How do the difficulty levels for Sudoku work?

Grading and Difficulty Levels

There are several different Sudoku puzzle variations available in Puzzle Maker Pro. One of the important things when solving, publishing and creating sudoku puzzles, is to set the difficulty level. In other words: Grading the puzzle.

When you look at sudoku puzzles from different sources, e.g. magazines or puzzle books from mainstream or KDP publishers, websites, or apps, you'll find that a 3 star or medium sudoku puzzle does not always have the same characteristics.

You may notice that sometimes you can easily solve a 6 star (very tough) sudoku puzzle, while you might get stuck at a 3 or 4 star puzzle in another puzzle book, or even in the same puzzle book.

2022 Changes

In earlier versions of Puzzle Maker Pro, we've used various websites to 'calibrate' our sudoku algorithms. At the start of 2022 we've released an update of Puzzle Maker Pro that has a brand new sudoku generation engine.

To help you, as a puzzle (book) publisher, we disclose the rules by which our sudoku puzzles are graded. We've determined these rules by examining puzzles from various sources, and we've tried to set up a balanced grading approach, that can also be logically explained.

Sudoku 9x9 Grading

First of all: Sudoku is a logic puzzle, not a math puzzle. Sudoku grading is based on the logical operations that are required to solve the puzzle.

Note: Some vendors use the number of clues (a.k.a. the number of digits in the puzzle when you start solving it) to grade the puzzle. Please be aware that this is very wrong: There are sudoku puzzles with only 17 clues that would be graded as Easy by all sophisticated grading systems.

Logical Operations

Although this article is not a Sudoku tutorial, if you want to understand difficulty levels, you'll need to understand the types of logical operations that can be used:

  • Naked Single: Only one number is possible in a given cell (square), all other numbers have been eliminated
  • Hidden Single: Even though there are several numbers in a given cell, that cell is the only location (in a row, column or box) where the number can be placed
  • Naked Pair, Naked Triple, Naked Quad: 2, 3, or 4 numbers that are the only possibilities for 2, 3 or 4 cells
  • Hidden Pair, Hidden Triple, Hidden Quad: Just like the Hidden Single, there are other numbers in the 2, 3 or 4 cells, but the 2, 3 or 4 numbers that form the Pair, Triple or Quad, can only be placed in these cells and not elsewhere.

In general, for Sudokus up to 5 star level (which is called very hard, in Puzzle Maker Pro), you can find Naked Singles, Hidden Singles, Naked Pairs, Naked Triples and Hidden Pairs.

  • Other operations can found on several websites and in sudoku books, such as X-Wing, Y-Wing, (Simple) Coloring, Tracing, Overlay Reduction (also called Pointing Pair / Line Box Reduction), etc etc. These operations can be needed for 6 * and up.

Difficulty Levels

  • Level 1: Only a 'few' Naked Singles and Removing Options (if a '7' is already present in a row, column or box, you can remove it from each cell in that row, column or box)
  • Level 2: Many Naked Singles
  • Level 3: Level 2 plus a few Hidden Singles
  • Level 4: Level 3 plus many Hidden Singles and possibly a Naked Pair or Naked Triple
  • Level 5: Level 4 plus more Hidden Singles and several Naked Pairs or Naked Triples
  • Level 6: Level 5 plus more Naked Pairs / Triples and possibly a 'Very Hard' technique (e.g. Hidden Triple or Overlay Reduction)
  • Level 7: Level 6 plus more 'Very Hard' techiques and possibly a 'Tough' technique (e.g. X-Wing, Coloring)

Is this a big change compared to the old sudoku puzzles?

No, but as part of the new (2022) sudoku 'engine', I decided to make the grading rules more explicit and to offer transparency about them.

Grading of other Sudoku Variations

For regular 9x9 Sudoku puzzles, there are quite a few websites and apps that can be used to solve and/or grade the puzzles. For other formats, ranging from 4x4 to 25x25 sudokus, and for the variations such as Twins, Triathlon, Marathon, Samurai and 3D sudoku, there is no such thing. (And, as far as I know, nobody else is transparent about their grading system. If there's any standard to be found anywhere, I'll be happy to use that for Puzzle Maker Pro)

The approach that's used in Puzzle Maker Pro is easy to use, but there are some caveats, which I'll explain after describing the main approach.

Main Approach

Basically, a 9x9 sudoku has 81 squares, while a 10x10 sudoku has 100 squares and a 16x16 sudoku has 256 squares. Sudoku variations have more squares as well, e.g. a Triathlon puzzle has 189 squares and a Samurai sudoku has 360 squares.

The approach chosen for Puzzle Maker Pro is to scale the difficulty level thresholds proportionally: When a 9x9 sudoku needs 7 hidden singles to be considered Level 3, a 10x10 sudoku would need 9 or 10, a 16x16 sudoku would need 22 and a Samurai Sudoku would need 32 hidden singles to be Level 3.

Small Sudokus 4x4 and 6x6

For the smaller sudokus, there are fewer difficulty levels, since certain options (quads) cannot occur, and even in Hard 6x6 sudokus, triples will be very rare.

Caveat

However, the occurence these patterns, such as naked or hidden triples or quads, does not increase proportionally, at least not linearly, but faster than that.

For example, when you have a 9x9 sudoku, where in a given row, 5 clues are given. In that case there are 4 squares empty, which 4 possible numbers for these squares. The chances that there is a (naked or hidden) triple or quad are very small. But if you compare this to a 25x25 sudoku, where 5/9 or (more or less) 60% of the clues are given, the result is very different. You'd have 15 clues and 10 empty squares. The odds of having triples or quads are still small, but a lot bigger than for a 9x9 sudoku.

Currently, Puzzle Maker Pro does not compensate for this additional complexity.

Like the older version, the difficulty levels 3 and 4 will be reasonably easy to solve, level 5 will be comparable to regular 9x9 level 5 difficulty, but levels 6 and 7 can be relatively much harder to solve. Of course this all depends on the actual puzzle format and variation.