Sudoku techniques, from singles to XY-wings

Every sudoku on Point Paper is solved by a program before it is published, using a fixed list of eleven techniques. The difficulty label is simply the hardest of those techniques the puzzle needs, and the Hint button explains its steps with the same names. This guide goes through the list in the order the levels introduce it, so that when a hint says “a pointing pair” or “an XY-wing”, you know what it means and can find the next one yourself.

A few words first

Rows are counted from the top and columns from the left. Boxes are numbered 1 to 9 left to right, top to bottom, so box 1 is the top-left 3×3 box and box 5 is the centre. Hints name squares the same way, as “row 4, column 2”, and call them cells.

Two squares see each other when they share a row, a column or a box. They can never hold the same digit. A square’s candidates are the digits it could still hold, given everything it sees. Nearly every technique below is a way of crossing candidates off until a square has one left, or a digit has one place left in some row, column or box.

How the levels work

• Gentle: hidden singles and naked singles only.
• Moderate: adds pointing pairs and triples, and box/line reductions.
• Tricky: adds naked and hidden pairs and triples.
• Fiendish: adds X-wings, swordfish and XY-wings.

The daily sudoku follows the week: Gentle on Monday and Tuesday, Moderate on Wednesday and Thursday, Tricky on Friday and Saturday, and Fiendish on Sunday.

Two things about the ratings are worth knowing. First, a harder label does not mean every step is hard. Even in a Fiendish puzzle, almost every digit comes from a single; the advanced technique is typically needed once or twice. Second, our solver checks that each puzzle can be finished with at most one elimination step between placements. So when you find a pointing pair or an X-wing, look straight at the squares it cleared: there is very often a digit waiting there.

What Gentle puzzles need

Hidden single

Pick a box (or a row, or a column) and a digit it is missing. Every square in that box that sees the digit somewhere else is ruled out. If only one square is left, the digit goes there. This is the workhorse of every level: the hint’s title for it reads like “Box 1 needs a 7”.

Spotting it quickly. Work digit by digit rather than square by square. Take a digit that already appears many times, and for each box that lacks it, run your eye along the rows and columns that pass through the box. On the site, selecting a square that holds a digit shades every other square with the same digit, which makes this sweep much faster.

Naked single

Here you look from the square’s side instead. Collect the digits in its row, its column and its box. If eight different digits appear among them, the ninth is the only one the square can hold. The hint phrases it as “Row 5, column 5 must be 3”.

Spotting it quickly. Naked singles hide in squares where a crowded row crosses a crowded column. They tend to appear late in a puzzle, when hidden singles in every direction have run out.

What Moderate puzzles need

Moderate puzzles introduce the first eliminations: steps that don’t place a digit, but remove a candidate so that a single appears. Both Moderate techniques use the overlap between a box and a line (a row or column).

Pointing pairs and triples

If, inside one box, every possible place for a digit lies in the same row, then the box’s copy of that digit will be in that row. The rest of the row, outside the box, can’t have it. The same goes for columns. With two possible squares it is a pointing pair, with three a pointing triple.

Spotting it quickly. While hunting hidden singles you will often find a box where a digit has two places, not one. Before moving on, check whether the two places share a row or column. If they do, follow that line out of the box and see what it blocks.

Box/line reduction

This is the same overlap read the other way. If, inside a row, every possible place for a digit lies in one box, then the row’s copy of the digit is in that box, and the other squares of the box can’t have it. Hints call this a box/line reduction. It turns up less often than the pointing kind, so it is easy to forget: when a row or column is nearly full, check whether its missing digits are squeezed into a single box.

What Tricky puzzles need

Tricky puzzles need subsets: groups of two or three squares, or two or three digits, that lock each other in. You will want notes for these, because the pattern is in the candidates, not the printed digits.

Naked pairs and triples

If two squares in the same row, column or box can each hold only the same two digits, those two digits will fill those two squares between them, in some order. No other square in that unit can hold either one. A naked triple is three squares whose candidates, together, are just three digits (each square need not have all three).

Spotting it quickly. Look for squares with only two candidates, then check whether a neighbour in the same unit has the identical two.

Hidden pairs and triples

The mirror image. If two digits can each go in only the same two squares of a unit, those squares must take those two digits, so any other candidates in them can be crossed out. The squares may look busy, with four or five notes each, which is why the pattern is called hidden. Triples work the same way with three digits confined to three squares.

Spotting it quickly. Count places, not candidates: for a unit that is resisting, list each missing digit with the number of squares it could use. Two digits that both have exactly two places, and the same two, are a hidden pair.

What Fiendish puzzles need

The Fiendish techniques look beyond one unit. They are rarer, but once you have seen each a few times the shapes become familiar. In our collection so far the XY-wing is the one Fiendish puzzles need most often.

X-wing

Choose a digit and find two rows where it has exactly two possible places, and those places are in the same two columns. Each row needs its copy of the digit, and the two copies can’t share a column, so one goes in each column, on opposite corners of the rectangle. Either way, both columns get their copy from these two rows, and no other square in those columns can hold the digit. The same works with columns and rows swapped.

Spotting it quickly. Pick one digit and, row by row, note which rows have it in only two places. If two of those rows use the same pair of columns, you have it.

Swordfish

A swordfish is an X-wing with three lines. If in three rows a digit can only go in three columns (each row using two or three of them), those three rows supply the digit for all three columns, and the rest of those columns loses it. It is the rarest technique on the list, and searching for it is slow, so try it last.

XY-wing

You need three squares with two candidates each. The pivot holds x or y. It sees two pincers: one holding x or z, the other y or z. If the pivot is x, the first pincer must be z; if it is y, the second pincer must be z. Either way one of the pincers is z, so any square that sees both pincers can’t be z.

Spotting it quickly. Start from squares with two candidates, which are easy to see if you keep notes. For each one, look along its row, column and box for another two-candidate square that shares one digit with it, then for a third that shares its other digit and the same new one.

A good solving order

1. Sweep for hidden singles digit by digit, 1 to 9, box by box. Repeat until a full sweep finds nothing.
2. Check the most crowded rows, columns and squares for naked singles.
3. Only now fill in notes. Placing a digit removes it from the notes of every square in the same row, column and box, so notes stay tidy once written (the controls are in How to play).
4. Look for pointing pairs and box/line reductions, then for pairs and triples, then for the Fiendish patterns.
5. After any elimination, go back to step 1 in the area it touched.

The order matters because easy steps are cheap to find and each one changes the board. Hunting for an X-wing while a hidden single is sitting in plain view wastes time.

Common mistakes

• Trusting stale notes. A note that was never correct will keep misleading you. The hint ignores your notes and works only from the digits on the board, so if it disagrees with your notes, check them.
• Reading a pair that isn’t one. A naked pair must be two squares in the same unit with the same two candidates. Two squares that merely share one digit prove nothing.
• Eliminating in the wrong place. A pointing pair clears its line outside the box; a box/line reduction clears its box outside the line. Mixing them up removes true candidates.
• Guessing when stuck. Every puzzle here can be solved without it. If nothing appears, go back to the start of the routine; the missed step is usually a single.

Using hints to learn

When a hint uses one of the techniques above, its explanation names it in brackets, highlights the squares involved and, usually, ends with the digit the step unlocks. Try to find that digit from the explanation before pressing Reveal, then look for the same pattern elsewhere on the board. Getting started has more on using hints without spoiling the puzzle.