W-Wing Strategy

W-Wings have been around for a long time [see 1 and 2] and I was late to the party even in 2014 when I coded them into the solver but I felt they were overlapping with several other chaining strategies so didn't include it in the main Sudoku solver until now (January 2026).

But it is helpful to the human solver to know as many stand-out strategies as possible and these are relatively easy and can go early in the 'tough' order of strategies. I'll explain the impact with some stats later in the article.

For the time being I'm introducing them just to the Sudoku Solver.
If anyone knows why they are called W-Wings, do let me know.

To explain this strategy it is worth discussing the exception rather than the rule. Consider this 'tough' puzzle at this point. There happen to be a lot of 4/9 cells on the board. Pairs like this don't appear in isolation - if there are two in the same row, column or box they would be Naked Pairs and we'd have used them to remove other 4s and 9s in those units. But we are past that point.

Instead we can check if there are two cells with 4 and 9 that can't see each other (not a conjugate pair) but can be connected to each other using other 4+9 cells. This is the case with A3 and H7. They are connected by strong links via A9 and G9. The solver is displaying an alternating strong/weak chain not just because weak links can be strong but because we alternate on and off. The solver is showing one of four possible chains logic:

• A3 is 4 implying H7 must be not 4, so 9.
• A3 is 9 implying H7 must be not 9, so 4.
• A3 is not 4 (must be 9) implying H7 must be 4.
• A3 is not 9 (must be 4) implying H7 must be 9.

Any of these are proofs. It means that any cell that can see both ends of the chain cannot have a 4 or a 9. Such as cell is H3

In the very next step is another elimination of 4s and 9s, this time in H2. A slightly different set of cells but I'm sure the pattern and logic is clear.

The name for this pattern is a Remote Pair Chain. A more convoluted explanation for this appears here as Remote Pairs. A long time ago I removed it from the solver as it was wholly a subset of XY-Chains. But length-4 version it is still simple to spot and worth using before the broader XY-Chain logic is invoked.

Using this pattern three times saves six Rectangle Eliminations. Untick W-Wings and try.

Note: Remote Pairs can be more than four cells long. But the next size is six and then eight - so even numbers. The coincidence of so many identical bi-values cells means they are increasingly rare at longer lengths but keep an eye out.

So what does this have to do with 'W-Wing'?

In this puzzle we've found two cells with 2 and 9 in them - cells B9 and J2. They are far apart and cannot see each other. We cannot connect them through 2 but we can connect them through 9 and two other bi-value cells.

If B9 is not a 2 it must be 9. That gives us a weak link from +9 to -9 in B1. -9 in B1 forces +9 in J1 - a strong link (this is the important - it must be strong). +9 in J1 removes 9 in J2 making it 2. Reverse the chain logic and we prove that 2 must be in one of those orange cells. There is one cell that can see both ends B2 and we can remove 2 from it.

We can go again in the very next step. This time it is {1/2} in J3 connected to C9 via the strong link on 1 between C3 and J1. Trace the logic in both directions to satisfy yourself that one end of the W-Wing (orange cells) must be 2.

It might be clear now that if all the cells share the same digits you have two mirrored Single W-Wings and therefore a Double or a Remote Pair Chain.

I've tested the strategy on two datasets. Andy Potvin has been a strong advocate for this strategy and I'm following his advice to put it just before Y-Wings. There is the 17-clue set containing all 31,512 sudoku puzzles with 17 clues. Most are easy but it is a fun and useful set to test with. The second is Ruud's top 50,000 from way back when. I might run some stats again on a randomly generated stock.

The solver distinguishes between Double and Single W-Wings. The double type I find about the same number, about 600 so it seems pretty rare. There were 1,156 proper W-Wings in Just17 and 14,775 leading to 17,911 eliminations in Ruud's set.

The largest declines in other strategies have been Y-Wings (8%), Rectangle Eliminations (6%), Simple Colouring (6%), XYZ-Wings (7%) and XY-Chains (6%), 3D Medusa (4%) but this is highly sensitive to ordering. Chute Remote Pairs also go down if evaluated before them.

The top 50k set with W-Wings required 6,561 more strategy steps than without, a 1.3% increase. That suggests a third of the 14,775 W-Wings are not contributing to a solve path and are just chipping away at the candidates. To be fair that might be true of other strategies.

In terms of score I've used the same values as Chute Remote Pairs and overall I see a score reduction in about 75% of puzzles that use against a 25% increase in score. So mixed results. This is inline with instinct that it is nice to have but not always a better performance.