Why every puzzle here has exactly one answer
A good Sudoku has one answer. Not “at least one”, exactly one. If a puzzle can be finished in two different ways, then at some point you will reach a square where two numbers both fit and nothing on the board can tell them apart. The only way forward is to guess. That is not a puzzle any more, it is a coin toss.
So every puzzle on this site is checked before you see it. Here is how that works.
Step 1: build a finished grid
The program starts with an empty 9 by 9 grid and fills it in, square by square, trying numbers in a random order. Whenever it gets stuck, it backs up and tries a different number. After a few thousandths of a second it has a complete, valid grid: every row, column and box holds 1 to 9 once each.
This finished grid is the answer to the puzzle you are about to play.
Step 2: take numbers away
Next the program removes numbers from the finished grid. After each removal it asks a solver a simple question: how many ways can this puzzle now be finished?
- If the answer is one, the number stays removed.
- If the answer is two or more, the number goes back in, because without it the puzzle would need a guess.
The solver does not need to find every solution. It stops as soon as it finds a second one, which keeps the check fast.
Step 3: keep it balanced
Numbers are removed in mirrored pairs where possible. If the square in row 2, column 3 is emptied, so is the one in row 8, column 7. That gives the starting grid the balanced look that Japanese puzzle makers made standard in the 1980s. It makes no difference to the logic, but most people find it nicer to look at.
What the levels change
The level you pick sets how many numbers are left at the start.
| Level | Starting numbers |
|---|---|
| Easy | about 42 |
| Medium | about 36 |
| Hard | about 31 |
| Expert | about 27 |
| Master | about 24 |
| Extreme | about 22 |
Fewer numbers means fewer easy first moves. On Easy you can usually find the next number just by scanning rows and columns. By Expert, most squares have several candidates and you need techniques like pairs and pointing pairs to narrow them down.
For Master and Extreme, the program makes several puzzles and keeps the one with the fewest numbers, because the digging step sometimes stops early when no more numbers can be removed.
Can a puzzle have fewer numbers?
Yes, down to a point. The fewest numbers a 9 by 9 Sudoku can have and still have one answer is 17. That was proved in 2012 with a huge computer search, and we wrote about it in Seventeen: the fewest clues a Sudoku can have. Puzzles with 17 to 20 numbers exist, but they are rare and hard to make on the spot, so our Extreme level stops at around 22.
What this means for you
If you ever feel you have to guess, you have missed something. Somewhere on the board there is a square with only one option, or a number with only one home. The strategy guide lists the patterns to look for.
More from the blog
When to start using pencil marks
Notes are the most useful tool in Sudoku and also the easiest to overdo. Here is when to use them, how much to write, and how to keep them honest.
Seventeen: the fewest clues a Sudoku can have
A Sudoku needs at least 17 given numbers to have only one answer. Here is what that means, how it was proved, and why our Extreme puzzles stop at around 22.