X-Wing
An X-Wing is four cells that form a rectangle: in two rows or columns a digit can only go in the same two places. Then the digit can be removed from the rest of the crossing lines.
Learn the techniqueKraken Fish
Kraken Fish is here an X-Wing or Swordfish with one fin, that is, an extra candidate that almost ruins the regular fish pattern.
See the technique in practice
Work through the examples step by step. Each step explains what you see on the puzzle and why the conclusion holds.
Example:
- We follow digit 4 through row 1 og row 4. It almost looks like a fish, but one of the lines has too many candidates.
- The extra candidate sits in row 1, column 3. We call it the fin, because it breaks the perfect pattern.
- Either the pattern works as normal without the fin, and then 4 must be removed from the rest of column 1 og column 2 just like in a regular fish.
- Or 4 sits in the fin itself, and then all cells that see the fin lose digit 4.
- Cells hit in both cases can be removed regardless. We remove 4 from the notes in row 3, column 2.
How to recognize the pattern
Kraken Fish is a fish with a fin. You find an almost perfect fish pattern, for example an X-Wing or Swordfish, but one extra candidate in one of the base lines breaks the pattern. That extra candidate is called the fin, and instead of discarding the fish you use the fin as a separate hypothesis.
The reasoning has two branches. If the fin is wrong, the fish works as normal, and the digit can be removed from the rest of the cover. If the fin is right, the digit sits in the fin cell, and all cells that see the fin lose the digit. Cells hit in both branches can be removed regardless, and in practice those are the cells that both lie in the cover and see the fin.
Step-by-step procedure
- Look for a fish pattern where one of the base lines has one candidate too many.
- Mark the fin, that is, the candidate that breaks the pattern.
- Find the removals the fish would give without the fin.
- Find the cells that see the fin, and keep only the removals that are hit in both cases.
Common mistakes
- Using the fish's full removals. With a fin, only the intersection of the two branches is safe to remove.
- Allowing multiple fins without expanding both branches. Each extra fin gives an extra branch that all removals must hold in.
- Forgetting the box around the fin. The cells that see the fin include its box, and that is usually where the valid removals lie.
When do you need the technique?
The hardest puzzles require techniques that follow long logical chains through the entire puzzle. They are in practice proof by contradiction: assume something, follow the consequences and see what does not hold. Try your way through the examples below, step by step, using the same tools the solver uses on your own puzzle.
Try it on your own puzzle
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