Logo
Rätsel und Puzzles

≡ ▲

Sudoku

Fortschrittliche Lösungstechniken

Mit den fortgeschrittenen Strategien lassen sich auch die meisten schweren Aufgaben lösen – und wenn nicht, gilt eine derartige Aufgabe zumeist als "unlösbar".

Grundlegende Strategien:
Naked Pairs, Naked Triples, Naked Quads, Hidden Pairs, Hidden Triples, Hidden Quads, Intersection Removal, Pointing Pairs/Triples, Box/Line Reduction
Fortschrittliche Strategien:
X-Wing, Swordfish, Jellyfish, Squirm Bag, Turbofish, Finned X-Wing, Sashimi Finned X-Wing
Meisterhafte Strategien:
Singles Chains (Simple Coloring), Multi-Coloring, Y-Wing (XY-Wing), Y-Wing Chain, XY-Chains, XYZ-Wing, WXYZ-Wing, Aligned Pair Exclusion, Remote Pairs, Unique Rectangles, Guardians (Broken Wings)

X-Wing

Bei X-Wing schauen wir nach einzelnen Zahlen in Zeilen und Spalten. Die Regel lautet:

When there are

  • only two possible cells for a value in each of two different rows,
  • and these candidates lie also in the same columns,
    then all other candidates for this value in the columns can be eliminated.

    The reverse is also true for 2 columns with 2 common rows.

    X-Wing Example

    The above picture shows a classic x-wing, this example being based on the number six. The X is formed from the diagonal correspondence of squares marked A, B, C and D. What's special about them? Well, A and B are a locked pair of 6's. So is C and D. They are locked because they are the only 6's in the first and last rows. We know therefore that if A turns out to be a 6 then B cannot be a 6, and vice versa. Likewise if C turns out to be a 6 then D cannot be, and vice versa.

    What is interesting is the 6's present in the two columns 6 and 9 directly between A and C and B and D. These have been highlighted with red boxes. Think about the example this way. A, B, C and D form a rectangle. If A turns out to be a 6 then it rules out a 6 at C as well as B. Because A and CD are 'locked' then D must be a 6 if A is. Or vice versa. So a 6 MUST be present at AD or BC. If this is the case then any other 6's along the edge of our rectangle are redundant.

    We can remove the 6's marked in the cyan squares. This is good news because this leaves only a 9 at G9 and we can complete.

    This strategy works in the other direction as well. If we had two pairs in two columns and those four numbers shared two rows, then we can eliminate any other occurrences of those numbers on the same rows.

    Generalisierung von X-Wing

    X-Wing is not restricted to rows and columns. We can also extend the idea to boxes as well. If we generalise the rule above we get:

    When there are

  • only 2 candidates for a value, in each of 2 different units of the same kind,
  • and these candidates lie also on 2 other units of the same kind,
    then all other candidates for that value can be eliminated from the latter two units.

    Now we have 6 combinations:

    1. Starting from 2 rows and eliminating in 2 columns
    2. Starting from 2 columns and eliminating in 2 rows
    3. Starting from 2 boxes and eliminating in 2 rows
    4. Starting from 2 boxes and eliminating in 2 columns
    5. Starting from 2 rows and eliminating in 2 boxes
    6. Starting from 2 columns and eliminating in 2 boxes

     
    Here is an example of combination 5. Starting from 2 rows and eliminating in 2 boxes, in this case the last two boxes in the Sudoku. The rows are 7 and 8 and they each have two 7s. Our x-Wing is now a trapeziod but the logic is the same. We can be certain that 7 can be elminated at X, Y and Z.

    X-Wing Example

    But HOLD UP one moment. There is a simpler strategy that does the same job!

    X-Wing Example

    A and B above are a pointing pair. This removes the same 7s in the same place. Combination 6 is also the complement of a pointing pair. Combinations 3 and 4 are also complements of the Line/Box Reduction. Our generalization of X-Wing to boxes hasn't profited us at all. We learn that

     

    X-Wings containing boxes are the inverse of the Intersection Removal strategies

    • Swordfisch

    With X-Wing we looked at a rectangle formed by four numbers at the corners. This allowed us to exclude other occurrences of that number in either the row or column. We can extend this pattern to nine cells connected by locked pairs. In the example below (concentrating on the number 5) we have three sets of locked pairs at AB, CD and EF. They are all horizontal pairs but they also lock each vertically in a staircase fashion (I guess this inspired the name).

    SwordFish Example

    The vertical pairing is between AF, BD and CE. Now, in this example we can clearly see that the green horizontal lines connect pairs of 5. Because 5 is also locked vertically the red lines represent columns where if a 5 is not on our grid of nice nodes it can be excluded. There is one such 5 on cell X (E2).

    Another way of looking at it is to consider any 5 on the Sword-Fish grid. Pretending for a moment its a real 5 the others in the row and column are repressed. What we're left with is an X-Wing. X-Wing logic then applies to exclude the 5s it can see.

    Jellyfish

    Jelly-Fish extends Swordfish one further row and column. We are looking for either

    If this configuration is found then we can look in the opposite direction (if by row then down the column, if by column then across the row. If any candidates are found they can be eliminated. After the elimination both conditions above will hold.

    Jelly Fish 1

    How does it work? Pick any red cell in the example above that contains a 4. Keeping an eye on it. Pretend the solution actually is a 4. All others 4s in the row and columns are repressed. What we're left with is a Sword-Fish. The Sword-Fish logic then applies. Pick any 4 in the Sword-Fish and it reduces to an X-Wing. Since any combination of 4s on the grid are possible there is no room for 4s outside the grid - that align on the grid rows and columns.

    Squirm Bag

    Jellyfish kann auf fünf Zeilen und Spalten erweitert werden. Ich (Andrew Stuart) kenne jedoch kein Beispiel dafür.

    Turbofish

    (wird ergänzt)

    Finned X-Wing

    Siehe auch "Hunting finned X-Wing and other fishy stuff":

    (wird ergänzt)

    Finned Sashimi X-Wing

    (wird ergänzt)