Technique Glossary

A house with only one empty cell left, which must take the one digit still missing from it.

Every row, column and box needs each digit exactly once. If eight of a house's nine cells are already filled, only one digit is missing overall, and only one cell is left to receive it -- no candidates need to be checked at all.

Result: Places one digit immediately, and clears that digit from the solved cell's peers.

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A cell left with only one possible candidate.

Once every peer in a cell's row, column and box has claimed a digit, only one digit has nowhere else to go in that cell. A naked single is often the visible end result of eliminations made earlier, rather than a discovery in its own right.

Result: Places one digit, which can then be eliminated from every peer of that cell.

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A digit that has only one legal cell left within a house, even though that cell still shows other candidates.

Fix one digit and one house, then rule out any cell that already sees that digit elsewhere in the house. If exactly one cell survives, the digit belongs there -- the deduction comes from counting the digit's remaining positions, not from counting the cell's candidates.

Result: Places the digit and clears every other candidate from that cell, plus the digit from its peers.

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A digit's remaining candidates inside a box all fall on one row or column, so the digit is barred from the rest of that line outside the box.

The box must place the digit somewhere, and every option for it already sits on one shared line. Whichever cell turns out to be true, it occupies that line -- so nothing else on the line, outside the box, can hold the digit.

Result: Eliminates the digit from the rest of the row or column, outside the box.

The mirror image of pointing: a digit's remaining candidates on a row or column all fall inside one box, so the digit is barred from the rest of that box.

The line must place the digit somewhere, and every option for it already sits in one box. The box's instance of the digit is therefore owed to those intersection cells, so nowhere else in the box can hold it.

Result: Eliminates the digit from the rest of the box, outside the shared line.

Two cells in the same house whose candidate lists are identical and contain exactly two digits.

Between them, the two cells must absorb both digits, one each. Because the pair already reserves both values for that house, no other cell in the house may use either digit.

Result: Eliminates both digits from every other cell of the shared house (and from a second house too, if the pair also shares a box-line intersection).

Two digits that, within one house, appear as candidates in only two cells, regardless of what else those cells contain.

Each digit needs a home in the house, and both are limited to the same two cells -- so those two cells are reserved for those two digits collectively. Any other candidate in either cell would leave one of the two digits with nowhere to go.

Result: Eliminates every other candidate from the two cells, which usually reveals a naked pair underneath.

Three cells in one house whose combined candidates span only three digits.

Three digits must be distributed across three cells -- one each. No single cell needs to show all three candidates, only their combined set across the three cells has to equal exactly three digits.

Result: Eliminates the three digits from every other cell of the house.

Three digits whose remaining candidate positions in one house are limited to the same three cells.

Each of the three digits must be placed somewhere in the house, and all three are confined to the same three cells, so those cells are collectively booked for them. Any other candidate there would leave one digit homeless.

Result: Eliminates every non-member candidate from the three cells, typically leaving a naked triple behind.

Four cells in one house whose combined candidates cover only four digits.

The same reasoning as a naked triple, one size larger: four digits spread across four cells consume them entirely, so the rest of the house cannot use any of the four.

Result: Eliminates the four digits from all other cells of the house.

Four digits whose candidate positions inside one house are limited to the same four cells.

The four digits must occupy those four cells between them, so no other candidate can survive there. This pattern is rare, and a solver checking naked subsets first will often surface its complementary naked subset instead.

Result: Eliminates all non-member candidates from the four cells.

One digit restricted to the same two columns across two rows (or the same two rows across two columns), forming a rectangle of four candidate cells.

Each of the two rows must place the digit in one of the two shared columns. Whichever corner is true in the first row forces the diagonally opposite corner in the second row, since neither a row nor a column can repeat the digit -- so between them, the two columns are fully accounted for.

Result: Eliminates the digit from the rest of the two columns (or rows), outside the four-cell pattern.

Follow one digit through a chain of houses that each have only two remaining positions for it, alternating two colors along the chain, then look for a contradiction.

A house with only two positions for a digit means exactly one of them is true -- alternating colors along a connected chain of these links produces one cluster where one color is entirely true and the other entirely false. If two same-colored cells share a house, that color is impossible (a wrap). If an uncolored cell sees both colors, it can't hold the digit either (a trap).

Result: Eliminates the digit from cells caught by a color trap, or from an entire contradicted color in a color wrap.

The three-line version of an X-Wing: one digit confined to the same three columns across three rows, or vice versa.

Each of the three base rows must place the digit in one of the three shared columns, and the three placements must land in three different columns -- so the columns are entirely spoken for by the base rows, even if not every intersection cell carries a candidate.

Result: Eliminates the digit from the rest of the three columns (or rows), outside the pattern.

The four-line version of the same fish pattern: one digit confined to four columns across four rows, or vice versa.

The same base/cover logic as X-Wing and Swordfish, one size larger. If a candidate outside the pattern were true, one of the four base lines would be left with no legal position for the digit.

Result: Eliminates the digit from the rest of the four columns (or rows), outside the pattern.

Three bivalue cells -- a pivot holding two digits X/Y, and two pincers holding X/Z and Y/Z, each sharing a house with the pivot -- that together force digit Z into at least one pincer.

The pivot must be X or Y. If it's X, the X/Z pincer is forced to Z; if it's Y, the Y/Z pincer is forced to Z. Either way, at least one pincer holds Z, so any cell that sees both pincers can't hold Z.

Result: Eliminates Z from every cell that sees both pincer cells.

Like an XY-Wing, but the pivot cell carries all three digits X, Y and Z instead of just two.

The pivot can be X, Y, or Z. If X, the X/Z pincer supplies Z; if Y, the Y/Z pincer supplies Z; if Z, the pivot itself supplies it. Z is guaranteed somewhere among all three cells, so an eliminating cell must see all three -- pivot included.

Result: Eliminates Z from any cell that sees all three cells of the wing.

Two cells holding an identical candidate pair that don't see each other, connected by a strong link on one of the two shared digits.

Say the strong link is on digit X: exactly one end of that link must be X. Each end of the link sees one of the two bivalue cells, so whichever end turns out to be X forces the bivalue cell it sees to drop X -- leaving the other digit, W, as its answer. So at least one of the two paired cells is W.

Result: Eliminates W from every cell that sees both of the bivalue cells.

One digit with exactly two positions in each of two rows (or columns), where one position from each shares a line, leaving two offset far ends.

Each base line gives a strong link on the digit. The two ends that share a line can't both be true, so working through both possibilities always forces the digit into one of the two far, offset ends.

Result: Eliminates the digit from any cell that sees both far ends.

One digit with exactly two positions on a row and exactly two positions on a column, where one end of each meets inside a shared box.

Each line is a strong link, so each contributes one true cell. The two ends that share a box can't both be true, so the reasoning always forces the digit into one of the two outer ends.

Result: Eliminates the digit from any cell that sees both outer ends.

A box where one digit's candidates are confined to a single row and a single column within that box, combined with a strong link elsewhere on the same digit.

Because the box's candidates for the digit fit inside one row-of-the-box and one column-of-the-box, the digit must eventually sit on one of those two lines. Pairing that with a strong link on the same digit elsewhere lets both possible outcomes point at the same target cell.

Result: Eliminates the digit from the one cell where the strong link's other end crosses the box's remaining line.

Four cells across two rows, two columns and exactly two boxes, where three are pure two-digit pairs and the fourth carries extra candidates.

If all four cells held only the same two digits, they could be swapped diagonally to create a second valid solution -- which a properly built puzzle can't allow. So whichever corner carries extra candidates is the only cell that can break the pattern, meaning it cannot be either of the two rectangle digits.

Result: Eliminates both rectangle digits from the corner with extra candidates.

A potential deadly rectangle where two corners on the same side each carry exactly one identical extra candidate.

The other two corners are pure pairs. If neither of the two remaining corners held its extra candidate, all four would collapse to the same two digits and create the impossible double-solution pattern -- so the extra candidate must be true in one of those two corners.

Result: Eliminates the extra candidate from every cell that sees both of the corners carrying it.

A late-game grid where every unsolved cell is bivalue and every candidate appears exactly twice per house, except for one cell carrying a single extra candidate.

A grid that is entirely bivalue with every candidate appearing exactly twice per house would have two equally valid completions -- impossible in a well-formed puzzle. If the current grid matches that pattern plus one extra candidate, removing that candidate would make the impossible pattern real, so the extra candidate must be the answer.

Result: Places the extra candidate in its cell, which often cascades into several more singles.

A single-digit chain that alternates strong and weak links and both starts and ends with a strong link, proving the digit sits at one of the two ends.

A strong link means at least one of two cells is true; a weak link means at most one is. Alternating them lets the argument “this end isn't the digit” propagate all the way to “the far end is the digit” -- and reading the chain backwards proves the same thing about the other end.

Result: Eliminates the digit from every cell that sees both chain ends. Length-4 chains are exactly the Skyscraper / 2-String Kite / Empty Rectangle family seen at Hard tier.

A chain of bivalue cells, each linked to the next by sharing a house, whose two end cells share a common digit.

Inside each bivalue cell, rejecting one candidate forces the other -- a strong link. Between consecutive cells, sharing a house on the linking digit forwards the implication -- a weak link. Following the chain from either end proves the shared end digit must be true at one of the two ends.

Result: Eliminates the shared end digit from every cell that sees both end cells.

Coloring one digit produces several separate two-color clusters; comparing two clusters against each other yields eliminations a single cluster can't reach alone.

Each cluster still obeys the rule that one of its two colors is entirely true. If a cell in one cluster shares a house with a cell in another, they can't both be true -- so any cell seeing the opposite color in both clusters is eliminated. If two same-colored cells in one cluster both see the two opposite colors of another cluster, that entire color is false.

Result: Eliminates the digit from cells seeing two designated opposite colors, or from an entire contradicted color cluster.

A backtracking solver doesn't reason about patterns at all. It searches. It picks an unsolved cell, tries a legal digit, and recurses into the resulting grid. If a later cell runs out of legal digits, the solver undoes the most recent guess and tries the next one, unwinding further back whenever every option at a level is exhausted. The search either fills the whole grid or proves no solution exists.

This is why it's a validator rather than a teacher: its output is an answer, not a reason, so it can never explain why a digit is forced. Sudoku University uses it behind the scenes only, to confirm every generated puzzle has exactly one solution, to double-check that the technique ladder's placements agree with the true solution, and as a last-resort fallback if the human-technique pipeline ever stalls. The difficulty rating and every hint you see are always derived from the technique ladder above, never from this solver.

Result: confirms or completes a solution; never produces a taught hint.