How to solve sudoku

Sudoku has a reputation for being intuition-based, but every well-formed puzzle has a deduction path. You don't guess your way through; you work through a small set of techniques in roughly the same order every time, and the path emerges underneath you as constraints tighten. This guide is about that order — what each technique does, when each one shows up, and how to know when to climb to the next.

The basic rule is the load-bearing constraint: every row, every column, and every 3×3 box has to contain the digits 1 through 9 exactly once. A grid is solved when those three constraints all hold at the same time, and "solving" means finding the placements that satisfy them. The puzzle is a logic problem, not a numerical one — the digits could be any nine symbols and the structure would be identical.

The four levels of techniques

The deduction path sorts into four levels, ordered by how hard the moves are to spot. The article walks through them in turn.

Level 1 — Singles. A cell where only one digit can legally go, or a unit where a digit has only one cell it can occupy. Naked singles and hidden singles. These solve most of an easy puzzle on their own and stay relevant to the very last move of an expert one.

Level 2 — Pair-shaped eliminations. Naked pairs and hidden pairs that lock two digits into two cells, plus pointing pairs and box-line reductions that lock a digit's only homes within a single unit. Triples are the three-cell version of the same logic. These usually don't place digits — they eliminate candidates, and the eliminations open up singles elsewhere.

Level 3 — Fish. X-wings across two rows or columns, swordfish across three, jellyfish across four. The shape is the same: a digit's candidate-cells in several rows happen to share the same columns, which forces the digit into those columns and eliminates it from elsewhere. Fish live across the whole grid rather than within a single unit, and that's most of why they're harder to spot.

Level 4 — Wings and chains. Y-wings, XY-wings, and the broader family of forcing chains. These are if-then deductions strung across three or more cells: if this digit is here, then that one is there, then this conclusion follows. Chains are where sudoku stops being purely pattern recognition and starts being conditional reasoning.

How to use the levels

The escalation rule: never reach for a Level 2 technique while Level 1 still has moves available, and never reach for Level 3 while Level 2 does. Puzzles are designed to be solved in roughly that order, and pattern recognition goes wrong reliably when you skip levels — you start chasing imagined X-wings while ignoring an obvious naked single in front of you.

The reset rule: after any successful move at any level, drop back to Level 1 and rescan. Almost every elimination at Level 2 cascades into singles that weren't there a moment ago. Almost every Level 3 elimination cascades into Level 2 patterns. The puzzle accelerates as you solve, and the acceleration is mostly the lower levels re-opening as the higher levels strip candidates away.

The escalation-and-reset rhythm is the framework. Beginners who skip it tend to fall into one of two failure modes: either they stay stuck at Level 1 long after it's exhausted, staring at the grid and waiting for inspiration, or they escalate too eagerly and trip over moves they could have made three levels down. Hold the rhythm and the puzzle solves itself in passes.

The rest of the article walks each level in turn — what the techniques look like, what they're for, and how to tell when it's time to climb.

Singles: where every solve starts

The two moves that count as Level 1 are the naked single and the hidden single. Together they solve most easy puzzles outright and stay relevant to the very last move of an expert one. They're the techniques you scan for first, last, and every time anything in the grid changes.

The two moves are extensions of the same constraint, asked from two different perspectives. A naked single asks a question about a cell. A hidden single asks a question about a digit. Knowing both, and switching between them on demand, is the core perceptual skill of sudoku. The longer piece on the two moves that solve most easy puzzles walks through them in detail; this section is the tour.

Naked singles

Pick a blank cell. The cell's legal candidates are the digits 1–9 minus everything already placed in the cell's row, the cell's column, and the cell's 3×3 box. If that subtraction leaves exactly one digit standing, the cell is a naked single and the surviving digit goes in.

Naked singles are findable by reasoning about a single cell at a time, which is why they're usually the first move beginners learn. The perceptual load is small — three units to check — and the payoff is a placed digit. At easy difficulty, the grid usually has several naked singles available before you've made a single move; placing one tightens the constraint pressure on neighbouring cells, and a previously two-candidate cell will often collapse into a fresh single. The cascade keeps going as long as the constraints keep narrowing.

The trap with naked singles is missing the ones that aren't fully naked yet. A cell with two candidates left isn't a single — but if you place a single elsewhere that removes one of those two, suddenly it is. The grid has to be rescanned after every placement, because the population of singles changes with the placement before. The reset rule from the framework is doing real work here: every move kicks Level 1 back up.

Hidden singles

The naked single is a question about a cell. The hidden single is a question about a digit. Pick a unit — a row, a column, or a 3×3 box — and pick a digit that hasn't been placed inside it yet. Walk through every blank cell in the unit and ask whether the digit could legally land there given the row/column/box constraints. If only one cell survives the walk, that's where the digit goes — never mind whether other digits could also have lived there.

That's a hidden single. The cell looked ambiguous when you scanned it cell-by-cell — its candidate list has more than one entry — but from the digit-first perspective, the unit has only one place that digit can go, so the cell is forced.

Hidden singles are what carry an easy puzzle the rest of the way home once the obvious naked singles have run out. They're also what unlocks medium puzzles, where naked singles run out earlier in the solve. The catch is that hidden singles don't look singled when you're scanning cell-by-cell; the cell holding the hidden single typically has several candidates listed, and there's nothing visually distinctive about any one of them. Surfacing the move means switching scanning modes — picking a digit you've placed several times already, walking through the units it's still missing from, and checking whether each unit has exactly one cell where the digit could legally land.

What makes singles dependable

Singles alone solve roughly two thirds of an easy puzzle, the bulk of a medium, and a meaningful share of even hard ones. Beyond expert tier the proportion drops, but singles never disappear — every higher technique above Level 1 eventually cascades into singles, which is what closes out the back half of the solve.

The flip is what makes singles dependable. Beginners who only scan one way — usually cell-by-cell — get easy puzzles done but stall on mediums, because half the moves require the digit-first view. Once the flip is comfortable, easy puzzles become close to mechanical and medium puzzles start feeling like easy puzzles with a longer back half.

Common ways singles get missed

Two failure modes show up reliably at the singles tier.

The first is scanning columns first. Beginners tend to scan rows because that's how the grid reads, but in practice boxes yield the most singles per scan, especially in the early game when constraint pressure is concentrated by box. Scan boxes first, then rows, then columns. The order isn't load-bearing, but most solvers settle on box-first within a few weeks of practice.

The second is missing hidden singles in completed boxes. If a 3×3 box has eight digits placed and one remaining, the missing digit is forced — but it's surprisingly easy to walk past, because once a box looks "done" the eye stops scanning it. The fix is mechanical: after any move that places the eighth digit in a box, immediately fill the ninth.

Once both moves are in your hands and the two failure modes are out, the singles tier becomes invisible. You stop thinking of them as moves and start thinking of them as the rhythm the rest of the article is set against. Level 2 is what you reach for when this rhythm runs out.

Pairs and triples

Level 2 begins where Level 1 stops yielding moves. The shape changes — instead of placing digits one at a time, you start eliminating candidates two and three at a time. The pieces are naked pairs, hidden pairs, and their three-cell extensions. The longer treatment lives in naked and hidden pairs and triples; this section is the framework view.

The conceptual move from Level 1 to Level 2 is small: you're still asking the same two questions (cell-first or digit-first?), just stretched across two cells at a time instead of one. Once singles feel automatic, pairs slot in with very little new mental scaffolding.

Naked pairs

A naked pair shows up when two cells in the same unit are pencilled with the same two-digit candidate set, and only that set. Suppose two cells in column 6 are both pencilled {2, 8} with no other candidates. The pair claims those digits between them: one cell will be 2 and the other 8, even though you can't yet say which is which.

The useful consequence isn't the placement — it's what the pair does to the rest of the unit. With the digits 2 and 8 already spoken for inside column 6, any other cell in column 6 that had a 2 or an 8 in its candidate list can drop them. Those eliminations don't place a digit directly, but they almost always cascade into Level 1 hidden or naked singles in the cells they touch.

Naked pairs are the most common Level 2 move and the easiest to spot, because their visual signature is distinctive: two adjacent or nearby cells in a unit, each showing exactly two pencil-marks, and the marks match.

Hidden pairs

A hidden pair is the same logical structure read from the digit side instead of the cell side. Take a unit; ask which cells in it could still legally hold each digit. If two of those digits each have only the same two candidate-cells in the unit, the pair is sitting hidden inside whatever other candidates those two cells happen to be carrying.

The deduction is the mirror of a naked pair. There, the cells named the digits and excluded everything else; here, the digits name the cells and exclude everything else. Either way, two digits and two cells get tied together, and the rest of the unit clears accordingly.

The reason hidden pairs are easier to walk past than naked pairs is that the cells holding them usually have several other candidates in their pencil lists. The pair's visual signature is buried in the noise of the other candidates, and the only way to surface it is the digit-by-digit scan that surfaced hidden singles at Level 1 — same scanning habit, applied at this tier.

Triples

The three-cell version of pairs follows the same structure with one wrinkle: the candidate sets inside the three cells don't have to match. A naked triple is three cells whose candidates collectively span exactly three specific digits, with each cell carrying some subset of those three. A unit could have cells pencilled {4, 6}, {4, 9}, and {6, 9}, for example — three different two-digit sets, but the union of the three cells is exactly {4, 6, 9}, so the three digits are claimed and excluded from the rest of the unit.

Hidden triples are the digit-first version: three digits whose candidate-cells in a unit collapse to the same three cells. Same logic as hidden pairs, one cell wider.

Triples are perceptually harder than pairs because the cell-level pattern is non-uniform. You can't just look for "three cells with the same three-digit candidate set" — sometimes a triple's cells are {4, 6, 9} / {4, 9} / {6, 9}, with the first cell carrying all three digits and the other two carrying subsets. The trick is to scan for naked pairs first, find the unit-level pattern that almost works but isn't a clean pair, and then check whether a third cell completes a three-digit collective.

The pencil-mark precondition

Pairs and triples are invisible without pencil marks. You can spot the occasional pair on an easy puzzle by mental tracking, but as soon as candidate sets get into three-or-four-per-cell territory, you have to write them down. The structural piece on reading pencil marks like a shape covers the visual habits that make these patterns surface from a fully marked grid; the short version is that pairs look like pairs once the marking is complete, and the eye starts flagging them faster than conscious thought can catch up.

The trap is half-marked grids. Pencil marks are only useful if they're complete and current — partial marks generate false patterns, and stale marks lock in real patterns that no longer hold after recent placements. Level 2 is the first tier where bookkeeping discipline starts mattering. Either fully mark the unit you're working in or don't pencil at all; the in-between produces wrong answers.

When pairs and triples show up

Easy puzzles rarely require pairs — most easies solve with singles alone. Medium puzzles have at least one pair somewhere mid-solve, often more. Hard puzzles tend to need pairs early and triples in the back half, often as the bridge to Level 3 (X-wings and the rest of the fish family).

A practical rhythm for medium and hard puzzles: solve every single you can find, then scan for naked pairs (easiest to spot), hidden pairs second, naked triples third, hidden triples last. The first one you find usually unlocks several singles in the surrounding cells, and you drop back to Level 1 to clear them before scanning for the next Level 2 move.

The common mistake at this tier is jumping at a partial triple. Two cells with {4, 6} and a third nearby cell with {4, 6, 9} aren't a triple unless the third cell's third candidate is shared with one of the first two cells — which it isn't, in this case. Confirm the three-digit collective before acting on what looks like a pattern.

Pairs and triples don't ever stop being useful. Even on expert puzzles where Level 4 techniques surface late, the back half is usually a stream of pairs and triples cleaning up the remaining cells.

Pointing pairs and box-line reduction

Pointing pairs and box-line reductions are also Level 2, but with a different shape from naked and hidden pairs. Where naked pairs lock two digits into two cells, pointing pairs lock a digit's only homes within a single unit by leaning on the relationship between a 3×3 box and the rows or columns that pass through it. The depth treatment is in pointing pairs and the snake; this section places them in the framework.

The technique is sometimes called box-line reduction or locked candidates. All three names describe the same constraint pattern.

Pointing pairs

Pick a 3×3 box. Pick a digit that hasn't been placed in it yet. List every cell in the box where the digit could still legally go. If those cells all sit in the same row, the digit is going to occupy that row inside this box — even if you don't know which cell. The consequence is that the digit can't appear in that row anywhere outside this box, because the box has already claimed the row's instance of that digit. Every cell in that row outside the box has the digit eliminated from its candidates.

The same logic works for columns. If the digit's candidate-cells inside a box all sit in the same column, the digit's row-or-column claim runs along the column instead.

The technique is called a pointing pair because two cells in a box pointing along a row or column is the most common shape — but the same constraint can come from three cells in a row inside a box, in which case the move is sometimes called a pointing triple. The shape of the pointer doesn't matter; the elimination follows from the same logic regardless.

Claiming (the inverse)

Claiming is pointing pairs run in the other direction. Pick a row. Pick a digit. List every cell in the row where the digit could still legally go. If those cells all sit inside the same 3×3 box, the digit will occupy that box inside this row, and therefore can't appear anywhere else in the box. You eliminate the digit from every cell in the box that isn't in your row.

The technique has the same shape as pointing pairs — just inverted. Pointing: a digit's box-cells sit in one row, so you eliminate from the row outside the box. Claiming: a digit's row-cells sit in one box, so you eliminate from the box outside the row. Same constraint, opposite reading. Some solvers treat the two as one move and don't distinguish them in conscious thought.

Why box-line eliminations unlock medium puzzles

Box-line is the technique that reliably opens medium puzzles where pure pairs and triples have stopped yielding. Two reasons.

First, the pattern is common in the early-mid game. Almost every medium puzzle has at least one pointing pair available within the first dozen moves, often more. Solvers who scan for naked pairs but don't scan for pointing pairs leave that progress on the table.

Second, the eliminations tend to cascade. Removing a single candidate from a cell that has three candidates left often turns it into a Level 1 hidden single elsewhere, which places a digit, which opens up the next pointing pair, which places more digits. Box-line eliminations don't place anything directly, but they trip a chain of placements through Level 1.

The perceptual habit that surfaces pointing pairs is paying attention to where a digit's candidates concentrate within a box. Most beginners scan boxes for naked or hidden singles and stop there, but the same scan, extended one step, surfaces pointing pairs as a side effect: when the candidate-cells for a digit cluster along one row or column inside a box, the box is pointing, and the elimination follows. Once the habit is in, pointing pairs don't even feel like a separate move — they're an extension of box-scanning that you already do.

Fish: X-wing, swordfish, and jellyfish

Level 3 starts when Level 2 has stopped yielding moves and the puzzle still isn't done. The techniques here all share a structural feature: the elimination involves cells across multiple units rather than within a single one. They're called fish, and the family scales by row count — two-row fish are X-wings, three-row are swordfish, four-row are jellyfish. The mechanics are the same; only the size changes.

Why the X-wing keeps tripping people up goes deep on the perceptual problem these techniques cause. This section covers the structural pattern.

X-wing

An X-wing forms when a single digit's candidates in two different rows have collapsed to exactly two cells per row, with both rows' candidate-cells living in the same pair of columns. The four cells outline a rectangle on the grid. Because the digit has to land somewhere in each row, and each row only has the digit available in those two columns, the digit must occupy one corner of the rectangle in each row — even though you don't yet know which two of the four corners.

That's the deduction. Once the rectangle is identified, the digit can be eliminated from every other cell in those two columns, because the columns' instance of the digit is already committed to one of the two rectangle corners.

The pattern is symmetric between rows and columns. Two columns where a digit's only candidates share two rows produces the same structure with the eliminations running along the rows instead. Most solvers learn to scan in both directions during a single pass, looking for the rectangle no matter which axis it sits on.

The thing that makes the X-wing perceptually awkward isn't the deduction itself; it's that you act on the elimination before the rectangle resolves. You commit to taking the digit out of other cells without knowing which two corners it actually occupies. That step — acting on a partially-resolved configuration — is the first time in the framework where the move runs ahead of the placement, and it's where intermediate solvers tend to hesitate.

Swordfish

A swordfish is the X-wing's three-row generalisation, with one wrinkle: each row's candidate-cells don't have to fill the same set of columns. As long as no row in the chosen three has the digit anywhere outside three specific columns, the swordfish holds. A row might have the digit in two of the three chosen columns, or all three, or even just one — what matters is that no row strays outside the column trio.

If three rows can be found whose digit-candidates all sit inside columns {2, 5, 8}, for example, then the digit must occupy three cells inside that 3-column band between the three rows, and it can be eliminated from columns 2, 5, and 8 in every other row.

The relaxation from "exactly two per row" (X-wing) to "any subset of three columns" (swordfish) is what makes swordfish hard to spot. You're no longer looking for a clean rectangle; you're looking for a column trio that contains all the candidate-cells across three rows, with rows allowed to skip columns inside the trio. The visual signature is messier, and most solvers don't see swordfish on the first scan.

Swordfish are uncommon in mediums and rare in the back half of hards. They become more frequent at expert and master tiers, where puzzles are constructed with fewer Level 1–2 moves available and the solver is forced to find Level 3 patterns to make any progress.

Jellyfish

A jellyfish is the four-row version of the same logic. Four rows whose candidate-cells for a digit all fall inside the same four columns, with rows allowed to skip columns inside the quartet. The eliminations follow the same shape: the digit can't appear anywhere else in those four columns, in any other row.

In practice, jellyfish are vanishingly rare in puzzles that don't bill themselves as expert or extreme. Most solvers go years between sightings. The reason to learn them isn't that you'll spot them often; it's that knowing the family scales — X-wing for two, swordfish for three, jellyfish for four — gives you the right mental model for reading any fish-shaped pattern when one shows up. The longer piece on swordfish and jellyfish when they actually help covers the practical question of when these techniques are worth the cost.

Why fish are hard to spot

Two structural things make fish techniques harder than pairs.

The first is scope. Pairs and pointing pairs sit inside a single unit, so finding them means scanning one unit at a time — which is the perceptual default. Fish span multiple rows or columns simultaneously, and surfacing the pattern means reading the grid with both axes in mind. The scanning habit that surfaces pairs doesn't surface fish, and most intermediate solvers never deliberately switch.

The second is that fish are about where the digit isn't. To recognise an X-wing, you have to notice that a digit is missing from every cell of two rows except for the four corners. Pairs and pointing pairs are about a digit's presence in particular candidate-cells; fish are about its absence everywhere else. That shift between presence-tracking and absence-tracking is what most intermediate solvers find uncomfortable, and it's why fish techniques require the deliberate retraining the deeper article walks through.

The fix isn't to read more about fish — the technique is straightforward once you've seen it work. The fix is to scan the grid digit-by-digit during the pencil-marking phase, looking at where each digit's candidate-cells cluster across rows and columns. When the same two (or three, or four) columns keep showing up across rows, the fish announces itself. The other fish techniques follow the same scanning habit, just with more rows.

Wings and chains

Level 4 is where pattern recognition stops being enough and conditional reasoning takes over. The moves at this tier are if-then deductions strung across three or more cells: if this digit goes here, then that digit goes there, then this conclusion follows, and the conclusion is an elimination or a placement somewhere else in the grid.

The most reliable Level 4 techniques are the wings — the Y-wing and the XYZ-wing — and below them, the broader family of forcing chains. The wings are the gateway. Once they feel comfortable, every harder technique above them is an extension of the same conditional reasoning, just longer.

This section is the orientation. The deep treatment of chains will live in a future article — the family is large enough that compressing it into 900 words would do more harm than good.

Y-wing

A Y-wing forms across three cells, each carrying exactly two candidates. The configuration: one cell is the pivot — say it holds digits A and B; one wing is a cell that shares a unit with the pivot and holds digits A and C; the other wing shares a different unit with the pivot and holds B and C. The two wings need to share visibility with at least one common cell, even if they don't share a unit with each other.

Now branch the reasoning. If the pivot resolves to A, the (A, C) wing is forced to C. If the pivot resolves to B, the (B, C) wing is forced to C. The pivot has only those two options, so one of the two wings ends up holding C either way — and any cell visible to both wings has C eliminated from its candidates, because the digit is already accounted for somewhere in the wing pair.

The Y-wing is the smallest technique that requires holding two if-then branches in mind at once. That's most of what makes it harder than the Level 2–3 techniques: you can't just look at the grid; you have to do a small piece of conditional reasoning before you can act on what you see. The longer piece on the Y-wing when it finally clicks walks through the recognition habit at length.

XYZ-wing

The XYZ-wing extends the Y-wing by giving the pivot a third candidate. The shape: pivot with {A, B, C}, one wing with {A, C}, one wing with {B, C}. The deduction is identical in structure — one of the three cells must be C — but with the pivot now in scope, the elimination of C only applies to cells that see all three of pivot, wing, and wing. That's a smaller eliminations footprint than the Y-wing, which is why XYZ-wings are usually harder to make pay than Y-wings.

In practice most solvers don't bother distinguishing the two during a solve. Once Y-wing scanning is comfortable, XYZ-wings show up as a slight variant — pivot has three candidates instead of two — and the elimination follows the same construction with the smaller footprint.

Forcing chains

A forcing chain generalises the if-then logic. Pick a cell with two candidates. Branch the reasoning: if this candidate, then that consequence; if the other candidate, then that other consequence. If both branches lead to the same conclusion in some other cell, the conclusion holds regardless of which branch is real, and you act on it.

Wings are the smallest forcing chain — three cells, one branch step. A general forcing chain runs across an arbitrary number of cells, with the same logical structure. Some chains are short and tractable. Some run twelve cells and require diagrammatic notation. The same family scales the same way fish do.

The reason chains live at the top of the framework is that the perceptual cost is high. Pairs are something you see; fish are something you see with the right scanning habit; chains are something you construct. The construction is what takes time, and what makes Level 4 feel like a different activity from Levels 1–3.

When to climb to Level 4

The escalation rule applies hardest at this tier. Don't reach for chains while Levels 1–3 still have moves available. The reason is partly the perceptual cost — chain reasoning is slow, and a missed pair is a fast move you've delayed by minutes — and partly that pattern recognition errors are more expensive at Level 4. A misread chain places a wrong digit and corrupts the rest of the solve in ways that aren't obvious for several moves.

In practice, Level 4 only fires on hard-and-above puzzles, and not on every one. Mediums almost never need chains. Hards need them maybe one in three. Expert and master puzzles need them most of the time, often as the bridge between two stretches of Level 1–3 cleanup. The frequency rises as the tier rises, but the shape stays the same.

The deeper guide to when chain reasoning is worth the cost — and how to construct chains without losing track — is for a later piece. For now, the Level 4 takeaway is twofold. Wings are the entry technique. And the entry technique is enough to solve most hard puzzles that need any chain reasoning at all. Climb to wings when nothing else moves. Climb past wings only when a puzzle's construction insists.

Common mistakes and when to use what

Most beginner mistakes at this point are framework violations rather than technique errors. The framework is "scan in passes, escalate when stuck, reset after every move," and the mistakes are escapes from that rhythm in different directions.

The four common mistakes

The first is escalating too early. Beginners who learn about X-wings or chains often start scanning for them on every puzzle, including easies, and end up missing obvious naked singles while looking for exotic patterns. The fix is the escalation rule: don't even consider Level 3 until Level 2 is exhausted, and don't consider Level 2 while Level 1 still has moves. Most easy puzzles never leave Level 1.

The second is missing the easy moves on completed parts of the grid. Hidden singles in nearly-completed boxes are the textbook case, but the same pattern applies across the framework — once a region looks resolved, the eye stops scanning it, even though Level 1 moves often hide there. The fix is mechanical: after any placement, re-scan the unit you placed in for the next single before moving on.

The third is trusting incomplete pencil marks. Pairs and triples (Level 2) and everything above only work on cells whose candidates are correct and current. Half-marked grids generate false patterns that cascade into wrong placements. The fix is the discipline of either fully marking the grid or not marking at all — there's no usable middle.

The fourth is skipping levels. Solvers who know about chains sometimes use them as a substitute for hunting Level 2 patterns, because chains feel decisive and pairs feel like bookkeeping. The trade is a bad one: chains take five times as long as pairs and are five times as error-prone. The escalation rule isn't about elegance; it's about per-move cost.

A decision tree

The decision tree is shorter than it looks.

Scan for Level 1 moves. Place every single you can find, in any order, in any unit. When the singles stop coming, move to Level 2 — naked pairs first, hidden pairs second, pointing pairs third, triples last. Place or eliminate. Drop back to Level 1 and rescan. When Level 2 stops yielding, move to Level 3 — X-wing first, swordfish if X-wings don't appear, jellyfish only on the hardest puzzles. Eliminate. Drop back. When Level 3 stops yielding, move to Level 4 — Y-wings first, XYZ-wings if a Y-wing doesn't fit, forcing chains only when the puzzle's construction forces them. Eliminate or place. Drop back.

The whole tree is "always at the lowest level that still has moves; always reset to the bottom after every move." Internalising both clauses is the practical core of mid-level work.

How far up you usually have to climb

Roughly, by tier.

Easy puzzles solve at Level 1 alone. The whole tier is naked singles and hidden singles, with maybe an occasional Level 2 move at the back end of the harder easies.

Medium puzzles need Level 2 — pairs, pointing pairs, occasional triples. Chains and fish almost never appear.

Hard puzzles need Level 2 throughout and Level 3 in the back half. X-wings appear in maybe one in five hards; swordfish are rarer.

Expert and master puzzles need every level, often as a sequence: Level 1 cleanup, Level 2 cleanup, a Level 3 break-through, Level 1–2 cascade, sometimes a Level 4 chain to bridge the next stuck point, more Level 1–2 cleanup. The puzzle alternates between the higher tiers (which break the deadlock) and the lower tiers (which capitalise on the eliminations).

Extreme puzzles need Level 4 reliably — usually multiple chains across a single solve, sometimes long ones. The frequency of Level 4 moves rises faster than the difficulty rating, which is most of what makes extreme puzzles feel like a different game from hards.

The pattern across all tiers is the same. The framework doesn't change with difficulty; the mix of techniques does. A solver who knows the framework well can pick up a tier they haven't tried and solve it on the first attempt, slowly. A solver who knows the techniques but not the framework will get stuck on easier puzzles than they should.

A note on the variants

The framework so far is for classic sudoku — the standard 9×9 grid with row/column/box constraints. Two common variants extend the constraint set without throwing the framework away. They reuse every level you've just learned and add their own at the upper tiers.

Killer Sudoku

Killer Sudoku keeps the standard 9×9 grid and the row/column/box rules, then overlays cages — irregularly shaped groups of cells with a target sum. Each cage's cells must add to its sum, with no digit repeated within the cage.

The classic levels of techniques still apply — singles, pairs, fish, wings — but Killer also gets its own cage-based moves. Cage singles (a cage with one cell remaining whose digit is forced by the sum), killer pairs (two cells in a cage whose candidates are constrained by the cage sum to the same two digits), and the 45-rule (the digits in any row, column, or box must sum to 45, which lets you compute "missing" cells from cage sums) are the new entries at Levels 1–2. The deeper killer techniques live above Level 4 and aren't relevant to most hards. Try Killer at medium difficulty once the framework feels comfortable on classic — the techniques transfer cleanly.

Kakuro

Kakuro shares the digit-placement-with-no-repetition rule but drops the 9×9 grid for a crossword-shaped layout. Instead of rows/columns/boxes, the constraint is runs — straight strings of cells with a target sum and no repetition. It looks different from sudoku at first glance, but the underlying logic is the same constraint reasoning, applied to runs instead of units.

The framework still maps. Singles, pairs, and pointing-style eliminations all have kakuro analogues, and the same scan-in-passes-and-reset rhythm carries over. The new ingredient is sum combinations — for any (run length, target sum), there's a finite set of digit combinations that fit, and the sum table is the kakuro-specific reference solvers learn over time. Try Kakuro at easy and the cross-pollination is immediate.

How the framework extends

Both variants treat the classic framework as a foundation, not as a separate skillset. A classic solver who picks up Killer or Kakuro and walks through them at easy or medium will find the moves familiar; the only learning is the variant-specific layer at the top. The reverse is also true — solvers who start with Killer often find classic sudoku feels under-constrained at first, until the perception flips and the absence of cages becomes its own kind of structure.

The takeaway is that the framework is broader than classic sudoku. The same four levels, the same escalation rule, the same reset rhythm work across the variant family. Each variant just adds its own moves at the top.

Practice and next steps

The framework is most useful when it feels automatic, and the only path to automatic is repetition. The fastest way to put it through its paces is to try it on a fresh puzzle now, while the four-level rhythm is still loaded in working memory.

If you want a steady supply of puzzles without thinking about it, the daily archive ships a fresh classic and a fresh killer every morning, with difficulty rotating through the week. Easier early in the week, harder by the weekend — the schedule is calibrated to give the framework a workout at every level.

For working at scale, the printable booklet hub collects PDF packs at each tier, including a 30-puzzle easy classic set if your goal is to drill Level 1 until singles feel automatic.

Where to go next

Once the framework feels reliable, three directions extend it.

Read the glossary for the encyclopaedic treatment of every technique mentioned here, plus the harder ones (skyscraper, two-string kite, empty rectangle, finned X-wing, ALS-XY, unique rectangles, BUG, and the rest). Glossary entries each have an interactive worked example for the patterns whose data is shipped, and a per-technique notes section for when each move tends to be worth searching for.

Read the strategy articles for editorial pieces on the perceptual habits behind each level — how scanning patterns develop, why some techniques resist learning, how to recover when stuck.

Try a variant — Killer or Kakuro at medium difficulty. The framework transfers cleanly, and the variant-specific moves are easier to learn against a base of solid classic technique than from scratch.

The one thing that doesn't speed up improvement is reading more about techniques without solving puzzles. The technique knowledge plateaus at about Level 3 if your scanning habits don't develop in parallel, and the only way scanning habits develop is by working through real puzzles where the framework is the answer.

Sit with the four levels for a few weeks. Easy puzzles will start feeling rhythmic. Medium puzzles will start feeling like easy puzzles with a longer back half. Hard puzzles will start feeling like puzzles, full stop. The framework is the thing that compounds.