Basic Fish

A fish is a grid structure formed by the intersection of $n$ rows and $n$ columns. These two kinds of houses can be divided into two groups: the base set and the cover set. If any $n$ rows are taken as the base set, and all occurrences of some candidate digit $d$ in those $n$ rows fall exactly within $n$ columns, then those $n$ columns are the cover set of the fish. The converse is also true: any $n$ columns can be taken as the base set, and if all occurrences of some candidate digit $d$ in those $n$ columns fall exactly within $n$ rows, then those $n$ rows are the cover set of the fish.

Because Sudoku requires that a digit may appear only once in any house (row, column, or box), the $n$ candidate occurrences of $d$ in the base set must always ensure exactly one occurrence in each house of the cover set. Therefore, according to Sudoku rules, candidate digit $d$ cannot appear in any other cells of the cover set; if it does, it can be eliminated.

Here, the number $n$ of houses (rows or columns) contained in the base set or cover set is called the fish's size or order. So when size matters, a fish may be called an "$n$-fish". Since the size of a fish affects how difficult it is to spot, fish of different sizes are usually treated as different techniques.

Fish size $n$	Technique alias
$2$	X-Wing
$3$	Swordfish
$4$	Jellyfish

In theory, $n$ can be any integer from $1$ to $9$. But if $n=1$, the intersection of one row and one column degenerates into a single cell, so no elimination is possible. If $n=9$, the cover set already contains at most all nine rows or columns, so there are no "other cells" left to eliminate from. Also, for practical human solving, fish with $n\ge 5$ are usually too difficult to spot, so other techniques should generally be considered to simplify the puzzle instead.

It is worth noting that candidate digit $d$ is not required to appear exactly $n$ times in every house of the base set. It is only required that whenever it appears, it falls within the cover set. In practice, however, candidate digit $d$ is usually expected to appear at least twice in every house of the base set; otherwise, in that house it would degenerate into a hidden single.