Finned X-Wing

In sudoku, a Finned X-Wing is an X-Wing that almost works. One of the two base lines carries an extra candidate or two (the fin) and as long as every fin cell sits in a single box, you can still eliminate, just from fewer cells than a clean X-Wing would give you.

What the pattern is

Start with the usual shape: one digit, two base rows, and two cover columns holding the digit in both rows. In a clean X-Wing every candidate in those two rows lies in the two cover columns, and the digit leaves both columns everywhere else.

Now allow one base row to have extra positions outside the cover columns. Those extras are the fin. The reasoning splits in two:

• If no fin cell holds the digit, the shape is a real X-Wing and its eliminations apply.
• If a fin cell holds the digit, everything the fin sees loses the digit instead.

Only the eliminations that are true in both cases survive. That means a target cell must lie on a cover column and see every fin cell, in practice, the cover-column cells inside the fin's box. This is why the fin has to be confined to one box: spread the fin over two boxes and nothing sees all of it.

When to look for it

Look whenever an X-Wing search fails by one or two candidates. If you keep noticing "this would be an X-Wing except for that cell", check whether the offending cells share a box with one of the four corners. That is a Finned X-Wing.

How to apply it, step by step

1. Pick a digit and a base row with exactly two positions.
2. Find a second base row that shares those two columns and has one or two extra positions.
3. Check that all the extra positions sit inside one box, and that the box also contains one of the four corners.
4. Take the cover column running through that box.
5. Remove the digit from the cells of that column which lie inside the fin's box, leaving the pattern cells alone.

A worked example

Digit 5, base rows B and F. Row B holds 5 in B4, B7 and B9; row F only in F4 and F9. The cover columns are 4 and 9, and B7 is the fin, in box 3 with the corner B9.

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Step 0 of 6

1. Easier techniques are used up. Look at the 5s in rows B and F.

2. 1

   Row B can hold its 5 in B4, B7 or B9.

   Row F can hold its 5 only in F4 or F9.

3. 2

   Without B7 the two rows would be an X-Wing on columns 4 and 9. B7 is the fin: it shares box 3 with the corner B9.

4. 3

   Case 1: B7 is not 5. Then rows B and F are a plain X-Wing, and the rest of columns 4 and 9 lose 5: C4, A9, C9 and E9.

5. 4

   Case 2: B7 is 5. Then every other cell of box 3 loses 5: A7, A9, B9 and C9.

6. 5

   A9 and C9 lose 5 in both cases: they sit in column 9 and in the fin box.

   C4 and E9 lose it only in case 1, so they keep their 5.

7. 6

   5 removed from A9 and C9 because either the X-Wing or the fin in B7 rules it out.

Easier techniques are used up. Look at the 5s in rows B and F.

The demo tests both branches:

• B7 is not 5: rows B and F form a plain X-Wing, so columns 4 and 9 lose 5 in C4, A9, C9 and E9.
• B7 is 5: box 3 loses 5 in A7, A9, B9 and C9.

Only A9 and C9 lose 5 in both cases, so those two are removed. C4 and E9 lose it only in the first case; they do not see the fin, so they keep their 5.

Sashimi is the same pattern with a hole: a corner is missing entirely, and the fin cell stands in for it.

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Step 0 of 4

1. Easier techniques are used up. Look at the 6s in columns 2 and 7.

2. 1

   Column 7 can hold its 6 only in A7 or D7.

   Column 2 can hold its 6 only in C2 or D2. Its corner A2 is already filled with 2.

3. 2

   Without C2 the columns would lock their 6s into rows A and D, but the corner A2 is missing. That makes C2 a sashimi fin, in box 1 next to where A2 would be.

4. 3

   If C2 is not 6, column 2 needs D2, so column 7 needs A7, and row A loses its other 6s.

   If C2 is 6, box 1 loses its other 6s. A3 is in row A and in box 1, so it cannot be 6 either way.

5. 4

   6 removed from A3 because row A and box 1 both rule it out.

Easier techniques are used up. Look at the 6s in columns 2 and 7.

Here the digit is 6 and the base lines are columns 2 and 7 (the pattern runs in columns this time). Column 7 holds 6 only in A7 and D7; column 2 only in C2 and D2. The corner A2 is a given 2, so C2 is a sashimi fin in box 1. If C2 is not 6, D2 is, so A7 is, and row A loses its other 6s. If C2 is 6, box 1 loses its other 6s. A3 is in row A and box 1, so it loses its 6 either way.

Common mistakes

• A fin spread across two boxes. Then no cell sees all of it and the technique yields nothing.
• Eliminating from the far cover column. Only the column passing through the fin's box is in play.
• Removing candidates from the fin cells or the corners. They are part of the pattern.
• Forgetting to try columns as base lines. Every fish runs in both directions.
• Treating three or more fin cells in different rows as a fin. Keep it to the cells of one base line inside one box.

Where you meet it

The Finned X-Wing is an extreme-tier pattern. It first becomes necessary in Master puzzles, where clean fish have usually been mined out and the near-misses are all that remain. On Hard and Expert grids, look for the unfinned version first.