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.
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 ≥ 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.