Sudoku Solving Techniques
Every high-quality puzzle should be solvable by logic alone – no guessing required. This guide covers all the techniques the puzzles on this website can demand, from the singles every solve is built on to advanced techniques. Each one is shown in action on a real puzzle, exactly as the in-game hint would present it: The green cells form the pattern, the orange cells lose candidates, and the struck-through pencil marks are the eliminations. Knowing these techniques, you can solve any puzzle published on this website and other high-quality sources, and you’ll have more fun with the process too.
Foundation – Singles
Every solve starts here. These two steps apply to every puzzle at every difficulty level.
Naked Single
Only one candidate remains in a cell.
Also called a sole candidate. After eliminating the digits already present in the same row, column, and box, one cell is left with a single option. That digit must go there. Most Easy puzzles reduce almost entirely to a sequence of naked singles once you pencil in your candidates.
Easy – Intersections
Easy puzzles are solvable with singles plus these two intersection deductions. Nothing harder is ever required.
Pointing Pair / Triple
All candidates for a digit in a box line up in one row or column.
Look at a 3×3 box. If every cell that could hold a particular digit sits in the same row (or column), the digit must come from inside the box, so it can be eliminated from every other cell of that row (or column) outside the box. The logic runs outward: box → line. Pointing and claiming are together known as Locked Candidates.
Claiming Pair / Triple
All candidates for a digit in a row or column lie in one box.
The mirror image of pointing. If every cell in a row (or column) that could hold a digit sits inside the same 3×3 box, the digit must land in one of those cells. Eliminate it from every other cell of that box. The logic runs inward: line → box. Also called box-line reduction (and, with pointing, Locked Candidates).
Intermediate – Subsets
Naked and hidden subsets are the backbone of intermediate solving. Once you see pairs, triples are just the same idea with one more cell.
Naked Pair
Two cells in a house containing exactly the same two candidates.
If two cells in the same house each contain only the same two digits {a, b}, those digits must occupy those two cells in some order. Eliminate both a and b from every other cell in the house. The cells do not need to be adjacent. Naked pairs, triples, and quads are collectively called naked (or disjoint) subsets.
Naked Triple
Three cells in a house whose candidates are drawn from the same three digits.
Three cells in a house whose candidates are all subsets of {a, b, c} – each cell can hold any two or all three – together must contain a, b, and c exactly. Eliminate those three digits from the rest of the house. Classic shapes: {ab, ac, bc}, {abc, ab, ac}, {abc, abc, abc}.
Naked Quad
Four cells in a house sharing four candidates between them.
Same idea as a triple, one size up. Four cells whose candidates lie entirely within {a, b, c, d} lock those four digits into those four cells. Eliminate a, b, c, d from the remaining cells of the house.
Expert – Fish, Wings, Chains & Uniqueness
Expert puzzles are guaranteed to need at least three techniques from this tier, including at least one that no intermediate technique can replace.
X-Wing
A digit restricted to exactly two cells in two different rows, forming a rectangle.
Find two rows where a digit X has only two possible positions, and those positions share the same two columns. The four cells form a rectangle. Whichever diagonal pair holds X, each of the two columns gets exactly one X – so X can be eliminated from all other cells of those columns. The same pattern works with rows and columns swapped. The X-Wing is the smallest "fish"; Swordfish and Jellyfish extend it to three and four lines.
Swordfish
The 3-row / 3-column generalisation of an X-Wing.
In each of three rows, digit X can only go in cells that together lie in just three columns (a row meets each column once, so that is at most three X-cells per row). Each of those rows still needs an X, and no two of them can sit in the same column, so the three X's land in three different columns – and with only three columns available, they fill all three, one apiece. Those columns are now full of X: it can't appear anywhere else in them, so eliminate X from every other cell of the three columns. The pattern also works with rows and columns swapped. In fish notation this is a 3-fish.
Jellyfish
The 4-row / 4-column version of Swordfish.
Same counting as a Swordfish, one size up. In each of four rows, digit X can only go in cells that together lie in just four columns (at most four X-cells per row). Each of those rows needs an X, and no two can share a column, so the four X's land in four different columns – and with only four available, they fill all four, one apiece. Those columns are now full of X, so eliminate it from every other cell of the four columns. The same works with rows and columns swapped. Jellyfish – a 4-fish – is the largest fish used here; any 5-row pattern mathematically reduces to a smaller fish.
Finned X-Wing
An X-Wing plus one or two extra candidates (the fin) inside one box.
Begin with an almost-complete X-Wing. For digit X, take two parallel lines (two rows or two columns) where X is confined to two cells each, lining up on the same two crossing lines – except one of the base lines has a third spot for X, the fin, tucked into the same box as one of its corners. The fin breaks the clean sweep, so X can't be cleared from a whole crossing line. One cell is still doomed either way: If X avoids the fin you have a true X-Wing and its eliminations hold; if X sits in the fin, it's trapped inside that box. The cell where a crossing line meets the fin's box loses on both branches, so X comes out of it. Finned fish are among the most common expert patterns.
XY-Wing
Three bivalue cells {XY}, {XZ}, {YZ} – a hinge and two pincers.
Also known as a Y-Wing. Find a hinge cell – also called the pivot – with candidates {X, Y}. Find two pincer cells (the wings) that each see the hinge: one holding {X, Z} and one holding {Y, Z}. Whatever value the hinge takes, one pincer is forced to Z. Any cell that sees both pincers cannot contain Z.
XYZ-Wing
XY-Wing with a three-candidate hinge {XYZ}.
The hinge (pivot) has candidates {X, Y, Z}; the pincers (wings) are {X, Z} and {Y, Z}. The same chain logic applies, but elimination cells must see the hinge and both pincers simultaneously – which makes the scope narrower than a standard XY-Wing.
W-Wing
Two bivalue cells with identical candidates {X, Y}, bridged by a strong link on one digit.
Two cells both holding {X, Y}. A strong link on Y (Y appears in exactly two cells of some house, and one of those cells sees each of the bivalue cells) ensures one bivalue cell must take X. Any cell that sees both bivalue cells cannot contain X.
Skyscraper
Two rows (or columns) where a digit has exactly two positions, sharing one column (or row).
In two rows, digit X is confined to two cells each, and one cell from each row shares the same column. The shared column is the "base"; the other two cells are the "roof". At least one roof cell must contain X, so any cell that sees both roof cells cannot contain X. The skyscraper is a single-digit chain, one of the Turbot Fish patterns.
2-String Kite
One strong link on a row and one on a column, sharing a cell in the same box.
Also called a Kite or Turbot Fish. Digit X has exactly two candidates in some row and exactly two in some column. One candidate from the row and one from the column share a 3×3 box. The other two cells – one from each string – are the kite tails. Any cell that sees both tails cannot hold X.
XY-Chain
A chain of bivalue cells that forces a digit at both endpoints.
Each step in the chain is a bivalue cell. Adjacent cells share one candidate value. Both endpoints share the same outgoing digit X. Whichever endpoint takes a different value, the other forces X somewhere. Any cell that sees both endpoints cannot contain X.
Nice Loop (AIC)
An alternating chain of strong and weak links – open or closed.
An Alternating Inference Chain alternates strong links (a digit with only two spots in a house, or a bivalue cell) and weak links (two candidates that can't both be true). An open chain ends on the same digit at both ends and proves at least one end holds it, so that digit can be removed from any cell that sees both ends. A closed loop additionally forces an elimination at every weak link around it. Because the links can switch between digits, Nice Loops generalise single-digit chains and subsume skyscrapers, 2-string kites, and many wing patterns.
Unique Rectangle
A pattern that would create two solutions – logic forces it to break.
Four cells forming a rectangle across two rows, two columns, and two boxes. If all four held only {X, Y}, the puzzle would have two completions – the "deadly pattern" a unique puzzle forbids. Type 1: When three cells are {X, Y} and the fourth has extras, remove X and Y from that fourth cell. Type 4: When one digit has a strong link along one side of the rectangle, eliminate the other digit from the extra-candidate cells.
BUG+1
All unsolved cells are bivalue except one – that cell's odd digit is forced.
BUG stands for Bivalue Universal Grimace. If every unsolved cell has exactly two candidates except one cell with three, the grid is one step from a deadly pattern that would permit two solutions. Since the real puzzle has only one, the three-candidate cell must break the pattern: It takes the one of its three digits that appears an odd number of times – three, not twice – across its row, column, and box. The other two candidates would leave the deadly pattern intact.
Extended AIC
A Nice Loop with no length cap – the same alternating logic, followed further.
Identical in kind to the Nice Loop above: strong links and weak links alternating along a chain of candidates. The only difference is how far the search may run. Peak Sudoku hunts for Nice Loops up to six links, which covers virtually every chain a solver spots by eye. When nothing that short exists, this search keeps alternating for as long as the logic holds. The payoff is unchanged: whichever end of the chain turns out false forces the other end true, so at least one end holds its digit, and every cell that sees both ends can drop it. A longer chain is harder to find but no harder to check, because each link still stands on its own.
Suppose R4C4 is not 1.
- Then R5C5 is 1, since the center box has nowhere else for 1.
- So R9C5 is not 1, since column 5 holds just one 1.
- So R9C7 is 1, since row 9 has nowhere else for 1.
Either R4C4 or R9C7 is 1, so any cell that sees both loses 1.
Grouped AIC
An alternating chain whose links can be whole groups of cells, not just single candidates.
An ordinary chain link is one candidate in one cell. A grouped link is every candidate for a digit inside a single box-and-line intersection: the two or three cells where a row or column crosses a box. Such a group behaves exactly like a lone candidate. It is "on" when the digit lands anywhere inside it and "off" when the digit avoids all of it, so it can carry a strong or weak link in its own right. That matters because a digit often has three possible spots in a box which all sit in the same row. An ordinary chain can do nothing with that, while a grouped chain treats the trio as one link and carries the inference straight out along the row. The Empty Rectangle is the best-known special case of this idea.
A whole group of cells acts as one link here. The group is "on" when its digit lands anywhere inside it.
Suppose R4C2 is not 6.
- Then R4C8 is 6, since row 4 has nowhere else for 6.
- So neither R8C8 nor R9C8 is 6, since column 8 holds just one 6.
- So R9C9 is 6, since the bottom-right box has nowhere else for 6.
Either R4C2 or R9C9 is 6, so any cell that sees both loses 6.
ALS Chain
Links built from almost-locked sets – N cells sharing N+1 candidates.
An Almost Locked Set is a group of N unsolved cells inside one house holding exactly N+1 different candidates between them. A single bivalue cell is the smallest example (one cell, two candidates); three cells sharing four digits is another. Since N cells can only ever hold N digits, exactly one of those N+1 candidates has to be the odd one out. That yields an unusually strong link: rule any single digit out of the set, and every remaining digit is pinned inside it. Chaining these sets together reaches eliminations that no wing, fish or bivalue chain can find, because the links no longer depend on a digit having just two homes in a house. ALS logic is what separates the genuinely hardest puzzles from merely hard ones, and it is the technique most often needed to break an Expert grid open.
Almost-locked sets do the work here. A set holding just one more candidate than it has cells must place all but one of those digits, so ruling a digit out of the set pins the rest inside it.
Suppose R7C3 is not 2.
- Then 9 sits in R5C3, R6C3 or R7C3, since R2C3, R5C3, R6C3 and R7C3 hold only five candidates between them and cannot lack both 2 and 9.
- So R8C3 is not 9, since column 3 holds just one 9.
- So 7 sits in R8C2, R8C3, R9C1 or R9C2, since R8C2, R8C3, R9C1 and R9C2 hold only five candidates between them and cannot lack both 9 and 7.
Either R7C3 is 2 or 7 sits in R8C2, R8C3, R9C1 or R9C2, so any candidate both of them rule out can go.
Forcing Chain
Both branches from a bivalue cell reach the same conclusion.
Pick a bivalue cell. Trace the logical consequences of each value separately. If both assumptions force the same digit into (or out of) some other cell, that conclusion is certain regardless of which branch is correct. Should be reserved for the absolutely trickiest puzzles, and in all cases kept shallow enough to follow by hand.
Ready to put them to use? Three new puzzles – Easy, Intermediate, and Expert – are published every day, and the in-game hint button explains the next logical step with the same visuals used on this page.