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What are the criteria for determining the difficulty of Sudoku puzzles?

Is it a subjective estimation based on the author's experience, or are there any precise guidelines?

For example, there are some sudokus, where in each step there's always at least one digit that, if you look on some line, must be exactly in that position. Some, however, require to determine that field A can hold 2 values, but assuming it contains X, there's a contradiction in next step, so it must be Y... in other words, require either memorizing or making notices in the corner of the fields.

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Modern sudoku puzzles are ranked according to the difficulty of the techniques required to generate a solution. It's important to note that there is no standardized system or metric, but rather, these are based on qualitative assessments.

playr.co.uk lists these requirements for its various ratings:

  • Level 1 puzzles can be solved simply by reducing the possibilities of a square to one, and filling it in.
  • Level 2 puzzles use the occurrence of pairs of numbers within a block (cell, row, or column). This includes pairs of numbers, number chains, and hidden pairs and chains.
  • Level 3 uses the X-wing and Y-wing techniques, as well as
  • Level 4 typically requires Nishio or Forcing Chains (which effectively involve making assumptions and tracing their results)
  • Level 5 requires flat-out trial and error or lookup tables.
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In my experience, it's generally What is the hardest technique required to solve it.

  • Example hardest: Single Position - Easy puzzle
  • Ex. Candidate Line - Medium Puzzle
  • Ex. Naked Pair - Hard Puzzle
  • Ex. Swordfish - Very Hard Puzzle
  • Ex. Forcing Chains - Master Puzzle

Definitions can be found here: http://www.sudokuoftheday.com/pages/techniques-overview.php

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    $\begingroup$ Could you explain this terminology $\endgroup$
    – Rory Alsop
    May 14, 2014 at 21:45
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    $\begingroup$ @RoryAlsop Maybe you aren't active anymore since it's two years later, but I'll go into detail in Sudoku solving strategies here. $\endgroup$
    – Kevin Cruijssen
    Aug 20, 2016 at 10:34
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    $\begingroup$ @RoryAlsop you should've waited another 2 years before replying $\endgroup$
    – Areeb
    Aug 20, 2016 at 23:24
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To be able to judge the difficulty of a Sudoku, you first need to define "difficulty". The most useful definition seems to rely on the question:

How long does it take an "average" solver to solve the puzzle?

Making this precise will involve fixing the population of solvers, and setting up some reference. E.g., take the 10 best Sudoku solvers at the last World Sudoku Championships and say a puzzle is of "medium" difficulty if they take 3 minutes on average.

In practice, assembling a representative group of a few solvers and having them solve your puzzles will give a reliable view of their relative difficulty.

Regarding the technical difficulty definitions, note that it is possible to make reasonably tricky puzzles with basic techniques (hidden singles can be hard to spot, but a skilled author can make them obvious). Similarly, X-wings can be in plain sight or hidden. So judging difficulty by required techniques alone is not particularly helpful.

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One approach would be doing a breadth-first brute force and noting the highest depth level visited before arriving at a solution. The higher the needed depth, the more difficult the puzzle is.

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I just want to mention this link: AI Sudoku.

Not only rates the difficulty level of a sudoku, but it also creates the hardest sudoku in the world by AI. You may find the formula to rate the difficulty there.

Excerpt of the Formula

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Hardest Sudoku in The World

enter image description here

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If you use a Sudoku solver, for example SudokuWiki.org, you can add up the number of times the solver uses a particular method, so:

1pt: Hidden Singles
2pt: Naked Pairs/Triples
3pt: Hidden Pairs/Triples
...
7pt: X-Wing
...etc.

Add them all up and this gives you a difficulty level.

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