Techniques

Naked Single A cell has only one candidate digit left — every other digit 1–9 is already used somewhere in that cell's row, column or 3×3 box. Whatever's left is the only digit that can possibly go there. Singles Hidden Single A digit can only fit in one cell within some row, column or box, even though that cell may still show other candidates too. Since the digit has to go somewhere in that unit, and this is the only cell that can take it, it belongs there. Singles Pointing Pointing pair or triple. Look at one 3×3 box: if every remaining cell that could hold a digit lines up in a single row or column, that digit must end up somewhere in that line inside the box — so it can be eliminated from the rest of that row or column outside the box. Locked candidates Box/Line Reduction Box/line reduction — the mirror of pointing. Look at one row or column: if every remaining cell that could hold a digit falls inside a single 3×3 box, that digit must end up somewhere in that box — so it can be eliminated from the rest of the box outside that row or column. Locked candidates Naked Pair Two cells in the same row, column or box carry only two candidate digits between them, combined. Whichever cell ends up with which digit, both digits are used up between those two cells — so neither digit can appear anywhere else in that unit. Pairs Hidden Pair Two digits, within one row, column or box, only fit into the same two cells — even though those cells may still show other candidates too. Since both digits have to go somewhere in that unit, and only these two cells can take them, every other candidate can be cleared from those two cells. Pairs Naked Triple Three cells in the same row, column or box carry only three candidate digits between them, combined — not necessarily three each. Those three digits must fill those three cells between them, so every other cell in that unit can have them eliminated. Triples Hidden Triple Three digits, within one row, column or box, only fit into the same three cells — even though those cells may still show other candidates too. Since all three digits have to go somewhere in that unit, and only these three cells can take them, every other candidate can be cleared from those three cells. Triples Naked Quad Four cells in the same row, column or box carry only four candidate digits between them, combined — not necessarily one each. Those four digits must fill those four cells between them, so every other cell in that unit can have them eliminated. The rarest of the naked subsets to actually need: by the time four cells narrow to four candidates, a smaller subset has usually already fired first. Quads Hidden Quad Four digits, within one row, column or box, only fit into the same four cells — even though those cells may still show other candidates too. Since all four digits have to go somewhere in that unit, and only these four cells can take them, every other candidate can be cleared from those four cells. Quads X-Wing Pick a digit and two rows where it has only two possible cells left. If those two pairs sit in the very same two columns, the digit takes one cell in each row — and between them they use up both columns. So the digit can be eliminated from those two columns everywhere outside the pattern. It works identically starting from two columns and eliminating along rows. Fish Swordfish An X-Wing stretched to three. Three rows where a digit's remaining cells all fall inside the same three columns — each row needs two or three of them, not necessarily all three — must place the digit once per row, one in each column. Every other cell of those three columns can therefore lose the digit. As with X-Wing, rows and columns swap freely. Fish Jellyfish The same fish one size larger: four rows whose remaining cells for a digit all fall inside the same four columns. Four rows means four copies of the digit, one per column, so nothing else in those four columns can hold it. Rare in practice — by the time a puzzle needs a Jellyfish, the smaller fish have usually already broken it open. Fish Skyscraper Find a digit with exactly two candidate cells in each of two rows (or two columns), where one cell from each row shares a column. If that shared cell isn't the digit, its own row forces the digit into the row's other cell — and since the two shared cells can't both be the digit (they share a column), at least one row's *other* cell always ends up holding it. Any cell that can see both of those other cells loses the digit either way. Skyscraper & Kite Two-String Kite Find a digit with exactly two candidate cells in one row and exactly two in one column, where one cell from the row and one from the column sit in the same 3×3 box. Those two can't both be the digit — a box only holds it once — so whichever one misses out leaves its own line with nowhere else for the digit but that line's other cell. Either way, one of the row's or column's 'other' cells always ends up holding it, so any cell that sees both of those other cells loses the digit. Skyscraper & Kite Simple Colouring Take one digit and find every unit where it has exactly two possible cells left — in each of those, one of the two is definitely the digit. Chain those pairs together and colour them alternately, and the chain splits into two groups: one group holds the digit everywhere, the other nowhere. Two things follow without ever learning which is which. If a colour would put the digit twice into one unit it must be the wrong group, so the digit goes from every cell wearing it; and any cell outside the chain that sees both colours loses the digit either way. Colouring Empty Rectangle Find a box where a digit's remaining candidates all fall inside one row and one column crossing through it — nowhere else in the box. That box has to place the digit somewhere in that row or that column. Now find a strong link for the same digit elsewhere: two cells on a line running the other way, one of them sharing the box's row or column. Whichever of those two ends up as the digit, the crossing cell where the box's other line meets the strong link's far line loses it — either the far cell takes it directly (same line as the crossing cell), or the near cell does, filling the box's shared line and pushing the box's digit into its other line instead, which the crossing cell also shares. Empty Rectangle Y-Wing Three cells with two candidates each: a pivot holding X and Y, and two pincers it can see, one holding X and Z, the other Y and Z. The pivot is either X or Y, so one pincer or the other is forced to be Z — we just don't know which. Either way Z lands in one of them, so any cell that sees both pincers can't be Z. Wings XYZ-Wing A Y-Wing whose pivot carries three candidates instead of two: a pivot holding X, Y and Z with pincers X-Z and Y-Z that it can see. Z now has to land in one of three cells rather than one of two, which makes it harder to spot and narrows the payoff — only a cell seeing all three, pivot included, can have Z eliminated. Wings W-Wing Two cells, not seeing each other, that both carry only the same two candidates X and Y. Somewhere else on the grid, find a pair of cells for X that only leaves room for X in one or the other — a strong link — with one end seeing the first cell and the other end seeing the second. Whichever candidate the first cell turns out to be, the argument still lands on Y being used up in one of the two starting cells, so any cell that sees both of them can't be Y. Wings WXYZ-Wing Four cells carrying only four candidate digits between them, spread across a box and a line rather than sitting in one unit — that would just be a Naked Quad. Three of those digits can be used only once each inside the set, because every cell holding them sees every other one; three digits cannot fill four cells, so the fourth digit Z has to be placed somewhere in the set. Any cell outside it that sees every set cell holding Z loses Z. Wings