Sudoku puzzles are solved entirely through logical deduction, never guessing. Every technique below removes candidate numbers from cells or places a digit with certainty. This guide organizes more than 30 proven methods into four difficulty tiers — beginner, intermediate, advanced, and expert — plus variant-specific strategies for Killer and Sandwich Sudoku. Whether you are solving your first 9x9 grid or tackling competition-level puzzles, the right technique at the right moment is what separates steady solvers from stuck ones.
Beginner Techniques
These foundational methods are enough to solve most easy and many medium Sudoku puzzles. They rely on the basic Sudoku constraint: each digit 1–9 appears exactly once in every row, column, and 3x3 box.
Naked Singles
A naked single occurs when a cell has only one remaining candidate after you eliminate every number that already appears in its row, column, and box. This is the most fundamental Sudoku technique and the one you will use most often. If a cell in row 3, column 5 can see the digits 1, 2, 3, 4, 6, 7, and 9 in its row, column, and box, then the only possibility left is 5 — place it immediately.
When to use it: On every scan of the puzzle. Naked singles should be your first check for any cell with few remaining options. After each placement, re-scan affected rows, columns, and boxes for new naked singles.
Hidden Singles
A hidden single exists when a particular digit can only go in one cell within a row, column, or box, even though that cell might still have multiple candidates. For example, if the number 8 can only fit in one cell within box 4 (because 8 already appears in the rows and columns intersecting the other empty cells), then 8 must go there regardless of what other candidates the cell holds.
When to use it: After checking for naked singles. Scan each row, column, and box asking "where can this number go?" rather than "what number goes here?" Hidden singles are especially powerful in boxes where many cells are already filled.
Scanning and Crosshatching
Scanning is a systematic visual technique where you pick a digit (say, 7) and trace its influence across the grid. For each row and column that already contains a 7, mentally draw a line through the corresponding cells in other boxes. Crosshatching extends this by checking both row and column constraints simultaneously within a box. When lines from existing 7s leave only one open cell in a box, you have found where that 7 belongs.
When to use it: At the very start of a puzzle. Pick the digit that appears most frequently on the grid and scan for it first. This method is fast, requires no pencil marks, and can place many digits quickly on easier puzzles.
Last Remaining Cell
When a row, column, or box has eight of its nine cells filled, the empty cell must contain the one missing digit. This is the simplest possible deduction. While it sounds obvious, keeping an eye on nearly-complete units prevents you from overlooking easy placements while focusing on harder logic elsewhere.
When to use it: Continuously. After every digit you place, check whether the affected row, column, or box now has only one cell remaining. This often triggers a cascade of further placements.
Intermediate Techniques
When basic scanning stalls, these candidate-elimination techniques break through. They require pencil marks (small candidate numbers written in each cell) to track possibilities. Most medium and hard puzzles need at least one of these methods.
Naked Pairs
A naked pair occurs when two cells in the same row, column, or box each contain exactly the same two candidates and no others. Since one cell must hold one value and the second cell must hold the other, those two numbers are "locked" into that pair of cells. You can safely remove both candidates from every other cell in the shared unit.
For example, if cells R1C3 and R1C7 both contain only {4, 9}, then 4 and 9 can be eliminated from all other cells in row 1. One of those cells will be 4 and the other will be 9 — you just don't know which yet, but you do know no other cell in that row can be either number.
When to use it: When you spot two cells in a unit with identical two-candidate sets. This is the most common intermediate technique and often triggers a chain of further eliminations.
Naked Triples
A naked triple extends the pair concept to three cells. Three cells in the same unit collectively contain exactly three candidate digits, with each cell holding a subset of those three (not all cells need to contain all three candidates). Those three digits can be eliminated from every other cell in the unit.
For instance, cells with candidates {2, 5}, {2, 8}, and {5, 8} form a naked triple on {2, 5, 8}. The key insight is that three numbers are distributed among exactly three cells, locking them in place.
When to use it: When naked pairs fail to produce progress. Look for groups of three cells whose combined candidates total exactly three distinct numbers.
Hidden Pairs
A hidden pair occurs when two candidates appear together in exactly two cells within a unit, even though those cells may contain additional candidates. Since those two numbers must occupy those two cells, all other candidates can be removed from both cells, often revealing further deductions.
For example, if in a box, the numbers 3 and 6 only appear as candidates in cells R4C1 and R4C3 (even if those cells also contain candidates like 1, 5, or 8), then R4C1 and R4C3 must hold 3 and 6. Remove every other candidate from both cells.
When to use it: When you notice two numbers only appearing in two cells within a unit. Hidden pairs are harder to spot than naked pairs because the paired candidates are "hidden" among other candidates in those cells.
Pointing Pairs (and Pointing Triples)
When a candidate within a box is restricted to a single row (or column), that candidate can be eliminated from the rest of that row (or column) outside the box. The logic is straightforward: the number must appear in the box, and since it can only appear in that one row within the box, it will occupy one of those cells — so it cannot appear elsewhere in that row.
For example, if the number 5 in box 1 can only appear in R1C1 or R1C3, then 5 can be removed from R1C4 through R1C9.
When to use it: During candidate analysis, when you notice a number within a box is confined to one row or column. This technique bridges box logic and line logic.
Box/Line Reduction
Box/line reduction is the reverse of pointing pairs. When a candidate within a row (or column) is confined to a single box, that candidate can be removed from all other cells in the box. Since the number must appear in that row and it can only be in the cells that intersect the box, it is "claimed" by the row, and the box's other cells are cleared.
For example, if 7 in row 5 can only appear in the cells within box 4, then 7 can be removed from all other cells in box 4 (in rows 4 and 6).
When to use it: Whenever a candidate in a row or column is limited to cells within one box. This commonly unlocks hidden singles in the affected box.
Advanced Techniques
Advanced techniques span multiple rows, columns, and boxes simultaneously. They are required for hard and expert-level puzzles. Good pencil-mark management is essential before attempting these methods.
X-Wing
The X-Wing pattern forms when a candidate appears in exactly two cells in each of two different rows, and those four cells line up in the same two columns (forming the corners of a rectangle). Since the candidate must appear once per row, it must fill one diagonal pair of the rectangle. Either way, both columns are accounted for, so the candidate can be eliminated from all other cells in those two columns.
The pattern works identically starting from columns and eliminating from rows. The name "X-Wing" comes from the X-shaped pattern formed by connecting opposite corners of the rectangle.
When to use it: When a candidate appears exactly twice in two rows and those occurrences share the same columns. X-Wings typically eliminate 2–6 candidates at once, often breaking open a stalled puzzle.
Swordfish
Swordfish extends the X-Wing concept from two rows to three. If a candidate appears in only two or three cells in each of three rows, and all those cells fall within the same three columns, the candidate can be eliminated from all other cells in those three columns. The candidate is "locked" into a 3x3 pattern across the rows, accounting for all three column placements.
Swordfish patterns are harder to spot because not every row needs exactly two candidates — some can have three — and the pattern does not always form a clean rectangle.
When to use it: When X-Wings don't resolve the situation and a candidate shows a restricted pattern across three rows (or three columns). Swordfish patterns are rare but powerful when they appear.
XY-Wing
An XY-Wing involves three cells, each containing exactly two candidates. One cell is the "pivot" and the other two are "pincers." The pivot shares a unit (row, column, or box) with each pincer, but the pincers do not need to share a unit with each other. If the pivot has candidates {X, Y}, one pincer has {X, Z}, and the other has {Y, Z}, then any cell that can see both pincers cannot contain Z.
The logic is simple: the pivot is either X or Y. If it is X, the {X, Z} pincer becomes Z. If it is Y, the {Y, Z} pincer becomes Z. Either way, Z appears in one of the pincers, so any cell visible to both pincers cannot be Z.
When to use it: When you find a bi-value cell (two candidates) that connects to two other bi-value cells, and those three cells share a common candidate that can be eliminated from cells seeing both pincers.
Simple Coloring
Simple coloring (also called single-number coloring) tracks where a specific candidate must and must not go by assigning alternating "colors" along chains of conjugate pairs. A conjugate pair exists when a candidate appears in exactly two cells in a unit — if one cell holds the number, the other cannot, and vice versa.
Starting from any cell in a conjugate pair, color it blue and the other green. Continue the chain: if a green cell forms a conjugate pair with another cell, color that cell blue, and so on. If two cells of the same color can see each other, that color is impossible and the other color is correct. If a non-colored cell can see both a blue and a green cell, that cell cannot hold the candidate.
When to use it: When a candidate forms long chains of conjugate pairs across the grid. Simple coloring is especially effective for eliminating a candidate that appears many times but is constrained by pairwise relationships.
Unique Rectangles
Unique rectangles exploit the fact that a valid Sudoku must have exactly one solution. If four cells at the corners of a rectangle across two boxes would create a "deadly pattern" (where swapping two digits produces another valid solution), at least one corner must contain an additional candidate to break the symmetry.
The most common type (Type 1) has three corners with only two candidates {A, B} and a fourth corner with {A, B} plus extra candidates. The extra candidates at the fourth corner must include the solution for that cell, so A and B can be removed from it. This prevents the deadly pattern and preserves the puzzle's unique solution.
When to use it: When you spot four cells forming a rectangle across two boxes, where most corners share the same two candidates. Unique rectangles are common in harder puzzles and can eliminate candidates that no other technique can reach.
Expert Techniques
These techniques appear in the most difficult competition puzzles and diabolical-rated Sudoku. They require deep candidate analysis and careful chain-following. Most solvers only encounter these in expert or evil difficulty settings.
Finned X-Wing
A finned X-Wing is an almost-X-Wing where one of the two rows has an extra cell containing the candidate (the "fin"). The standard X-Wing elimination applies, but only to cells that can also see the fin. The fin limits where eliminations are valid, but the technique still removes candidates that simpler methods cannot touch.
When to use it: When you find an X-Wing pattern with one extra candidate cell (the fin) disrupting it. Check if the fin's influence still allows useful eliminations in the intersecting box.
Alternating Inference Chains (AICs)
AICs are generalized chains of logical inferences that alternate between strong links (exactly two candidates in a unit, so one must be true) and weak links (candidates in the same cell, so at most one can be true). If a chain starts and ends with strong links on the same candidate, eliminations follow from the endpoints. AICs subsume many named techniques (X-Wings, XY-Wings, remote pairs) as special cases.
When to use it: When specialized techniques fail. AICs require practice to spot but are the most versatile advanced solving tool. Start by tracing short chains (4–6 links) before attempting longer ones.
Remote Pairs
Remote pairs form when a chain of bi-value cells all containing the same two candidates {A, B} extends through the grid, connected end-to-end via shared units. Cells an even number of links apart in the chain have opposite values. Any cell outside the chain that can see both ends of such an even-length segment cannot contain either A or B.
When to use it: When you see a string of cells all containing the same two candidates stretching across multiple units. Count the chain length — even-length connections create elimination opportunities.
Almost Locked Sets (ALS)
An almost locked set is a group of N cells within a single unit that contain exactly N+1 distinct candidates. If one candidate is removed (by logic from outside the group), the remaining N candidates are locked into N cells. Pairing two ALSs that share a "restricted common candidate" allows eliminations from cells that see both sets. ALS techniques are among the most powerful solving methods, capable of cracking the hardest published puzzles.
When to use it: As a last resort on expert puzzles when all other techniques fail. Look for compact groups of cells with one more candidate than cell, then check if linking two such groups produces useful eliminations.
Variant-Specific Techniques
Sudoku variants add extra constraints that enable unique solving strategies not available in classic Sudoku. Below are techniques specific to Killer Sudoku and Sandwich Sudoku, two of the most popular variants available in Sudoku - Brain Puzzles.
Killer Sudoku: Cage Sum Combinations
Every cage in Killer Sudoku has a target sum, and the digits within a cage cannot repeat. The first step in any Killer puzzle is analyzing which digit combinations can produce each cage's sum. For example, a two-cell cage summing to 3 can only contain {1, 2}. A two-cell cage summing to 17 must be {8, 9}. Knowing the limited combinations immediately restricts candidates in those cells, often more than standard Sudoku constraints alone.
When to use it: Before starting any other analysis. Compute possible combinations for every cage, especially small cages (2–3 cells) and cages with extreme sums (very low or very high), as these have the fewest possible combinations.
Killer Sudoku: The Rule of 45
Every row, column, and box in Sudoku contains the digits 1 through 9, which sum to 45. In Killer Sudoku, if all cages within a row, column, or box are fully contained within that unit, you can calculate the value of any remaining cells. This is the Rule of 45: the sum of all cells in a unit is always 45, so subtract the cage sums you know to find the values of cells that don't belong to fully-contained cages.
When to use it: When most cages in a row, column, or box are contained within that unit. The Rule of 45 is especially powerful in rows or columns where a single cell is not part of an otherwise fully-contained set of cages.
Killer Sudoku: Innies and Outies
Innies are cells inside a unit (row, column, or box) that belong to cages extending outside the unit. Outies are cells outside the unit that belong to cages extending into it. By applying the Rule of 45, you can calculate the difference between innie and outie cells, often determining exact values or tight candidate ranges for cells that seem unconstrained.
For example, if cages crossing the border of box 1 have known sums, the sum of innie cells minus the sum of outie cells must equal 45 minus the total of fully-contained cage sums.
When to use it: On medium to hard Killer puzzles when cage combinations alone are not enough. Focus on units where one or two cells cross cage boundaries.
Sandwich Sudoku: Boundary Placement
In Sandwich Sudoku, the clue for each row and column tells you the sum of digits between the 1 and 9 (the "bread" numbers). The first technique is determining where 1 and 9 can be placed. A clue of 0 means 1 and 9 are adjacent with nothing between them. A clue of 35 means all digits 2 through 8 are sandwiched between 1 and 9, so they must be at opposite ends of the row with everything else between them.
When to use it: At the start of every Sandwich puzzle. Extreme clue values (0, 35, and values near these) constrain the positions of 1 and 9 significantly and should be analyzed first.
Sandwich Sudoku: Sum Analysis
Given a sandwich clue, determine which combinations of consecutive inner digits produce the target sum. The gap between 1 and 9 determines how many digits are sandwiched. For a gap of N cells, the N digits between 1 and 9 are drawn from {2, 3, 4, 5, 6, 7, 8} and must sum to the clue value. Enumerating valid combinations tells you exactly which digits are inside and outside the sandwich.
When to use it: Alongside boundary placement. Once you narrow down the possible gaps between 1 and 9, check which digit subsets sum to the clue. Small sums with small gaps and large sums with large gaps have the fewest valid sets.
Sandwich Sudoku: Zero-Sum Lines
A sandwich clue of 0 is the strongest constraint in the variant: it means 1 and 9 are directly adjacent with no cells between them. This immediately creates a "domino" of {1, 9} in the row or column. Combined with standard Sudoku constraints, a zero-sum clue often places both 1 and 9 within the first few deductions, rippling across the grid to constrain other sandwich clues.
When to use it: Always handle zero-sum clues first. They provide the most information per clue and frequently determine the placement of 1 and 9 in intersecting rows or columns as well.