How can I spot an X-Wing?

Question

I was doing a Sudoku puzzle (in my morning newspaper on the train to work) and got stuck at this point:

_ _ _   _ 4 _   7 2 1 
4 _ 2   6 _ 7   _ 8 9
7 _ _   8 _ 2   _ _ _

2 9 5   4 6 8   _ _ 7
3 4 6   5 7 1   8 9 2
8 _ _   _ _ 9   6 4 5

_ 2 _   7 _ 4   _ _ 3
5 _ _   1 _ 6   2 7 _
1 _ _   _ _ _   _ _ _

Having made no further progress by the time I got home, I plugged the numbers into HoDoKu for help, and got told that there was an X-Wing. I looked at the hints but couldn't find it, so eventually gave up and got it to tell me which number.

It's 9:

Now I've had it pointed out to me I can see it, but what strategies are there for "seeing" it? Bear in mind I want to be able to do this on the train using paper & pen, without the aid of a computer.

Answer 1

I've highlighted the X-Wing and the 9's involved in this X-Wing:

See how's there's two 9's in row 3 and also two 9's in row 8? (highlighted in blue)

Since every row must contain exactly one 9, either A and D must both be 9 OR B and C must both be 9.

In either case, these 4 squares will provide the 9's for rows 3 and 8, and because of that, they will also provide the 9's for their columns (3 and 5).

This means that any other squares in the same column as these squares can NOT be 9. So any 9's in these squares can be deleted. (highlighted in red)

Try making any of the red ones a 9, for instance row 1, column 3. If this were a 9, then A and C cannot be 9. But then, for rows 3 and 8 to both contain a 9, Both B and D would have to be 9 and this is not possible, because then column 5 would have two 9's.

Imo, X-Wings are not easy to spot without highlighting all occurrences of a number. I usually don't look for them when solving with pen & paper.

You'd have to look for 2 numbers occurring in 2 rows or columns. It is even more difficult to find 3 numbers in 3 rows/columns, or 4, etc.

There can also be an X-Wing involving a row and a 3x3 box, but they are even more difficult to see. Computers have no problem spotting them, of course.

Answer 2

I am rehashing this question because I was going through some historical puzzling.se and it was poorly answered and understood before. The issue is not only in the 'x-wing' shown before, but also in the three lowest boxes (7-9) in the shown columns 3 and 5.

The trick is that two different numbers are possible in each box involved in the x-wing (not just one, as the previous answer supposed - then, it may still become the only possible number, which is a crucial solving technique in sudoku). So in either of the cases where the combos are 3&9 or 8&9 in the boxes, the x-wing forms (think about it - an x is made up of two lines) and the puzzles become unsolvable.

I have circled some of the other numbers involved in the x-wing predicament in the following image.

I recognize that this post is more complicated than the previous answer, but it is important to note that when identifying x-wings more than one number must be involved in order for it to actually be a standstill/unsolvable situation

Notice also that each of the locations highlighted in the original answer are in separate 3x3 boxes and may be solvable within those boxes.

Quite simply, I have yet to find an easy way of spotting an x-wing without simply identifying four squares in a rectangular pattern and at most two 3x3 boxes that each have the same 2 possible numbers.

If this post needs further clarification please let me know.

Answer 3

An X-wing is where a forced number in two rows/columns form a rectangle, which then forces out the digit in the respective column/row.

In your example, the 9's are forced in row 3 and row 8, and this forces them out of column 3 and column 5.

The best way to spot one is to look for rows/columns where a digit can only appear twice, and then see if the same can be said about that digit in any other (respective) row/column, bearing in mind that you only need to look for a rectangle.