Sudoku Solving Techniques: From Singles to X-Wings

Difficulty in sudoku isn’t about how many cells start empty. It is about which logical techniques you need to solve the puzzle. An easy puzzle yields to two or three of the simplest techniques. A master-level puzzle requires you to chain four or five together, sometimes hunting for a single deduction across the whole board.

This is a working solver’s guide. Each technique is explained with the reasoning, the recognition pattern, and where it slots into your overall approach. The techniques are ordered roughly by difficulty, but more usefully, they’re ordered by how often you’ll actually use them. The first three will solve every easy and most medium puzzles. Add the next three and you can clear hard puzzles. The last few are for master difficulty.

The two single-cell techniques

Naked singles

A naked single is a cell that has only one possible candidate. You arrive at it from the cell’s perspective: look at the digits already in its row, column, and box, eliminate them from the possibilities, and check what’s left. If only one digit survives, that’s the answer.

Naked singles are usually rare at the start of a puzzle and increasingly common toward the end. The strongest hunting strategy is reactive: every time you place a digit, glance at the four neighbouring cells in the row, column, and box, in case the placement just turned one of them into a naked single.

Hidden singles

A hidden single is a digit that has only one possible cell within a row, column, or box. You arrive at it from the digit’s perspective: pick a digit, look at a row (or column or box), and ask which empty cells could possibly take that digit. If only one empty cell qualifies, that digit goes there.

Hidden singles are far more common than naked singles in the early and middle game, and they’re the technique that solves most easy and medium puzzles. The fastest hunting strategy is cross-hatching: pick a digit, find every box that doesn’t yet contain it, mentally project lines from the rows and columns where the digit already appears, and look for the box cells that survive. Most beginner puzzles can be solved with cross-hatching alone.

Pair-based techniques

Naked pairs

Two cells in the same row, column, or box that both have the same two candidates form a naked pair. Whichever way the digits split between the two cells, one of them is digit A and the other is digit B, so neither A nor B can appear in any other cell of that row, column, or box.

For example, if cells R3C1 and R3C2 both have candidates {3, 7}, then 3 and 7 are claimed by those two cells alone. You can erase 3 and 7 from every other cell’s candidate list in row 3. This often unlocks a hidden single elsewhere.

Naked pairs extend to naked triples (three cells sharing exactly three candidates between them) and naked quads, but these are increasingly rare. In practice, the named techniques you’ll find most useful are the naked pair and the hidden pair.

Hidden pairs

Two digits that can only appear in the same two cells of a row, column, or box form a hidden pair. Even if those two cells have other candidates besides the pair, those extras can be eliminated, because the pair is locked into the two cells.

Hidden pairs are harder to spot than naked pairs because the relevant pencil marks are surrounded by noise. The cleanest way to find them is to scan a single unit (a row, column, or box) and ask, “which digits have only two possible cells in this unit?” If two digits share the same two cells, you’ve found a hidden pair.

Locked candidates (pointing pairs and box-line reduction)

This is the highest-value intermediate technique in sudoku, the one that turns easy solvers into hard solvers.

Pointing pairs

Within a 3×3 box, a digit’s candidate cells sometimes all fall in the same row or column. When that happens, the digit is “pointing” out of the box: it must be placed in that row or column, but somewhere inside the box. Therefore, you can eliminate that digit from the rest of the row or column, outside the box.

Example: in box 4 (the middle-left box), suppose the digit 6 can only go in the two cells R4C2 and R6C2. Both are in column 2. Therefore 6 must end up in column 2 within box 4, which means 6 cannot appear anywhere else in column 2 outside box 4. Erase 6 from those cells’ candidates and a chain of further deductions usually opens up.

Box-line reduction

The mirror image. If a digit’s candidates within a row or column all fall inside the same 3×3 box, the digit must be placed inside that box at one of those cells, which lets you eliminate it from all other cells of the box.

Pointing pairs and box-line reduction are sometimes lumped together as locked candidates. They’re both applications of the same idea: in a single unit, a digit’s only valid placements are confined to a smaller intersecting unit, which lets you eliminate it from the larger one.

X-Wing

The X-Wing is the first technique that operates on rectangles across the whole grid, not on a single row, column, or box. It’s the technique that separates competent solvers from advanced ones.

Find a digit, say 4. Look at each row and identify which rows have exactly two candidate cells for that digit. If two such rows have their candidates in the same two columns, you have an X-Wing.

Why it works: in those two rows, the digit must appear in one of the two shared columns. The two valid placements form opposite corners of a rectangle. Whichever diagonal you choose, you fill both columns. Therefore the digit cannot appear elsewhere in those two columns, anywhere on the grid. Erase the digit from any other candidate cell in those columns.

X-Wings work in either direction: spot them by row and eliminate from columns, or spot them by column and eliminate from rows. If you’re stuck on a hard puzzle and singles, pairs, and locked candidates have run dry, the X-Wing is the next thing to scan for.

Swordfish

The Swordfish is the X-Wing’s bigger cousin: a 3×3 version of the same idea. Find three rows where a digit has candidates only in the same three columns. The digit must occupy one cell per row, and those cells must use all three columns between them. As with the X-Wing, you can eliminate the digit from any other candidate cell in those three columns.

Swordfish are rare but unmistakable once you know to look for them, and they appear regularly at master difficulty. The general family is called fish patterns: X-Wing (size 2), Swordfish (size 3), Jellyfish (size 4). Above size 4 the patterns are mathematically symmetric to smaller fish on the complement digit, so size 4 is effectively the practical limit.

Y-Wing (XY-Wing)

The Y-Wing operates on three cells with two candidates each. Pick a “pivot” cell with candidates {A, B}. Find a second cell that shares a unit with the pivot and has candidates {A, C}. Find a third cell that shares a unit with the pivot (a different unit from the second) and has candidates {B, C}. Now whichever digit the pivot takes (A or B), one of the other two cells must take C.

Therefore, any cell that “sees” both wing cells (i.e. shares a row, column, or box with each) cannot contain C, and you can erase C from its candidates. Y-Wings are particularly useful on puzzles where pencil marks are dense and no single-unit technique seems to apply.

Which technique to try when

A practical sudoku-solving order, suitable for any puzzle:

1. Cross-hatch by digit until no hidden singles remain.
2. Scan empty cells for naked singles.
3. Pencil-mark a single near-full unit and look for naked or hidden pairs.
4. Hunt locked candidates: any digit confined to one row or column within a box, or to one box within a row or column.
5. Place every cascade these eliminations create.
6. If still stuck, fully pencil-mark the board and scan for X-Wings.
7. If still stuck, scan for Y-Wings, then Swordfish.

Most hard puzzles fall to step 4 or 5. Step 6 is the line where master-difficulty puzzles begin. Below that line, you’re in the territory of expert solving techniques like the XYZ-Wing, coloring, and forcing chains: valuable but specialised tools that we’ll cover separately.

A note on why guessing is a trap

At a certain point in any hard puzzle, beginners are tempted to guess. A 50/50 between two candidates is a coin flip, after all. Why not just try one and see?

Because the alternative is always a deduction, and that deduction is what you’re actually trying to learn. Guessing skips the education. The cells that look like a 50/50 to a beginner are usually one X-Wing, one Y-Wing, or one chain away from a forced placement. Find that, and the next puzzle’s version of the same situation will be obvious. Guess, and you’ll be coin-flipping for years.

A reputable puzzle, including every puzzle in Sudoku Samurai (all of which are solver-verified to require pure logic), guarantees that the deduction exists. Your job is to find it.

How to actually internalise these techniques

Reading about sudoku techniques is twenty percent of learning them. The other eighty is recognising the pattern in a real puzzle, under the soft pressure of being stuck. The fastest way we know to do that:

• Pick one technique per week. Pencil-mark every puzzle that week and look for that technique specifically.
• When you find one, place the deduction immediately. Don’t continue searching for “a better move.”
• When you don’t find one, double-check that one isn’t there. Half the time it is, you just missed it.

Within a month of disciplined practice, the named techniques stop feeling named. They become a single fluid scanning habit. That’s the goal. The fastest sudoku solvers don’t consciously identify naked pairs or X-Wings; they just see the constraint and move.