku grids). You will not run out.

Should beginners start with 6x6 or 9x9?

Start with 6x6. The rules are identical, but there are fewer squares to scan, and a well-graded easy 6x6 can be solved with the three questions above alone. Move up to 9x9 once you can finish a 6x6 without notes.

Your first week of sudoku

A gentle plan for the first week:

1. Days 1 and 2: solve two or three 6x6 grids a day using only the three questions. Say each reason out loud if it helps.
2. Days 3 and 4: try a 6x6 with fewer clues. You will lean on cross-hatching more.
3. Day 5: try your first 9x9. Use the scanning routine above and take notes only in the box you are working on.
4. Days 6 and 7: read the sudoku strategies guide up to pointing and claiming, and look for those patterns in your next 9x9.

If you want to know where the puzzle came from, from Dell's Number Place to Nikoli's name, read the history of sudoku. When you are ready to play, Puzzle Picnic's sudoku has 6x6 grids at Easy and Medium and classic 9x9 at Hard and Expert. Every puzzle there has one solution, and a hint explains the next logical step. More guides are on the sudoku topic page.

Sources

• Nikoli: Sudoku rules and history
• Wikipedia: Sudoku
• Wikipedia: Mathematics of Sudoku
• Cornell University: counting sudoku grids
• HoDoKu: Singles