Swordfish practice sheets

X-Wing with a third line

A swordfish is the three-line version of an X-Wing. Take one digit. Find three rows in which that digit has at most three possible cells — two or three each, not four. If every one of those cells falls inside the same three columns, then the digit can be eliminated from the rest of those three columns.

The reasoning is the same counting argument as the X-Wing: three rows each need the digit placed once, that is three placements, and the digit has only three columns available to it. So those three columns are fully occupied by those three rows, and nothing else in them can hold the digit.

That is why the X-Wing article says "anything you can say about X-Wing generalises to swordfish". It is literally the same proof with the numbers changed.

Why the sheet is slow to build, and why that is fine

Swordfish is rare. In an ordinary mixed puzzle set you can go a long time without meeting one, which makes it one of the techniques people read about and then never practise. To find puzzles that genuinely need one, the generator keeps drawing boards and discarding them — the first puzzle of a set sometimes takes a second or two, and the status line above the sheet shows it working.

Every puzzle is replayed with the cheapest technique first and kept only if a swordfish appears on that path. Nothing is cached between visits; the set is new each time you load the page.

What the grader guarantees

Every puzzle needs at least one swordfish. Because it sits near the top of this site's difficulty ladder, some puzzles also need one step beyond it — a W-Wing or a chain. That is not a flaw in the sheet; it is what the top of the ladder looks like.

Keeping the scan manageable

• One digit at a time. Pick a digit and count its homes in each row. Rows with four or more candidates are out immediately; you only care about rows with two or three.
• Write the three columns down. For each candidate row, list which columns its cells sit in. The pattern is there when the union of three rows covers exactly three columns.
• Eliminate in the columns, not the rows. If the base was rows, the digit is removed from the rest of the columns. It is never removed from the base rows themselves — those cells are the ones doing the work.

A pencil helps more than a screen here: three rows, three lists, compare the columns. That is the whole method, and it is why one puzzle per page is the version to learn on.

Related practice

The full guide to swordfish has a worked position and the reason a four-row version (jellyfish) exists but is almost never needed.

Build up to it: X-Wing for two lines, skyscraper for the missing-corner case, XY-Wing for the completely different kind of argument that swords never use.