How Many Sudoku Puzzles Are There? The Maths of Sudoku, Explained

Every Sudoku starts with the same empty 9×9 grid and the same nine digits, so you might expect the game to run out of puzzles one day. It won’t. The maths behind Sudoku is full of numbers so big they need new words, plus a few surprisingly small ones. Here is how many Sudoku grids and puzzles there are, explained for curious people who don’t need to be mathematicians.

How many Sudoku puzzles are there? The short answer

The numbers at a glance

Start small: counting 4×4 Sudoku

Big numbers are easier to trust when you’ve seen a small one worked out. So let’s shrink Sudoku to a 4×4 grid: four rows, four columns, four 2×2 boxes and the digits 1 to 4. (It’s the size we use in our Sudoku for kids guide.) Here is a real 4×4 puzzle with 6 clues, next to its one and only solution.

How many completed 4×4 grids are there? You can count them with a pencil:

1. Top-left box: put 1, 2, 3 and 4 in its four cells in any order. That’s 4 × 3 × 2 × 1 = 24 ways.
2. Rest of the top row: the two missing digits can go in either order: 2 ways.
3. Rest of the left column: the same again: 2 ways.
4. Everything else: once those are fixed, it turns out there are exactly 3 ways to finish the grid.

Multiply them: 24 × 2 × 2 × 3 = 288. And here’s the twist: if you count two grids as “the same” when one is just the other with the digits renamed, rows swapped or the grid flipped, only 2 truly different 4×4 grids are left. Hold on to that idea, because the 9×9 story works the same way, just with much bigger numbers.

6.67 sextillion completed 9×9 grids

For the full 9×9 grid, counting by hand is hopeless. In 2005 Bertram Felgenhauer and Frazer Jarvis used a computer and a clever shortcut. Instead of building every grid, they looked at the ways to fill the top three rows (the top “band”), grouped band patterns that are bound to have the same number of completions, and only counted the completions of one example from each group. The answer was:

6,670,903,752,021,072,936,960 completed grids, or roughly 6.67 × 10²¹.

How big is that? If you could check one grid per second, without sleeping, it would take you more than 200 trillion years to see them all. If every one of the world’s Sudoku fans started today, they still wouldn’t make a dent.

Only 5.5 billion really different grids

Many of those 6.67 sextillion grids are the same grid in disguise. Take any finished Sudoku and try these tricks: every one of them gives you another valid grid.

• Rename the digits. Swap every 1 for a 7 and every 7 for a 1. Sudoku digits are just labels, so there are 9! = 362,880 ways to relabel.
• Shuffle rows inside a band (for example, swap rows 1 and 2), or shuffle the three bands themselves.
• Shuffle columns inside a stack, or shuffle the three stacks.
• Flip the grid along its diagonal, so rows become columns.

Combined, these tricks give about 1.2 trillion ways to disguise a single grid. Ed Russell and Frazer Jarvis then used a piece of group theory called Burnside’s lemma to count the grids that are left once all disguises are treated as one (their paper, “Mathematics of Sudoku II”, appeared in 2006): 5,472,730,538 essentially different grids.

That’s still a lot. At one grid per second, you’d need about 173 years to look at each of them once. Think of it like recipes: 5.5 billion genuinely different dishes, each of which can be served on more than a trillion different plates.

Sudoku as a Latin square and a colouring puzzle

Mathematicians knew Sudoku’s ancestors long before the puzzle had its name. A Latin square is a grid in which every symbol appears exactly once in each row and each column. The mathematician Leonhard Euler studied them in the 1780s, as our history of Sudoku explains. A finished Sudoku is a Latin square with one extra rule: each 3×3 box must also contain every digit.

That box rule is strict. There are about 5.52 × 10²⁷ Latin squares of size 9×9, but only about one in every 828,000 of them is also a valid Sudoku grid. On the 4×4 grid the rule is much gentler: there are 576 Latin squares and exactly half of them, 288, are Sudoku grids.

There’s another way to see Sudoku: as colouring a map. Imagine each of the 81 cells as a dot, and draw a line between any two cells that share a row, column or box. Every dot ends up joined to 20 others, and there are 810 lines in all. Now replace the digits 1–9 with nine colours. Solving a Sudoku means colouring every dot so that no two joined dots have the same colour, starting from the colours the clues give you. Mathematicians call this the Sudoku graph, and it needs at least nine colours, because every row is a group of nine dots that are all joined to each other.

Grids vs puzzles: why they’re not the same thing

This is the part most articles get muddled. A grid is a finished, fully filled 9×9 square. A puzzle is a grid with most digits removed, leaving a set of clues that leads back to exactly one grid. Curious how a grid becomes a puzzle? Here is how Sudoku puzzles are made, step by step.

One grid can produce a huge number of different puzzles, because there are countless ways to choose which digits to keep. Keep 30 clues one way and you get a gentle puzzle; keep a different 30 and you might get a fiendish one. So the number of puzzles is much bigger than 6.67 sextillion. As far as we know, nobody has published an exact count of all valid Sudoku puzzles, and any figure you see quoted as “the number of Sudoku puzzles” is either the grid count or an estimate.

Difficulty belongs to the puzzle, not the grid. The same grid can hide an easy puzzle and an expert one; what changes is which clues are given and which solving techniques you need. Our guide to Sudoku difficulty levels explains how that works.

Why a puzzle must have exactly one solution

A proper Sudoku has exactly one solution. That isn’t just tradition; it is what makes the puzzle fair. If two different grids both fit the clues, then at some point no amount of logic can tell you which digit goes in a cell, and you are reduced to guessing. Even worse, you could finish correctly and still be marked wrong because the setter had the other answer in mind.

A unique solution also gives you a quiet promise: every cell can be worked out, one deduction at a time. Expert solvers even use that promise as a tool, ruling out patterns that would allow two answers. On Sudoku.my, every puzzle is checked to have exactly one solution that can be reached by logic alone, without guessing, from Easy to Expert.

The 17-clue minimum

So how few clues can a puzzle have and still have just one solution? For years, solvers had found plenty of puzzles with 17 clues and none with 16. The mathematician Gordon Royle of the University of Western Australia collected them: his list grew to 49,151 different 17-clue puzzles.

In January 2012, Gary McGuire, Bastian Tugemann and Gilles Civario announced a proof that no 16-clue Sudoku exists. Their method was to take one grid from each of the 5,472,730,538 essentially different grids and search it, exhaustively, for a 16-clue puzzle. A simple brute-force search would have been far too slow, so they designed a faster algorithm based on “unavoidable sets”: groups of cells where at least one clue must be given, or the digits could be swapped to make a second solution. Even so, the search took about a year at the Irish Centre for High-End Computing, roughly 800 processor-years of computing. It found nothing. The full paper appeared in the journal Experimental Mathematics in 2014.

Will we ever run out of Sudoku?

No. Even if you only count essentially different grids, there are more than 5 billion of them, and each one can be turned into a great many distinct puzzles. One new grid a day would keep you busy for about 15 million years, and that’s before counting the different puzzles each grid can hide.

The surprising part is how this huge space feels in your hands: one empty grid, nine digits, one rule. That’s the charm of Sudoku. Somewhere among those sextillions there’s a puzzle that suits your mood today, and on Sudoku.my it’s waiting at the level you choose, with a new Daily Challenge that starts gentle on Monday and gets tougher towards the weekend.

#Maths#Fun facts#Sudoku grids