Difficulty of a Sudoku

What makes a Sudoku difficult?

There are four main factors that make a Sudoku difficult:

• how difficult the necessary techniques are
• how those techniques interact with each other
• when those techniques are needed ("early game", "late game")
• how much can be spotted simultaneously (tediousness)

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The Number of given digits is no indicator for the difficulty of a Sudoku. More givens will make it more likely that a randomly generated Sudoku is easier, but it is no guarantee. A Sudoku with only few givens can be extremely simple, and a Sudoku with many givens can be extremely difficult.

Technique difficulty

There are many different techniques and they can all be grouped by difficulty:

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Most of the Sudokus you will encounter in newspapers and books will only need basic and simple techniques.

These are all the techniques that are integrated into sudoku.coach:

Technique interactions

The difficulty of a Sudoku can also be rated by the need to notate candidates. The more candidate notation is needed, the harder the Sudoku is perceived by people.

Some techniques go well together and can be combined easily (doing the steps in our heads), while other techniques might not.

This example is basically a Locked Candidate that leads to a Naked Single.

• Locked Candidate: There cannot be a 1 in any of the cells. Since there must be a 1 in that row, it must be in any of the two cells. Therefore there cannot be a 1 in the cell.
• Naked Single: Since the cell sees all numbers except 1 and 9, and since we just ruled out a 1 from it, we know it must be the 9.

Even in a real Sudoku where it isn't as clear as in this example, such a combination can be easily spotted by an experienced Sudoku solver (without the need to notate candidates).

Here, we still have the same Locked Candidate, but this time it leads to a Naked Pair.

Both cells can only be 7 or 9.

In a real Sudoku, this combination can be much harder to spot than the previous one. And not just because a Naked Pair is in itself more difficult to spot. This combination's difficulty is greater than the sum of its parts.

The first example is much more likely to not require candidates to be notated.

Technique timing

Easier in later stages

Almost all techniques become easier the further you get in the Sudoku.

This will probably not be a surprise to anyone. The more candidates are already eliminated, the easier it is to spot something.

The first X-Wing is much easier to spot than the second one.

Easier in early stages

Some techniques are easier, though, on grids with few given digits, namely the hidden techniques (Hidden Single, etc...).

The first Hidden Single is much easier to spot than the second one.

Tediousness

Knowing all the techniques, there is still one factor left that is crucial for determining a Sudokus perceived difficulty and fun: Tediousness.

In general

In general, when designing a game (video game, board game, etc.) you need to be aware of the difference between actual difficulty and tediousness.

• Difficulty: How challenging is something? How much do you need to know to succeed? How precise do you need to push the controller's buttons? Etc.
• Tediousness: How much do you need to do to succeed? How repetitive does it feel? How minimal is the effect of an action?

Generally you want to maximize difficulty, and minimize tediousness. The former is mostly translated to fun, while the latter mostly means frustration and work.

In Sudoku

sudoku.coach uses the term "tediousness" as

• a large amount of time needed to find basic things, or in other words:
• only very few digits can be found simultaneously

So on sudoku.coach, the term is used a bit differently than usual. It usually means being forced to repeat something that is roughly of fixed effort or duration. In Sudokus, it is not really fixed in duration, because the limit is always just your brain and how efficient it is.

That is why we cannot clearly define difficulty as good and tediousness as bad. A Sudoku that is not tedious (where many digits can be found simultaneously) can be unpleasant, as a tedious Sudoku with a very straight solving path can be very enjoyable. In fact, many hand-crafted Sudoku puzzles have a very narrow solving path and can be fun because of it.

Still "tediousness" is a good term for saying "how many digits can be found simultaneously" and so sudoku.coach uses it thusly.

It contributes to a Sudoku's overall difficulty, and can be found as a metric in

• the sudoku.coach Solver summary and
• the difficulty info popup (question mark on the "play" page above the "quick game" buttons)

How many Sudoku solving techniques are there?

There are dozens of techniques, one more difficult than the next. Some find enjoyment in using those extremely difficult ones, some hate the idea of spending three hours to spot a "Finned Swordfish", just to be able to continue.

I have integrated several techniques on sudoku.coach, but there are far more than that.

A definitive answer to the question "how many techniques are there?" can unfortunately not be given, because

• Sudoku techniques overlap each other
• some techniques are variants of others
• some techniques are completely contained by others, and yet, we want to count them both, because they might have different strengths (e.g. one being much easier to apply).
• there are still techniques found or altered, or observations made, to this day

The most comprehensive compilations of techniques can be found on

• Andrew Stuart's website sudokuwiki.org.
• Bernhard Hobiger's website hodoku