What Makes a Sudoku Easy or Hard?
Hiroki Obara · Developer
Sudoku books and apps almost always label their puzzles Easy, Hard and so on. So what decides the level? Many people assume that the fewer numbers a puzzle starts with (the givens, or clues), the harder it is.
To put the conclusion first: the number of givens has little to do with it. What decides the level is which techniques you need to solve the puzzle to the end. In this post I go through that, checking it against the 1,200 puzzles in Sudoku365.
The number of givens barely changes
Sudoku365 has 300 puzzles at each of four levels, Easy, Medium, Hard and Expert, 1,200 in all. I counted the givens in all 1,200, level by level.
| Level | Givens (average) | Fewest to most |
|---|---|---|
| Easy | 25.1 | 25 to 27 |
| Medium | 25.2 | 25 to 27 |
| Hard | 24.0 | 21 to 27 |
| Expert | 24.7 | 22 to 28 |
Every level averages about 24 to 25 givens. The gap between Easy and Expert is only 0.4 on average. Expert even has a puzzle with 28 givens, more than any Easy puzzle.
There is a lower limit. According to the English Wikipedia article on the mathematics of Sudoku, a proper sudoku with a single solution needs at least 17 givens, which was proved in a 2014 paper by McGuire and colleagues using an exhaustive computer search. Above 17, the same number of givens can make an easy puzzle or a hard one.
The techniques needed decide the level
Sudoku techniques fall roughly into steps:
- Only one place for a digit in a box. Pick a digit and rule out cells using rows and columns
- Only one place for a digit in a row, column or box. Step 1 extended beyond boxes
- Only one digit for a cell. Look at one cell and rule out every digit in its row, column and box
- Boxes and rows or columns together. If a digit can only go in one row within a box, you can remove it from that row in the other boxes
- Reservations, and sets of two or three cells. These use relations between pencil marks (candidates)
- X-Wing, Swordfish, XY-Wing and so on. The named advanced techniques
- Patient reading when even those do not settle it
Steps 1 to 4 are in How to Solve Sudoku: The First Techniques Beginners Should Learn, and 5 and 6 in Advanced Sudoku Techniques.
Steps 1 to 3 can be done in your head without pencil marks. From step 4 on, you only see them once you have pencil marks. I think a puzzle’s level is decided by how far down this list you must go to solve it to the end.
The key point is that difficulty is set by the single hardest spot. If almost every cell falls to easy techniques but one place needs an X-Wing, you cannot solve that puzzle without knowing the X-Wing. The other way round, a puzzle with few givens and many cells to fill is not hard if every next cell can be found with steps 1 to 3. It just takes longer.
According to the English Wikipedia article on Sudoku, Wayne Gould, who brought sudoku to British newspapers, spent six years developing a computer program to produce puzzles and rate their difficulty. Measuring how hard a puzzle feels to a person from the solving side, not from surface features like the number of givens: I think that is at the heart of it wherever puzzles are made.
What each Sudoku365 level needs
I have written in earlier posts about solving the Sudoku365 puzzles technique by technique. In summary:
- Easy: all 300 puzzles can be solved to the end with steps 1 to 3 alone. No pencil marks needed
- Medium: none can be finished with steps 1 to 3 alone. Adding step 4, about 70% can be solved to the end
- Hard: all 300 can be solved to the end with techniques up to step 5 (reservations and sets of two or three cells)
- Expert: even with everything up to step 6, only about one in ten can be finished. Most need further techniques or patient reading
The number of givens is almost the same, yet the techniques needed are this different. I think it is fair to say the level names describe this difference, not the number of givens.
When to move up a level
If the level is decided by techniques, I think the clearest guide for moving up is also technique.
- Easy to Medium: when you can use steps 1 to 3 to finish without getting stuck. Medium is where pencil marks begin. How to write them is in How to Write Sudoku Pencil Marks
- Medium to Hard: when you can find step 4, boxes with rows and columns, on your own. Hard needs reservations and sets
- Hard to Expert: when you can spot reservations and sets in your pencil marks without hesitation
Stats show the number cleared, best time and average time for each level. When your average time stops falling, I take that as a sign that you have the techniques for your current level. On the other hand, if you keep getting stuck at the same kind of place on a higher level, that is a sign a technique is missing. Going back down one level and practising the technique you needed at that spot is, I think, the quickest route. How to look again when you are stuck is in Stuck on a Sudoku? How to Find the Next Move.
About Sudoku365
Sudoku365 is a sudoku app I make, for iPhone, iPad and Android. It has 300 puzzles at each of the four levels, 1,200 in all, and I have checked that every one of them has exactly one solution.
You start with 10 hints, and get one more for each puzzle you clear and one for watching an ad. The app is free with ads, and a one-time Remove Ads purchase turns them off.
