How to solve
an Evil sudoku?

The rest of the article walks through each step, with figures measured on our grids and a real example.

People often think a grid is hard because it starts with few digits. Our Evil grids have 24 to 27 givens, 26 on average: barely fewer than our Medium, Hard or Expert grids, which sit around 27-28. One or two digits less doesn't make a level. What does is the techniques needed to finish — that's the rule behind our level ranking. A grid is Evil when it requires chain techniques: XY-Chains, Simple Coloring or X-Chains.

Here is what the engine does across all our Evil grids:

Two things stand out. First, almost every digit is placed with the simplest technique there is: the hidden single. The hard work doesn't place digits, it removes candidates so that singles show up again. Second, the first block comes early, with half the grid still empty — and it isn't the last one. A lock, here, is a moment when no digit can be placed without an Evil technique; it sometimes takes several in a row to open it.

1. Sweep for hidden singles. For each digit, look for the box, row or column where it has only one place left. No notes needed at this stage.
2. Pencil in every candidate. As soon as nothing else comes, write every digit still possible in each empty cell. At Evil level this isn't optional: chains are read from the notes.
3. Exhaust the lower levels. Locked candidates, pairs and triples, then the Expert patterns (X-Wing, Skyscraper, Swordfish, XY-Wing). More than two Evil grids out of three need at least one along the way.
4. Look for a chain. Start with the cells that have only two candidates: that's where XY-Chains hide, by far the most useful. Details just below.
5. Go back to the start. Every elimination can reveal a single elsewhere. Do a full sweep before looking for the next chain.

Then repeat steps 4 and 5 as many times as needed. The good news: once the last lock is open, the end is easy. In nearly three grids out of four, only singles remain right up to the last cell.

It's the Evil technique that gets used most: it shows up in every one of our Evil grids, and on its own it accounts for more than three quarters of the Evil techniques played. It links cells that have only two candidates each, like the links of a chain: two neighbouring cells (same row, column or box) share a digit, and if one doesn't take it, it must take its other candidate — which forces the next one, and so on.

The example

Here is one of our Evil grids, with 26 givens, after 27 moves. The coloured cells are the four links of the chain, with their two candidates.

A four-cell XY-Chain. Both ends share the 9: any cell that sees both ends loses its 9.

Start from the top end, R2C7 (6 or 9), and suppose it is not a 9:

• then R2C7 is a 6, so R8C7, in the same column, is not a 6: it's a 4;
• so R8C8, in the same row, is not a 4: it's a 6;
• so R4C8, in the same column, is not a 6: it's a 9.

Conclusion: either R2C7 is a 9, or R4C8 is a 9. We don't know which, and we don't need to. Any cell that sees both ends cannot hold a 9. There are four here: R4C7 and R6C7 (column of R2C7, row or box of R4C8), R1C8 and R3C8 (box of R2C7, column of R4C8).

Immediate effect: in the middle-right box, the 9 has only one place left, R4C8. A hidden single, on the very next move.

For the full version of the technique, see XY-Chains in our guide. A three-cell XY-Chain even has its own name: it's the XY-Wing.

This chain isn't alone. In the same grid, the engine plays eight of them, spread over five locks:

• two chains in a row — the example is the second one — then a single;
• one chain, then a single;
• three chains in a row, then two singles;
• one chain, then four singles;
• one last chain, and the 27 remaining cells fall with singles.

That's the typical rhythm of our Evil grids. After an Evil technique, one of two things happens: either a digit can be placed right away, or you need another chain — which is the case almost one time in two. Hence two reflexes:

• A chain that places nothing is not a failure. It removed candidates; the next one will benefit.
• One open lock doesn't mean a finished grid. After each run of singles, review your notes: the next lock sometimes comes a few moves later.

Two other techniques take over. They rest on the same logic — "if it's not here, it's there" — but stay on a single digit.

• Simple Coloring (about one grid in two): when a digit has only two places in a row, column or box, it's one or the other — a strong link. You connect these pairs by colouring the cells alternately, one colour out of two. A cell that sees one cell of each colour loses the digit. Learn more.
• X-Chains (about one grid in four): still a single digit, but the chain may also go through weak links, as long as it starts and ends with a strong link. Learn more.

Our engine tries Simple Coloring before XY-Chains. But on our grids, XY-Chains get used five times more often than Coloring: if you're searching by hand, start with the two-candidate cells.

And what if a newspaper grid resists even that? It goes beyond the Evil level. Our solver goes further, with "Extreme" level techniques.

1. Incomplete notes. A forgotten candidate turns a three-candidate cell into a fake two-candidate cell: the chain looks right, and the elimination is wrong. Before following a chain, double-check the notes of each of its links.
2. Looking for a chain too early. As long as a single or a pair is available, a chain is a detour. The engine only resorts to it last, and you'll save time by doing the same.
3. Thinking one chain is enough. Only about one grid in five has a single lock. Plan to look for several, and keep your notes up to date in between.
4. Guessing. It's never needed on our grids: all of them can be solved by deduction. A random guess often only shows up ten cells later, when it's too late to tell where the mistake came from.

Do you have to guess to finish an Evil sudoku?

No. All our Evil grids can be solved by deduction, without trial and error. Chain techniques are enough, on top of those from the previous levels.

How many givens does an Evil sudoku have?

On our grids, 24 to 27, so 26 on average: barely fewer than our Medium, Hard or Expert grids. That's not what makes the level: a grid is Evil because it requires chain techniques, not because it has few digits.

Which technique should you learn first?

XY-Chains: they are used in every one of our Evil grids. Simple Coloring comes next (about one grid in two), X-Chains are rarer (about one in four).

On the Sudoku 9×9 page, pick Evil from the level list. To pencil in candidates, use the Notes button (or right-click on a cell to fill). If you get stuck, the Hint button shows the next deduction and the name of the technique, step by step, without giving away the whole solution.

Got a grid from a newspaper or magazine? Copy it into the solver: it explains every move, chains included, and can give you a progressive hint — where to look, then which technique, then the answer.