How to Solve Sudoku Without Guessing, Even on Extreme

Do you ever have to guess in Sudoku?

Any Sudoku with exactly one solution can be solved by trial and error: put in a digit, carry on, back up when you hit a contradiction. A computer does that in milliseconds. For a person it is the most tiring way to play, because you have to remember every branch, and a single missed contradiction makes twenty entries worthless.

So the real question is not whether a puzzle can be solved, but whether it can be solved with techniques a person can apply in their head or with pencil marks. A technique is simply a pattern that proves a digit must go somewhere or cannot go somewhere. Guessing means writing in a digit without any such reason.

There is a grey zone in between that is worth knowing about. On very hard puzzles, many experienced solvers assume a digit and follow the consequences in their head until a contradiction appears. Experts call these chains. That is demanding and perfectly fair, but it is not what a puzzle in an app should require. If you are playing on Hard, named techniques should be enough to get you through.

What we measured in our own generator

At the start of October we built a solver that works the way a person does: it only uses a fixed list of techniques, with no trial and error and no backtracking. When it gets stuck, it says so instead of guessing. Then we ran it on levels 1 to 150 of every difficulty our generator produced at the time.

The result was uncomfortable. Easy and Medium went through completely. On Hard the solver got stuck in 22 of 150 cases, roughly one in seven. On Extreme it was 133 of 150, which is 89%. Almost every Extreme puzzle needed more than the eight techniques we consider fair. Stuck does not mean the puzzle can only be solved by guessing. It means the puzzle lies beyond this ladder, and you need chains or, in practice, trial and error.

We turned that into a rule. Since October the generator checks every classic Sudoku before it is served: the solver has to finish it, and the hardest technique it needs has to match the difficulty. A puzzle that fails the check is thrown away and replaced by the next attempt. Since then, “every puzzle solvable without guessing” is no longer a promise for us but a property the code proves for every level.

The ladder: eight techniques in a fixed order

The solver knows exactly eight techniques and always tries them starting with the simplest. As soon as one of them makes progress, placing a digit or removing a candidate, it starts again from the top. That is the best strategy for people too, because each bit of progress usually opens up simple steps again, and those are quicker to find than the hard ones.

1. Singles: a cell has only one candidate left, or a digit fits in only one place in a row, column or box.
2. Pointing: within a box, a digit can only go in one row, so it disappears from the rest of that row.
3. Pairs: two cells that share two digits, as a naked or a hidden pair.
4. Triples: the same idea with three cells and three digits.
5. X-Wing: in two rows, a digit can only go in the same two columns.
6. Swordfish: the X-Wing with three rows and three columns.
7. XY-Wing: three cells with two candidates each that rule out a digit in a fourth place.
8. XYZ-Wing: the extension where the middle cell has three candidates.

Most of them have their own page with an example: the naked pair, the hidden pair, the naked triple, the X-Wing, the Swordfish and the XY-Wing. The overview of all of them is on the solving techniques page.

The technique people miss most often

The first rung of the ladder sounds trivial, yet most dead ends happen right there. The reason is that there are two kinds of singles. The visible kind is a cell with only one candidate left. The invisible kind is a digit that has only one possible place in a unit, even though that cell still has several candidates. It is called a hidden single.

If you only ever look at cells, you will never see these. You have to ask the question the other way round: where can the 6 still go in this row? In the example below the highlighted cell has three candidates, but the 6 has nowhere else to go in the whole row. Step through it:

The habit that fixes this is simple: on every pass, go through the digits 1 to 9 once and ask, for each box, where that digit can still go. It takes time the first few times and becomes routine after a handful of puzzles.

Which techniques each difficulty level needs

Because the generator checks every puzzle, we can say exactly what you need on each level. On Easy, singles are enough and nothing else is needed. The same goes for Medium, with noticeably fewer clues; pointing is allowed but rarely comes up. On Hard, every puzzle needs pairs or triples. On Extreme, every puzzle needs at least one X-Wing, Swordfish, XY-Wing or XYZ-Wing.

One number surprises a lot of people. On average our puzzles have about 45, 35, 28 and 28 clues across the four levels. So Extreme does not have fewer digits in the grid than Hard. The whole difference lies in which technique is needed at the tightest spot. If you find Extreme harder than Hard, you are right, but not because of the empty cells. It is because of one single step you have to find.

That gives a clear order for practising. If you are stuck on Hard, get comfortable with pairs before you worry about the X-Wing. If you are stuck on Extreme, do not look for more digits but for a pattern, usually a digit that has only two places left in each of two rows.

When nothing seems to work

Even with the right ladder, there comes a moment when you cannot find anything. A checklist helps more than a brave entry. First: are your pencil marks complete? One missing candidate can hide a pair or turn a real pattern into a false one. Second: have you removed every candidate that a placed digit rules out? Third: look for singles once more, this time digit by digit instead of cell by cell.

Only then is it worth looking for the hard patterns. For the X-Wing, pick one digit and mark in every row where it can still go. Rows with exactly two places are your candidates. If two such rows use the same two columns, you have found it. For the XY-Wing, look for cells with exactly two candidates and check whether three of them form a triangle.

Some Sudokus go beyond even that. The best known example is called Easter Monster and has been passed around among puzzle fans for years. Our solver cannot find a single step in it, because the puzzle needs long chains far beyond the eight techniques. We deliberately do not serve puzzles like that. So if one of our Sudokus feels impossible, what is missing is not luck but a pattern you can find.

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