Sudoku Logic: The Reasoning Patterns That Solve Every Puzzle

Every sudoku idea worth learning is one of three patterns: a cell with a single candidate left, a digit with a single legal home, or a box that locks a digit into one line. The famous advanced patterns — X-Wing, swordfish, chains — get all the ink, but they are rare guests next to these three. We tested that claim instead of repeating it.

We generated 80 puzzles per difficulty with this site's engine and ran a strict singles-only sweep over all 320: fill every cell that has one candidate; when nothing is left, fill every cell a digit claims alone in a row, column, or box; repeat. No pairs, no elimination tricks, no search. It is the two-pattern baseline you can run with pencil marks. Then we logged where the sweep stalled and what the stuck board looked like.

Naked Singles: The Pattern That Carries the Early Game

A naked single is a cell with exactly one legal digit left. Write it, re-mark the grid, and find the next one. The sweep placed an average of 40.8 naked singles per easy solve and 24.4 per evil one. Easy grids never stalled across the full 80-puzzle sample — this one pattern, applied over and over, closed every single one of them.

Real talk: beginners underrate the scan because it feels unglamorous. They want a technique that looks clever. The data says the opposite — on easy puzzles, the naked single scan essentially is the solve. That is not a boring finding; it is the boring finding that saves you from wasting weeks on the wrong curriculum.

Hidden Singles: The Pattern That Hides Inside Pencil Marks

A hidden single is a digit with exactly one legal cell left in a row, column, or box — even though that cell still shows several candidates. The digit is decided; the cell is not. This is the first pattern that requires pencil marks, because you cannot see it in the grid alone.

Hidden singles escalate fast as difficulty climbs: 0.2 per puzzle on easy, 3.7 on medium, 9.0 on hard, 12.3 on evil. They are the difference between easy and hard solves. And here is the number that changes how you look for them: across our sweeps, 57–78% of hidden single placements landed in cells carrying three or more candidates. Scan for cells with one candidate and you will never see these. You have to watch the digits — pick a digit, walk it through a box, then a row, then a column, and ask where it can still go.

Where do hidden singles tend to hide? A box could justify 64–70% of the placements we logged, while rows and columns each justified a bit over half. Almost none of them, under 7%, were visible only in a box. So the efficient habit is to scan boxes digit by digit — you get the widest view per scan, and you are nearly never doing it for nothing.

Pairs and Pointing Pairs: The Most Common Exits From a Stall

When no single exists anywhere, the sweep stalls. That happened in 3 of 80 medium puzzles, 13 of 80 hard ones, and 36 of 80 evil ones. On evil, the stall typically left about 40 empty cells — the worst case left 55, meaning plain singles had filled just 26 of 81 cells before drying up.

Here is what those stuck boards look like, because it tells you exactly where to aim next. Each empty cell carried about three candidates on average, and 12–16 cells per board held exactly two — the raw material of naked pairs. A naked pair is two cells in one unit holding the same two digits; those digits are locked into those two cells, so you can erase them from every other cell in the unit. Pointing pairs work along a different axis. A digit shows up in only two spots of a box, and both spots share one line. One of those two cells takes the digit, which leaves the rest of that line outside the box with no room for it — erase it there.

The stall census has one quiet surprise. Naked pairs were everywhere: four to six per stuck board, on average. But only about one of them actually eliminated something directly. Most visible pairs are already quiet — their digits no longer appear anywhere else in the unit, so there is nothing left to erase. Pairs pay off when the same two cells lock digits across two units at the same time, which is why pencil marks matter: the pair you can see in the row becomes a pair you can act on only when you also see the column and box.

Pointing pairs, not naked pairs, turned out to be the workhorse exit. A pointing pair sat ready in 69% of stalled hard boards and 83% of stalled evil boards — eight of ten times the sweep ran dry on the hardest tier, a locked-candidate move was already available in the pencil marks. When you stall, check boxes first: pick a digit, see if it hugs one line, and follow it outside the box.

The Advanced Toolkit Is a Rare Guest

X-Wing, XY-Wing, swordfish, and chains are real techniques and they are beautiful on paper. They are also the tail of the distribution. Stalls themselves were minority events — 16% of hard puzzles, 45% of evil ones — and when our sweep stalled, a pointing pair usually reopened the board without any of those tools.

That reframes the learning order completely. The three patterns in this article decide the overwhelming majority of placements on this site's puzzles. Learn them until they are automatic, and the rare grid where a true X-Wing appears will announce itself by the length of the stall — you will feel the difference between "no singles" and "no ideas." When you reach that wall, our pattern distribution article shows exactly how often each technique earns its place.

Train Your Logic

The fastest training loop is the one the sweep follows: fill single candidates, hunt hidden singles by digit, and when both dry up, scan boxes for locked candidates.

• Generate a puzzle at your level in the sudoku maker and run this three-step loop on it.
• Stuck anyway? The hints tool on the homepage walks your grid forward one logical move at a time — compare its move against what you missed.
• See the whole drill on a real board in our step-by-step walkthrough, or start from zero in the sudoku guide.

Sudoku logic is a short ladder. The three patterns above carry you through nearly every placement you will ever make, and the data backs it. Fill in the candidates, trust the singles, and save the swordfish for the day it actually shows up.