Mastering Sudoku: Tips for Identifying X-wing Patterns

Are you a Sudoku enthusiast looking to up your game? Ever heard of the X-wing strategy but unsure how to spot it in a puzzle? Look no further!

In this article, we will break down what Sudoku is, explain the concept of an X-wing, and guide you through the steps of identifying and using this powerful Y Wing strategy.

Discover the benefits and limitations of employing the X-wing technique to solve puzzles efficiently and effectively. Let’s dive in and master this advanced Sudoku strategy together!

What is Sudoku?

Sudoku is a logic-based combinatorial number-placement puzzle. The objective is to fill a 9×9 grid so that each column, each row, and each of the nine 3×3 grids1 to 9 exactly once. The puzzle was developed by the American puzzle maker Howard Garns in 1979 and introduced in Japan by puzzle publisher NIKOI initially known as Number Place.

Today it is known as Sudoku, and there are an infinite number of variations in the size of the grid, the number of given numbers, constraints on the puzzles, and the shape the solved puzzle should take. In the classic form of Sudoku, no arithmetic is involved. Which cells contain which numbers is deduced purely using logic to restrict the possibilities at different moments down to only one possibility for each cell. At least one cell in the puzzle must have only one possibility to start solving a sudoku puzzle.

What is an X-wing?

An X-wing is a special case of a Swordfish that represents three parallel bi-value cells that align in three separate rows and columns. This can be any three such units, with natural groupings forming from elimination along rows or columns rather than possible values. In the swordfish pattern found on the rows, the three red cells have possible values of 7 and 8 which cancel each other out and highlight the two yellow cells which must contain 7.

How to Spot an X-wing in Sudoku?

An X-Wing is easily spotted in Sudoku when there are two groups of four cells that share only two possible values. The two groups must be from different rows and/or columns. The two possible values must have alternatives along the rows and columns, but only appear once in the appropriate row or column of the other section. The X-Wing strategy solution is that the two sets of four cells can only have their desired digits positioned in one way to solve the possibilities arising from the pattern.

Step 1: Look for Rows or Columns with Only Two Possible Positions for a Number

Scan and select potential rows and columns with only two empty cells. Position a candidate number in A1. Imagine selecting A1 and positioning the possible number.

The Jellyfish technique is the closest algorithm to an X-Wing. It looks for columns or rows with two possible positions and the same candidate number on corner (a). If there are only two possible rows that this number can occur in and you can find two or three more empty cells on the corners of the four intersection boxes, there are only a small number of solutions for the corner squares. You have found an X-Wing. Remember to eliminate the candidate from the locations it does not appear that are in the same row or column as the X-Wing.

Petra Gargano and Markus Goetz, the writers of Sudoku-Nr.1, only use the X-Wing technique as a “last resort because utilizing this technique does not actually give any benefit to guessing step-by-step towards a complete or failed solution. Regardless it is an interesting rule to learn as it adds a new dimension to the game.

Step 2: Identify the Common Number in the Two Rows/Columns

The two rows 4, 6, 9, and 3 and 4, 6, 9, and 8 reveal that 4, 6, and 9 are the only common numbers. This means that if an additional cell is revealed, the 4, 6, and 9 cannot exist in both the first and second rows due to a Sudoku rule. Our added premise was that we have been using the cells in row 6 in our discussion as the two shown numbers of 4 and 9 can only be on the edge of our X-Wing diagram. The only remaining number that we can assign to row 6 is 1.

This is why we remove the 12378 from cell B2 in the first row. Mathematically this proves that we found an X-Wing formation, but this could potentially be invalidated by the newly revealed B2. So while it’s a step in the right direction, it is not a definitive proof yet. This is an abbreviated demonstration of the X-Wing technique, but this is a good time to introduce an S-Wing formation. Or, at least introduce what could one day be called the S-Wing.

There is no formal designation for such a formation in Sudoku as yet, but we introduce this in a slightly more advanced section for those who already have a grasp of the fundamental concepts. In the problematic part of the example (Star Trek Sudoku), a potential X-Wing formation never gets a chance to fully develop with a single complex number and its solution because it gets resolved one step before it would qualify. We will likely see more complex formations in the future, leading to more upsets in the sports entertainment world of Sudoku solving.

Step 3: Look for the Same Number in Another Row/Column

If at this point the strategies for locating an X-wing have yielded no results, the next step of the logical process in Sudoku is to look for the same number in the same rows or columns where the weak implications begin. By this, the goal is either to eliminate a row or column from contention as to which could or could not be home to the 4 number, or in the best case scenario to confirm the location.

It is now only necessary to examine the two former home-row/columns designated in step 1. This is the least helpful step in determining an X-wing configuration for a number, as its success rate is almost exactly 50-50 as far as either confirming or ruling out the X-wing in question. As such, successfully following this path does not efficiently speed up the deduction process and is always an essential step before moving on to the next.

Step 4: Eliminate the Common Number in the Other Row/Column

As we introduce the term common number. This is a similar process as above but with a different elimination logic. Now that we made the X-wing configuration of the column highlighted above by graying out the unrelated numbers. We simply take the common number in the different highlighted row of the X-wing, and eliminate it from the row of the main column of the X-wing. It’s that simple. Let’s work it out with the same 25th of June 2005 scenario in row six of column three. Act on the Trace.

Step 5: Check for the Eliminated Number in the Remaining Rows/Columns

• Once you have eliminated every other number in the row or column apart from the X spaces, you need to rule out that number from the further rows and columns. If the number appears in these rows, columns, or other blocks, then you do not have an unambiguous solution, which can sometimes happen but is usually incorrect.
• Remaining X spaces in row 2 means the number 1 is eliminated in row 1, 3 and 6, and remaining X spaces in column 1 means the number 9 is eliminated from columns 3 and 8.

How to Use X-wing Strategy in Solving Sudoku?

You can utilize X-wing on the remaining empty cells to solve this Impossible Sudoku which needs both the Two-String Kite Theory and the Y-wing Technique.

The simplest case of spotting an X-wing is when offshoots of both candidates in each of the columns are found as they are in columns 2 and 8 in the below screenshot. 9 is excluded from all cells within the X-wings, and thereby from the shared row 1, so the x-wing cells (R3C2, R8C8, R3C8, R8C2) must receive 9s to complete the pattern making these 9s impossible outside the x-wing.

Step 1: Identify the X-wing Pattern

The X-wing pattern is a Sudoku pattern that describes multiple instances of the same candidate number that are aligned in exactly two columns and two rows. These must be such that the squares lie in two opposite corners of a rectangle, forming an X shape. Let’s take a look at how one fits this description.

Step 2: Eliminate Numbers in the X-wing Rows/Columns

Once you have identified the 4 corners of your sudoku grid’s internal X-wing, you need to check the intermediate rows or columns to see if they only contain the desired numbers. In our example, after having rows 1 and 3 do not contain the desired numbers, but have columns 1 and 3 as the only columns with 1 and 3. Check these columns to see if their rows have the internal candidate cells with only the desired number.

Step 3: Check for the Eliminated Numbers in the Remaining Rows/Columns

LOCEN defines two Darth Maul X-wings in columns 5 (r3 and r7) and 7 (r3 and r9), which resolve the square on r9c5 to a 4. Column 5 only contains a 6 as the excluded number, so a 6 can be placed in the remaining square on r7c5.

Column 7 only contains a 6 as the excluded number, so a 6 can be placed in the remaining square on r3c7.

Column 9 contains both a 4 and a 6, so neither of these numbers can be placed there. These results match the previous confirmations of these numbers as excluded and the eligibility of 4 and 6 as the candidates for placing in r9c9.

What are the Benefits of Using X-wing Strategy in Sudoku?

The benefits of using x-wing strategy in sudoku are that x-wings can be used to identify the weak links between certain cells which allow moderately fast elimination of possibilities. This makes the puzzles faster and easier to complete.

An applied example was provided in the earlier data where using the x-wing strategy we could eliminate the possibilities of the number 9 from row 3, column 1, and column 9. This is because with the x-wing alternate cells must contain the true number, which was actually the case. Any cell that has an x-wing cannot contain 9 – these are ‘If-then’ logical statements. The block of cell 9 was the weak link that could easily be eliminated through the use of an x-wing strategic analysis.

Helps to Solve Difficult Puzzles

Recognizing an X-Wing Pattern early during the puzzle-solving process can help to solve challenging puzzles by orienting the player towards the most effective sequence of moves. For example, in the case below, use recognition of the 7’s in columns 3 and 6 to start unlocking the puzzle, and then notice column 3’s 5 and column 6’s 8 as the more easily pre-removing the 4 in cells a3 and a6.

Once the pre-removes in a3 and a6 are recognized, the player can avoid the net increase in the puzzle patterns that would have occurred if they had instead retains the 4’s in f3 and f6. At the point shown in the solved puzzle where 26’s are added to rows 5 and 6 and removal of the 26’s in a6 needs to happen for the X-wing to remain valid, the net result of having completed the series of 4 pre-removals is a reduction in the net decrease from the previous phase. Therefore, the X-wing is preserved and the puzzle can easily be solved from that point.

Saves Time and Effort

The more efficient method for spotting an X-Wing as opposed to seeing how to cancel out the values of blocks that have the fish shape is to take a look at blocks where you have two candidate values in two different cells. According to Kenneth Hamer’s coercion rule, when a candidate cell block has two candidate values, this allows the solver to either establish one of the two as given or rule them out entirely.

Increases Chances of Solving the Puzzle Correctly

Spotting an x-wing pattern in sudoku means the solver has engaged with the puzzle and is seeking to understand the relationships between numbers in order to solve the puzzle. If they properly identify the x-wing, it means they have correctly identified and marked candidate solutions to the puzzle. It increases the chances of solving the puzzle correctly by providing them alternative steps which apply to the elimination of possibilities. Solving the puzzle correctly is relevant to all sudoku solvers, but particularly for those solving them as printed puzzles in publications or books. They proceed under the assumption they are correctly solved according to a single answer model. This improves their chances of correctly solving the puzzle in a shorter number of steps, something particularly relevant for those with limited time.

What are the Limitations of Using X-wing Strategy in Sudoku?

1. It has a higher difficulty grade and without using a solver, you might not even be sure that an X-wing exists.
2. It will identify fewer single cells as having alignment and structure.
3. It is based on the same logical principle as locked candidates so it offers no additional solving insights or help beyond them.
4. It is only used for lines and being good at lines without using lock candidates does not synergistically work directly with any other strategy.

Only Works for Certain Types of Puzzles

Shadow X-Wing can only be found in sudoku puzzles which have two bands of the same number in two blocks. For the design reasons, it has to be a colored or painted band on a diagonal in order for all 4 patterns to be possible. These are puzzles which have been designed to use the shadow X-Wing in solving them.

These screenshots reveal designs and not actual puzzles. They are taken from a set of creative sudoku puzzles which use different patterns of colored bands. In the BuzzFeed Sudoku app, a different pattern of colored bands is used in the daily challenges backgrounds every day. The X-Wing formation can still be seen in the spaces between regions, but they are not so obvious as in screenshots from these puzzles.

Typically when you find it in these regions in a puzzle, you can find it in the same numbers outside these regions. This is due to the limitations in the number grids which sudoku puzzles are built on – either a 3×3 9-puzzle grid or a larger square with this being a component of the overall larger, solvable puzzle.

For Buzzfeed’s sudoku puzzles, this is often seen in the 3-4 and 7-8 bands.

Can Be Confusing for Beginners

The X-wing strategy can be confusing for beginners because the spaces do not line up in the center of the cross formation. Sudoku grids are made up of 81 spaces divided up into rows, columns, and boxes. Beginners learning the game will often go row or column by row, filling in each open space from 1-9 until they have the board filled.

When searching for X-wings, they need to remember to look for the special cross pattern in rows and columns even if they are not precisely in the middle of the row/column. When they see the cross pattern in a row or column at the outer edges of the pattern, they should trace the outside paths. Excluding all X values on these paths narrows down the effective paths for the cross to pass through.

Unfamiliarity with the process of superfluous decision-making is another reason beginners can be confused by spotting X-wings. This means making decisions and mapping out your strategy based on what answer you already know (in this case two). The ability to make guesses that do not trigger an incorrect domino effect by following through the scenario in your head is a key skill in sudoku.

May Not Always Guarantee a Solution

Because symmetry and transposition in which the clues have been flipped or rotated on the grid further refine what can be classified as an X-Wing and these symmetrically equivalent grids give multiple solutions, they often cause some confusion. Some sudoku variants explicitly allow multiple solutions. However, Classic Sudoku does have one and only one solution. A classic form should have a single one-point solution, but classifying some possible x-wings as one-point possibilities can sometimes show at least one solution when multiple ones are allowable.

Future research may be necessary for symmetrically unique (SU) X-wings in symmetrically unique sudoku grids to confirm whether they always allow recognizing a type of X-wing solution pattern which will ultimately limit the puzzle(s) results to a single genuine one.

Frequently Asked Questions

What is an X-wing in Sudoku?

An X-wing is a specific pattern that can occur in a Sudoku puzzle which allows you to eliminate certain numbers and solve the puzzle more efficiently.

How does an X-wing work in Sudoku?

An X-wing occurs when there are two rows and two columns in a Sudoku puzzle that each contain only two possible occurrences of a number. These two rows and columns intersect at four cells, creating an “X” shape. If the same number appears in the same position in each of the two rows or two columns, it can be eliminated as a possibility from the other cells in those rows and columns.

What does an X-wing look like in Sudoku?

An X-wing is represented by four cells in an “X” shape, with two rows and two columns intersecting. The cells will typically have only two possible numbers in each, and those numbers will be the same in the same position in each row or column.

How do I spot an X-wing in Sudoku?

To spot an X-wing in Sudoku, look for two rows and two columns that intersect at four cells, creating an “X” shape. Then, check to see if there are only two possible numbers in each of those four cells. If the same number appears in the same position in each row or column, then you have found an X-wing.

Can an X-wing occur with any number in Sudoku?

Yes, an X-wing can occur with any number in Sudoku. The key is that the number appears in the same position in both rows or both columns, allowing it to be eliminated as a possibility from other cells in those rows and columns.

What is the benefit of spotting an X-wing in Sudoku?

Spotting an X-wing in Sudoku can help you solve the puzzle more efficiently. By eliminating a certain number as a possibility in multiple cells, you can narrow down the options and make it easier to fill in the remaining squares. This can save time and frustration when solving difficult Sudoku puzzles.