Swordfish

Fix your attention on one digit — say 2. A Swordfish lives in three base lines: three rows, or three columns, in which every candidate for that digit is confined to the same three cross-lines. If your bases are rows, the cross-lines are columns; if your bases are columns, the cross-lines are rows.

The count is what matters, not the shape. Each base line may hold two or three candidate positions for the digit. What you are checking is that, added together, all three base lines touch only three distinct cross-lines between them. When that holds, the pattern is locked.

An X-Wing is the two-line version. Two rows, each with the digit's candidates pinned to the same two columns, force that digit onto those two columns — so you can clear it from the rest of both columns. The reasoning never mentions "two" as anything special. It only needs the digit to be trapped in a closed grid of lines.

Push that from two to three and you have a Swordfish. Three base rows whose candidates for the digit all fall within the same three columns must, between them, use each of those three columns exactly once. There is nowhere else for the digit to go in those rows. So the three columns are spoken for across the base rows, and the digit can be removed from those columns everywhere outside the three bases.

One difference is worth flagging. An X-Wing is always a tidy rectangle: two rows, two columns, four corners. A Swordfish need not be. A base row might carry the digit in only two of the three columns rather than all three. The cells can sit in a ragged, gap-toothed arrangement — and the logic holds regardless, provided no fourth column ever appears.

The grid above shows a row-based Swordfish on the digit 2. Look at rows 1, 7 and 9 — the base rows. In each of them, every remaining candidate for 2 sits in one of just three columns: columns 1, 5 and 9. Those blue-outlined cells are r1c1, r1c5, r7c1, r7c9, r9c5 and r9c9.

Notice the shape. Row 1 offers 2 in columns 1 and 5. Row 7 offers it in columns 1 and 9. Row 9 offers it in columns 5 and 9. No row uses all three columns, and no two rows use the same pair — yet across the three of them, only columns 1, 5 and 9 ever appear. This is the ragged, non-rectangular case: there is no clean box of corners, but the three-column ceiling is intact.

Now the conclusion. Row 1 must place its 2 in column 1 or 5. Row 7 must place its 2 in column 1 or 9. Row 9 must place its 2 in column 5 or 9. However these three rows resolve, they collectively fill columns 1, 5 and 9 — one 2 in each. That leaves no room for a 2 in those three columns anywhere else. So any 2 sitting in column 1, 5 or 9 outside rows 1, 7 and 9 can be removed. Those are the red, struck-through candidates, such as r3c9 and r2c5.

The guarantee comes from counting. Three rows each need one copy of the digit, and by assumption those copies can only fall in three columns. Three copies distributed over three columns, with at most one per column, must use every column exactly once. It is a pigeonhole argument, nothing more.

Because all three columns are consumed by the base rows, a copy of the digit anywhere else in one of those columns would be a second copy in that column — which the rules forbid. So the eliminations are not a guess or a likelihood. They follow with certainty, the same way an X-Wing's do. Swap rows for columns and the argument runs identically: three base columns confined to three rows let you clear the digit along those rows.

You will not stumble onto a Swordfish by staring at the whole grid — there is too much going on. Hunt it deliberately, one digit at a time. Take a digit and, for that digit alone, tally how many candidate cells sit in each row. Rows with two or three are your candidates for a base; rows with four or more cannot take part. Ignore everything else.

Now look at the columns those thinned-out rows use. If you can pick three such rows whose candidates, pooled together, touch only three columns, you have found the pattern. If the pooled columns spill to four, that trio will not work — try swapping one row for another, or start again with a different digit. Then run the same sweep with columns as the bases and rows as the cross-lines, since a Swordfish can sit either way up. It is patient, bookkeeping work, and a pencilled candidate grid makes it far easier than trying to hold the counts in your head.

Be honest with yourself about how often this comes up. A true Swordfish is uncommon. If you solve by always taking the plainest available move first — a naked single, a hidden single, a locked candidate, an X-Wing — you will usually find something easier before a Swordfish is your only option. Much of the time the same eliminations arrive through a shorter route.

So treat the Swordfish as a last logical resort: the pattern you reach for when the ordinary techniques have run dry and the alternative is opening up longer chains. When it is genuinely there, it is precise and clean, and it can break a position that looked stuck. Scan for it methodically — pick a digit, ask whether its candidates in three lines are penned into three cross-lines — rather than hoping to spot one at a glance. It rewards patience more than flair.