Sudoku puzzles are astonishing brain teasers, but once you find yourself bored in solving easier puzzles, you may have thought to try tough and more advanced sudoku puzzles. But beware because difficult sudoku puzzles will require you to learn advanced sudoku strategies. These strategies are used in the hardest levels of these puzzles and can either help reduce candidates or find the solution for a specific cell. Regardless, their application always demands high levels of concentration from the player as they work by deduction. In this article, I will discuss some advanced tips that will greatly help you to solve tough sudoku puzzles.
Difficult Sudoku Puzzles - The X-Wing Method
You can use this strategy when there is one candidate repeated in four cells that form a square or rectangle when mentally connected by row and column. By making an X linking diagonally the two opposite extremities of this rectangle, the player finds only two possible sets of positions for that digit.
Consider the puzzle above. All candidates have been completely and properly filled in. Let’s now focus on value 7. Notice that candidate 7 appears in only two cells in column 2. It also appears in the exact same two spots in column 6 as shown by the red circles above.
As value 7 has to be in either Row 3 Column 2 (R3C2) or R6C2, there are two possible scenarios. If it is in R3C2 then it cannot also be in R3C6 and so it has to be in R6C6. On the other hand, if value 7 is in R6C2 then it cannot also be in R6C6, so it has to be in R3C6.
The four cells form an X, which is why this solving technique is called X-Wing. Now let’s focus on the rows of the puzzle. We have just discovered that value 7 is either in R3C2 or in R3C6. Therefore we can erase the candidate from all other cells in Row 3. The same goes for cells R6C2 and R6C6. We can remove candidate 7 from all other cells in row 6. With the X-Wing method, we are left with single candidate 3 in cell R6C3.
Therefore we can erase the candidate from all other cells in Row 3. The same goes for cells R6C2 and R6C6. We can remove candidate 7 from all other cells in row 6. With the X-Wing method, we are left with single candidate 3 in cell R6C3.
A Swordfish in Sudoku is a particular pattern of cells in an arrangement where the same candidate is found in three different rows which align to form three columns (or vice versa).
With X-Wing, we had two rows (or columns) that had a candidate occur in the same two columns (or rows). The candidate could then be eliminated from all other rows (or columns). The same basic idea works for more than two rows or columns. Consider the puzzle above.
Have a look at rows 3, 5, and 9. Notice how in these three rows candidate 8 happens to be confined to the same three columns. So we have three rows. Every one of them has to contain value 8. A single column cannot contain a value more than once. So in these 3 rows, the value has to be distributed across all three columns.
Now look at the columns 2, 5, and 6. As shown in the puzzle above, every one of them has 8 in one of the rows we originally looked at. Therefore candidate 8 can be eliminated from every other cell in these columns. This means that this example candidate 8 can be removed from cells R4C2 and R8C6. With the swordfish technique, it helped us to have fewer candidates in a certain cell and can then be used to solve the puzzle through other techniques.
XY-Wing (sometimes called Y-Wing) is another advanced technique for eliminating candidates. It starts by finding a cell with only two candidates (a bi-value cell), called the pivot. These two candidates are called X and Y.
Consider the puzzle above. All candidates have been completely and properly filled in. Let’s focus on cell R5C2. This cell has only two candidates, 6 and 7 (the X and Y). One of these values has to be the correct one but we don’t know which one it is. Now let’s examine both cases one at a time. If the actual value is 6 (see red highlights), then the same value cannot also be in R5C6. With 6 removed, the only possible value of R5C6 is 5.
The other possibility is that the actual value of R5C2 is 7 (see green highlights). If that is the case, then that value cannot also be in R6C3. With 7 removed, the only possible value of R6C3 again is 5. This means there is a 5 in either R5C6 or R6C3, depending on which of the two possibilities is actually true. But in any case, value 5 can be removed from any cell that shares a region with both cells (see blue highlights). This means that in this example candidate 5 can be removed from cell R6C6 and will leave us a single candidate 2 in that cell. And that’s how the XY-Wing method works.
I hope that by using some of the three advanced techniques that I have discussed in solving difficult sudoku puzzles, you will be able to solve them faster and easier, and it will make you enjoy and crave more advanced sudoku puzzles in the future.