Y-Wing Sudoku Technique

The Y-Wing Sudoku technique, also called the XY-Wing, is built from three cells that hold only two candidate digits each. It is the first pattern most solvers meet that spans three different digits, and it is worth learning early because the wings above it are all variations of the same idea.

What the pattern is

You need three bivalue cells and exactly three digits between them:

• The pivot holds {x,y}.
• One pincer holds {x,z} and sees the pivot.
• The other pincer holds {y,z} and also sees the pivot.

The two pincers do not have to see each other. What matters is that each of them shares the pivot's row, column or box.

Now follow the pivot. If the pivot is x, the first pincer cannot be x, so it must be z. If the pivot is y, the second pincer cannot be y, so it must be z. The pivot has no other options, so one of the two pincers is always z.

That is the whole payoff: z is guaranteed to sit in one of the pincers, so any cell that sees both pincers can never be z.

When to look for it

• When the grid is stalled and you have several cells with exactly two candidates left.
• When two of those cells share a digit that a third bivalue cell does not have — that third cell is your pivot candidate.
• Especially around box corners, where a row peer and a column peer of the same cell often land in a shared box.

How to apply it, step by step

1. Pick a bivalue cell as the pivot and note its two digits x and y.
2. Scan its row, column and box for another bivalue cell containing x but not y. Call its second digit z.
3. Scan the pivot's peers again for a bivalue cell holding exactly {y,z}.
4. If you find both, you have a Y-Wing. Ignore the pivot from here on.
5. List the cells that see both pincers and still carry z.
6. Delete z from every one of them, then re-check for new singles.

A worked example

Three cells, with the rest of the grid left out:

          c2          c7
  r2    {5,8}  ...   {3,5}      <- pivot at c2, pincer at c7 (same row)
   .      :            :
  r6    {3,8}  ...  {1,3,9}     <- pincer at c2 (same column), target at c7

The pivot r2c2 is {5,8}. The pincer r2c7 is {3,5} and shares row 2 with it. The pincer r6c2 is {3,8} and shares column 2 with it. The three digits involved are 5, 8 and 3, so z is 3.

Run the two cases. If r2c2 is 5, then r2c7 loses its 5 and becomes 3. If r2c2 is 8, then r6c2 loses its 8 and becomes 3. One of the pincers is a 3 either way.

The cell r6c7 sees r6c2 along row 6 and r2c7 down column 7. So r6c7 cannot be 3, and it drops to {1,9}.

Common mistakes

• Four digits. If the three cells use four different digits, there is no shared z and no deduction. Count the digits before anything else.
• A pincer that does not see the pivot. Both pincers must be peers of the pivot. Two pincers linked only to each other prove nothing.
• Eliminating everywhere. Only cells that see both pincers are affected. A cell seeing one pincer and the pivot is safe.
• Using cells with three candidates. All three cells must be bivalue. A three-candidate pivot makes it an XYZ-Wing, which eliminates less.
• Missing the second target. Two pincers often share more than one common peer. Sweep the whole intersection, including the box.

Where you meet it

The Y-Wing belongs to the tough tier, and it is one of the most frequently needed patterns at that level. Expect it on hard classic Sudoku, where a stalled grid full of two-candidate cells is almost always hiding one.