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# Finned fish

In the page on matching we saw Principle C: a given digit defines a matching between rows and columns, and if the matching is tight for a subset, this may allow eliminations. The resulting argument was called 'X-wing' (on two lines), 'Swordfish' (on three lines), 'Jellyfish' (on four lines). However, the argument can be made much more widely applicable.

## Rule

A finned N-fish with elimination * for a given digit d is a set of N rows together with a cell * not in one of these rows, such that, after removal of all cells that 'see' * (i.e., lie in the same row, column or box as *), the remaining cells in these N rows that have d as a candidate lie in N-1 columns. Since N digits d cannot lie in N-1 columns, it follows that the cell * cannot be d, and we can eliminate this candidate.

(Or the same with rows and columns interchanged.)

If, not counting the row or column of *, all candidates for N rows lie in N-1 columns, then all candidates for the remaining 9-N columns lie in the remaining 8-N rows, also a contradiction. So one has a finned N-fish iff there is a finned (9-N)-fish, and it is never necessary to look at N larger than 4.

## Example

After some work, the given puzzle has progressed to the second diagram. The solution follows immediately from the finned 4-fish (jellyfish) indicated. Look at the digits 5 in columns 2,3,4,7. They may occur in rows 1,2,3,4,6. If (1,9)5, then the candidates in (1,4) and (3,7) are impossible, and only rows 2,4,6 are left. But four digits 5 cannot be in three rows, so (1,9)!5. The complete solution now follows quickly.