Learn the patterns that unlock better solves.
Start with clean scanning and singles, then move into pairs, box-line interactions, fish, wings, chains, and uniqueness. Each technique below is practical enough to use in today's puzzle.
Strategy library
A quick reference for the techniques used across daily puzzles.
Naked singles: place the only candidate
A naked single appears when one empty cell has only one legal digit left. You find it by checking the cell against its row, column, and 3×3 box.
Open step-by-step guidePencil marks: keep only useful candidates
Pencil marks are temporary possibilities, not permanent clutter. Add them when a cell has several plausible values, then remove them as soon as a peer placement rules one out.
Open step-by-step guideCross-hatching: block the impossible cells
Cross-hatching is a visual way to find where one digit can go. Choose a digit and a 3×3 box, then use matching digits in intersecting rows and columns to rule out cells inside that box.
Open step-by-step guideNaked pairs: reserve two candidates
A naked pair appears when two cells in one row, column, or box contain exactly the same two candidates. Those two digits must occupy those two cells in some order.
Open step-by-step guidePointing pairs: let the box point down a line
A pointing pair or triple appears when every candidate for one digit inside a 3×3 box lies on the same row or column.
Open step-by-step guideBox-line reduction: let the line claim the box
Box-line reduction is the reverse of pointing pairs. All candidates for one digit in a row or column fall inside a single box.
Open step-by-step guideNaked triples: reserve three digits in three cells
A naked triple uses three cells in one unit whose combined candidates contain exactly three digits. A cell does not need to show all three.
Open step-by-step guideNaked quads: reserve four digits at once
A naked quad is four cells in one unit whose candidate union contains exactly four digits.
Open step-by-step guideX-Wing: lock one digit into a rectangle
An X-Wing uses one digit that appears in exactly the same two columns across two rows, or the same two rows across two columns.
Open step-by-step guideSkyscraper: compare two offset strong links
A Skyscraper uses two parallel strong links for one digit. One end of each link aligns to form the base; the other two ends form the roofs.
Open step-by-step guideTwo-string kite: join a row and column through a box
A two-string kite uses one row strong link and one column strong link for the same digit, connected by a weak link inside a box.
Open step-by-step guideSimple coloring: alternate one digit through strong links
Simple coloring follows one digit through units where it has exactly two positions, assigning alternating colors to each conjugate pair.
Open step-by-step guideSwordfish: extend fish logic to three lines
A Swordfish uses one digit across three base rows whose candidates all fall within the same three cover columns.
Open step-by-step guideJellyfish: four base lines, four cover lines
A Jellyfish is a size-four fish: one digit in four base rows is confined to four cover columns.
Open step-by-step guideFinned X-Wing: keep the fish and account for the fin
A finned X-Wing is an X-Wing with an extra base candidate outside the two cover columns.
Open step-by-step guideFinned Swordfish: three-line fish with one escape
A finned Swordfish has three base rows and three cover columns, plus one extra base candidate outside the covers.
Open step-by-step guideXY-Wing: connect two wings through a pivot
An XY-Wing uses a pivot with candidates XY and two visible wings XZ and YZ. The wings do not need to see each other.
Open step-by-step guideXYZ-Wing: make all three cells share one digit
An XYZ-Wing uses an XYZ pivot with two wings XZ and YZ. Both wings see the pivot, and all three cells contain Z.
Open step-by-step guideMulti-coloring: compare separate color chains
Multi-coloring extends simple coloring by keeping separate two-color components for the same digit until they interact.
Open step-by-step guideRemote pairs: alternate the same pair along a chain
Remote pairs form a chain of bivalue cells that all contain the same two digits. Consecutive cells must see each other.
Open step-by-step guideWXYZ-Wing: find one restricted common digit
A WXYZ-Wing is four cells whose combined candidates are exactly four digits, with one digit restricted so all of its occurrences see each other.
Open step-by-step guideUnique rectangle: prevent a two-solution loop
A Type 1 unique rectangle has four cells in two rows, two columns, and two boxes. Three cells contain the same pair; the fourth contains that pair plus extras.
Open step-by-step guideAvoidable rectangle: stop solved values from forming a deadly loop
An avoidable rectangle is the uniqueness counterpart found after some rectangle corners are solved. One remaining candidate would complete an interchangeable pattern.
Open step-by-step guideSue de Coq: partition a line-box intersection
Sue de Coq starts where a row or column intersects a box. The intersection candidates can be partitioned into one set in the box and another on the line.
Open step-by-step guideAlmost locked sets: one candidate away from a locked set
An almost locked set is N cells in one unit containing exactly N+1 distinct candidates. Removing any one candidate locks the remaining digits into those cells.
Open step-by-step guideALS-XZ: connect two almost locked sets
ALS-XZ uses two ALSs that share a restricted common candidate X: every X in one ALS sees every X in the other.
Open step-by-step guideALS chains: pass a restriction through several sets
An ALS chain joins multiple almost locked sets. Consecutive sets use restricted common candidates to pass a forced alternative from one end to the other.
Open step-by-step guideNice loops: close an alternating inference chain
A nice loop is a closed chain of candidate nodes joined by strong and weak links. The link sequence must remain logically continuous around the loop.
Open step-by-step guideForcing chains: compare every branch
A forcing chain begins from a candidate choice and follows its logical consequences. A conclusion is safe only when every possible branch reaches it.
Open step-by-step guideFinned Jellyfish: four-line fish with an escape
A finned Jellyfish is a size-four fish plus one or more extra base candidates outside the four cover columns.
Open step-by-step guideKraken fish: connect every fin to the target
A Kraken fish is a fish pattern whose fins are linked to a target through inference chains rather than direct visibility alone.
Open step-by-step guideTrial and error: test one candidate to contradiction
Trial and error assumes one candidate temporarily, follows only forced moves, and checks whether the branch creates a contradiction.
Open step-by-step guide