How to Solve a (Standard) WXYZ-Wing in Sudoku in 10 Steps

This is an assignment for my technical writing class. The goal was to create a set of instructions that break down a complex task into a series of easy to follow steps. If you would like to try it for yourself and offer feedback I have included an unsolved puzzle that you can try at the end of the post. I would certainly appreciate any input you may choose to leave in the comments section.

Advanced Sudoku puzzles can include some very difficult patterns that take a lot of time and effort to learn. One of these patterns is known as a WXYZ-Wing and can be particularly painful to find if you don’t know what to look for. By following the ten steps below you should be able to solve this complex pattern without too much difficulty. 

What Is a WXYZ-Wing?

Note: hover over blue words for definitions.

• Think of it as a family of four unsolved cells, one of which is the pivot.
• Besides the pivot, the other three family members each have only two candidates.
• Family members must share a row, column, or block with the pivot.
• Here is where it gets tricky. Each cell in the family contains some combination of a certain group of four numbers. Two or three of these numbers will already be known because they are included in the pivot, the remaining one or two must be found. For example, if the pivot contained the digits 1, 2, and 3, other family members could be [1,2], [2,3], [1,3], [1,x], [2,x], and [3,x] where x is the fourth unknown digit. This is the most difficult part of finding families and requires some practice. Two examples are shown below, each with the pivot shown in green and the other family members in yellow. 
• Crossed out cells can be family members, but cannot be the pivot.
• Warning: Don't fall into the trap of thinking that every family member must share a digit with the pivot. If the pivot has only two candidates then this might not be the case. However, at least two family members will certainly share a digit with the pivot. 

This family is made up of 2, 3, 5, and 7.

This family is made up of 1, 3, 7, and 8.

Identify Possible “Pivots”

[1] Examine each row, column, and block and in each case look for pairs of cells that share the same two candidates and have no more than two candidates. (For instance, if two cells shared a column and both had only the numbers 4 and 7 as candidates.) Lightly cross these cells out.

"Twin" pairs of cells with only two candidates are crossed out if they share a row, column, or block.

[2] For each block, count the number of unsolved cells that have not been crossed out. If the number is two or fewer, lightly cross out the entire block.

3 blocks remain.

[3] In those blocks that have not been crossed out, pick a cell with two or three candidates and follow steps 4-10 below to see if it is the "pivot." Note: crossed out individual cells cannot be the pivot.

The pivot is in green.

Finding Other Family Members

Using one of the possible pivots you identified in step 3, follow the steps on the next page to find the other three family members. If the answer to any of the questions below is 'no' then the pivot you chose in step 3 was wrong, and you need to start over at step 4 using a different one.

[4] Look in the column, row, and block occupied by the pivot for cells with two candidates.

Cells with two candidates that share a row,
column, or block with the pivot are in cyan.

[5] With the pivot as the first family member, can you make a family of four? Note: at least one family member needs to share the same block as the pivot and at least one needs to share a row or column with the pivot.

Green and yellow highlighting shows the complete family.

[6] "Children" cells will be able to see the pivot and each other. The remaining cell in the family is the "spouse" and although it sees the pivot, it cannot directly see the children (there is a ongoing custody battle). Is there a spouse and two children?

Children are shown in orange, the spouse remains yellow.

[7] Do the pivot and its spouse share one digit that both children lack? Think of the spouse's other digit as “A”.

The pivot and the spouse both contain the
digit 4 as a candidate, but neither child does.

[8] Does at least one of the children also have digit “A”?

The spouse shares the digit 3 with one of the
children but not with the pivot.

[9] In the block with the pivot, is there an unsolved cell or cells which can be seen by all four family members?

The magenta cell is unsolved and seen by all family members.

[10] If you answered 'yes' to all of these questions, then congratulations! You have solved the WXYZ-Wing and can eliminate “A” as a possibility in the unsolved cell(s) which the whole family sees.

You're all done!



Below is an unsolved puzzle that you can print off and use to test the instructions.