Is Sudoku Math or Logic? (And Why It Is 9×9)

Sudoku is a logic puzzle, not a math puzzle. You never add, subtract or compare sizes. The digits 1 to 9 are just nine different symbols, and the puzzle would work exactly the same with letters, colours or shapes. What sudoku does share with mathematics is the kind of thinking: stating what must be true, ruling out possibilities, and following a chain of deductions to a conclusion.

The rules contain no arithmetic

Nikoli, the publisher that named the puzzle, states the rules as: place a number from 1 to 9 in each empty cell, so that each row, column and 3×3 block contains all the numbers from 1 to 9 (Nikoli sudoku rules). "Contains all the numbers" is about membership, not value. No step of a solve ever depends on 7 being bigger than 3. The Wordoku variant, which uses nine letters, proves the point: it is solved in exactly the same way.

Some variants do add arithmetic. Killer sudoku groups cells into cages with a target sum, and sandwich sudoku gives the total between the 1 and the 9. Those are different puzzles. Our guide to sudoku variants covers them.

What kind of thinking sudoku uses

Every sudoku technique is an "if, then" argument. A hidden single says: every other cell in this box already sees a 4, so the 4 must go here. A naked pair says: these two cells can only be 2 or 6, so no other cell in the row can be 2 or 6. An X-Wing combines four cells into one argument by cases. That is deductive logic, the same reasoning used in a mathematical proof, applied to a very small world.

Sudoku Royale's puzzle grader works with 13 such logical techniques and no arithmetic at all. It solves 91% of the puzzles we generate. Across the 457,048 graded puzzles in our library, 77.8% need nothing more than naked and hidden singles.

Where mathematics does come in

Sudoku is not arithmetic, but mathematicians study it seriously.

• Combinatorics: there are 6,670,903,752,021,072,936,960 valid grids, counted by Felgenhauer and Jarvis in 2005 (how many sudoku puzzles are there).
• Complexity theory: generalised sudoku is NP-complete (is sudoku NP-complete).
• Latin squares and graph colouring: a solved sudoku is a Latin square with an extra rule, and the puzzle can be written as colouring a graph with nine colours. See the mathematics behind sudoku.

Why is sudoku 9×9?

Because nine is a square number. The boxes need to be squares that tile the grid exactly, which works when the side of the grid is n×n for some n. With n = 3 you get a 9×9 grid made of nine 3×3 boxes, and the symbols fit neatly into the single digits 1 to 9. With n = 2 the grid is 4×4, which is too small to be interesting for adults. With n = 4 it is 16×16 and needs 16 symbols, so publishers use letters or two-digit numbers. The 9×9 size has been standard since Howard Garns's Number Place in 1979 (history of sudoku), and Nikoli's rules are written for it. Smaller 4×4 and 6×6 grids (the 6×6 uses 2×3 boxes) and larger 16×16 grids exist as variants.

The number nine also gives sudoku its unofficial holiday: 9 September is listed as International Sudoku Day by National Day Calendar. We could not find a puzzle federation or publisher behind the date, so treat it as a fun calendar entry.

Do you need to be good at math to play?

No. If you can tell nine symbols apart and keep track of where you have looked, you can solve sudoku. People who find arithmetic stressful often enjoy it for that reason. Start with the sudoku rules and beginner tips.