W-Wing Sudoku Technique

The W-Wing Sudoku technique joins two identical bivalue cells with a strong link and turns the pair into a guaranteed placement. It is easier to spot than most wings because two of the three ingredients look identical, and it often fires on grids where a Y-Wing finds nothing.

What the pattern is

You need three things:

• Two cells holding exactly the same two candidates, {x,y}, that do not see each other.
• A unit — a row, column or box — in which the digit x has exactly two possible cells. Two positions in one unit is a strong link, sometimes called a conjugate pair.
• One end of that link sees the first bivalue cell, the other end sees the second.

Because the link is strong, one of its two cells really is x. Whichever end it is, that end knocks x out of the bivalue cell it sees, and that cell falls to y.

So at least one of the two matching cells is y. Every cell that sees both of them loses y.

When to look for it

• Whenever you notice two cells with an identical candidate pair sitting far apart in the grid.
• When a digit is nearly placed and several units are down to two spots for it — those units are your strong links.
• After the easy patterns dry up but before you start colouring or chaining, because a W-Wing costs one quick scan.

How to apply it, step by step

1. Collect the bivalue cells and group them by their digit pair.
2. Take two cells from the same group that do not share a row, column or box.
3. Pick one of the two digits, x. Look for a unit where x has exactly two candidate cells.
4. Check the link: one of its two cells must see the first bivalue cell, the other must see the second. One end per cell — not both ends on the same one.
5. If it matches, at least one bivalue cell is the other digit y.
6. Delete y from every cell that sees both bivalue cells. Then try the same pair again with the roles of x and y swapped.

A worked example

Only the relevant cells are shown. Row 5 has just two places left for a 4:

           c2          c6
  r1     {4,7}      {2,7,9}    <- target, loses the 7
  r5    4 possible  4 possible <- row 5: the only two spots for a 4
  r8    {1,7,8}      {4,7}     <- target at c2, pair cell at c6

The pair cells are r1c2 and r8c6, both {4,7}. They share no row, no column and no box, so neither can see the other.

Row 5 is the strong link on 4: it must be at r5c2 or r5c6. The cell r5c2 sees r1c2 down column 2; the cell r5c6 sees r8c6 down column 6.

If the 4 in row 5 is at r5c2, then r1c2 cannot be 4, so it is 7. If instead it is at r5c6, then r8c6 cannot be 4, so that one is 7. Either way, one of the two pair cells is a 7.

Now find the cells that see both. r1c6 sees r1c2 along row 1 and r8c6 down column 6, so it loses its 7 and becomes {2,9}. r8c2 sees r8c6 along row 8 and r1c2 down column 2, so it drops to {1,8}.

Common mistakes

• Pair cells that see each other. Then they are a naked pair, and the strong link adds nothing new.
• Both link ends aimed at one cell. The link must reach one bivalue cell from each end, or the case split collapses.
• Eliminating the linked digit. The link is on x; the digit you erase is always y.
• Weak links. The unit must have exactly two positions for x. Three positions is not a strong link and the argument fails, though it may still work through simple colouring.

Where you meet it

The W-Wing sits in the tough tier alongside the other wings. You will meet it on hard classic Sudoku, typically on grids that are rich in bivalue cells but have no clean fish or Y-Wing available.