This is very similar to using X-Wings, in that it will allow you to use knowledge about rows to remove candidates from columns, and vice versa.
Make sure you're happy with why X-Wings work before moving on to Swordfish!
The complexity here is that you're using knowledge from 3 rows at the same time - and that's what makes them harder to spot. Unlike X-Wings, they don't form a simple rectangle.
This puzzle is mostly solved - but we've reached a point where simpler methods aren't helping.
There's actually a Swordfish in 4s in this puzzle, so we'll explain what it is and how it works.
To begin with, highlighting all of the places where 4 is still a candidate will help to make things easier.
What we're looking for are sets of values that we can use to make a chain - just like in an X-Wing being a closed chain of four values, a Swordfish needs a closed chain of 6 (or more) values.
The Swordfish here is in three rows (3, 5 and 8). We'll remove the other values for now to make it a little clearer.
Just like in the X-Wing example, a value in one position forces the other in the same row to not be that value. Lets put in some arrows to help to show that.
See that each of the arrows end in a column that matches one of the other rows?
This makes a fairly neat closed chain - and that means we can be sure that every one of those columns is occupied. To show the links, here are the arrows.
There really are only two possibilites for the positions of the 4s within this loop:
Either way the values were arranged, you can see that these three columns are occupied by the contents of those three rows.
Once again highlighting the columns, you know that you can remove candidates for 4 from anywhere in those columns other than the three Swordfish rows.
That's a great deal of work just to remove one candidate - but any progress helps when you're in the toughest puzzles!
This only works because the loop is closed! This makes it easier to search for – you know that if you follow a chain and find yourself back at the start, there’s a closed loop! It might not mean you can remove any candidates every time, though, which means you have to carry on searching.
Here’s another example – there’s a Swordfish in rows for 1s:
Hang on... so this works for any closed loops?
Yup - and it doesn't have to be limited to lines - its possible to connect values that share the same box, but it really does get incredibly complicated! Chances are that you can probably find a simpler method to help you.
Isn’t an X-Wing just a closed loop?
Again, yup! An X-Wing and Swordfish are really the same thing - an X-Wing with 2 rows and columns, and a Swordfish on 3 rows and columns.
If you can see where this is heading... yes, it means that it is possible to have a Swordfish-4, which means it uses the connections between 4 lines! (These are sometimes called Jellyfish.) These are incredibly rare indeed, and usually another technique will work without you having to rely on them!