Somewhere past the subset techniques, Sudoku vocabulary takes a turn for the aquatic: X-Wing, Swordfish, Jellyfish, and — for the size-five pattern — the magnificently named Squirmbag. The names are whimsical, but the family they describe is the most elegant idea in intermediate solving: a single counting argument about rows and columns that works at every size. This guide explains the logic once, properly, then the taxonomy, then the curious history of the names.
The core argument, using the X-Wing
Fish are single-digit techniques: you reason about one digit at a time, ignoring all others. Suppose the digit 7 has exactly two candidate cells in row 2, and exactly two candidate cells in row 6 — and these fall in the same two columns, say columns 3 and 8. The four cells form a rectangle.
Row 2 must contain a 7 somewhere, and its only options are columns 3 and 8. Likewise row 6. The two rows cannot use the same column (that would put two 7s in one column), so between them, rows 2 and 6 will place their 7s in columns 3 and 8 — one each, in some order. We do not know which order, and we never need to. Either way, column 3 gets its 7 from row 2 or row 6, and so does column 8. Both columns' 7s are accounted for — so every other candidate 7 in columns 3 and 8 can be eliminated. That is the X-Wing.
The general statement: if a digit's candidates in n rows are confined to the same n columns, then those n columns' copies of the digit must come from those n rows — and the digit is eliminated from those columns everywhere else. Solvers call the n rows the base sets and the n columns the cover sets. Everything works with rows and columns swapped, and if the pigeonhole flavor feels familiar, it should: this is the same k-things-in-k-places counting as the subset techniques, lifted from one unit to the whole grid.
The family by size
| Size (n) | Name | Pattern |
|---|---|---|
| 2 | X-Wing | 2 rows confined to 2 columns (or vice versa) |
| 3 | Swordfish | 3 rows confined to 3 columns |
| 4 | Jellyfish | 4 rows confined to 4 columns |
| 5 | Squirmbag | 5 rows confined to 5 columns — never needed (see below) |
One important relaxation at sizes three and up: the rows need not each contain the digit in all of the cover columns. A Swordfish needs each of its three rows to have candidates in only (a subset of) the three columns — two per row is fine, and the classic “2-2-2” Swordfish is actually the easiest to spot. What is forbidden is a base row having a candidate outside the cover columns.
Why you never need a fish bigger than a Jellyfish
Here is the family's most satisfying theorem. For any digit, consider the rows: some contain the digit already; suppose k of the remaining rows form a fish with k cover columns. A complementary-counting argument — the same one that pairs naked subsets with hidden ones — shows that a size-k fish in rows always coexists with a size-(m − k) fish in columns, where m is the number of rows/columns still missing the digit, and the two produce identical eliminations. With at most nine units in play, whenever a Squirmbag (5) or larger exists, a Jellyfish (4) or smaller exists pointing the other way. Squirmbags are hunted for sport, not necessity.
Where the names came from
The vocabulary was coined in the early 2000s on online solving communities — principally the Sudoku Players' Forum and its predecessors — where enthusiasts were naming patterns as fast as they discovered them, and whimsy was a feature, not a bug.
- X-Wing is generally understood as a Star Wars reference: connect the four corner cells and the crossing diagonals draw an X, like the fighter's wings. (It is a rectangle pattern, but “Rectangle” was destined for the uniqueness family instead.)
- Swordfish is the name that started the fish theme, and its exact origin is genuinely murky — early forum posters themselves debated it. Once “Swordfish” stuck for size three, the community ran with the joke: size four became Jellyfish, and the family as a whole became fish. The retronym then swallowed the X-Wing, which is nowadays classified as the size-two fish despite its aeronautical name.
- Squirmbag — the size-five fish — is fisherman's slang for a bag of live bait, a wink at the fact that the pattern is more wriggle than substance, since a smaller complementary fish always exists.
Later refinements extended the taxonomy with finned fish (a fish spoiled by one or two extra candidates — the “fin” — which still eliminates in the cells that see the fin) and sashimi fish (a finned fish that is missing part of its body), plus franken and mutant fish that use boxes as base or cover sets. Those lie beyond this site's ladder, but they are all the same cover-set argument with looser geometry — once the X-Wing proof above feels obvious, none of them will frighten you.
How to actually find fish
Fish live or die on single-digit vision — scanning the grid for one digit and seeing only its placements and candidates. The efficient search:
- Pick a digit placed five or six times. Few rows remain, so qualifying base rows are easy to enumerate.
- List rows where the digit has exactly two or three candidate cells. Rows with two candidates are X-Wing material; note their column pairs.
- Look for repeats. Two rows sharing the same column pair is an X-Wing. Three rows whose candidate columns union to three columns is a Swordfish.
- Repeat with columns as the base. Fish hide in both orientations, and by the complement theorem, a hard-to-see row fish is often an easy column fish.
Pencil-mark hygiene matters doubly here, since a single stale candidate can fake or hide a base row — the maintenance habits are a prerequisite. When you are ready to train the vision, the walkthroughs show real X-Wings and Swordfish with the eliminations drawn on the board, and the technique picker will generate puzzles guaranteed to require one. Ten fish in a row will do more for your eye than any amount of theory — including this article.