In this puzzle, this means 1 becomes sole candidate in the second row; 2 becomes sole candidate in row 6; and thus, 6 is sole candidate for row number 4. You can also use this technique if you have more than two candidates. For example, let us say the pairs circled in red were instead triple candidates of the numbers 1, 4, 7.
This would mean those three numbers would have to be placed in either rows 1, 2 or 5. We could remove these three numbers as candidates in any of the remaining cells in the column.
Hidden subset This is similar to Naked subset, but it affects the cells holding the candidates. In this example, we see that the numbers 5, 6, 7 can only be placed in cells 5 or 6 in the first column marked in a red circle , and that the number 5 can only be inserted in cell number 8 marked in a blue circle. Since 6 and 7 must be placed in one of the cells with a red circle, it follows that the number 5 has to be placed in cell number 8, and thus we can remove any other candidates from the 8th cell; in this case, 2 and 3.
X-Wing This method can work when you look at cells comprising a rectangle, such as the cells marked in red. In this example, let's say that the red and blue cells all have the number 5 as candidate numbers. Now, imagine if the red cells are the only cells in column 2 and 8 in which you can put 5. In this case you obviously need to put a 5 in two of the red cells, and you also know they cannot both be in the same row. Well, now, this means you can eliminate 5 as the candidate for all the blue cells.
This is because in the top row, either the first or the second red cell must have a 5, and the same can be said about the lower row. Swordfish Swordfish is a more complicated version of X-Wing. In most cases, the technique might seem like much work for very little pay, but some puzzles can only be solved with it. So if you want to be a sudoku-solving master, read on! Example A In example A, we've plotted in some candidate cells for the number 3.
Now, assume that in column 2, 4, 7 and 9, the only cells that can contain the number 3 are the ones marked in red. You know that each column must contain a 3. Example B Look at example B. We can eliminate 3 as candidate in every cell marked in blue.
The reason for this is that if we consider the possible placements of the number 3 in the red cells, we get two alternatives: either you must put 3s in the green cells, or in the purple cells, as example C shows. In any case, each of the columns 2, 4, 7 and 9, must contain a 3 in one of the colored cells, so no other cell in those rows can contain a 3.
Personally, I write all the possibilities for 1, then 2, then 3, etc then move on to the next tricks. Take care to not miss a cell or else you could end up making the puzzle harder for yourself. Not one I have ever put into practice, but the logic is sound.
Whenever you have a row or column where a specific number appears within a single block only, then that number can be removed from the other 2 rows or columns making up that same block. In the example, there is a pair of spaces where a 5 could be on the middle left block in the third row, marked in blue, so all the candidates marked in red can be erased. These would not occur if you did the optional counting trick, but if you went straight into doing pencil marks on a fresh puzzle then you will find some of these, simply put, they are cells with only one possibility, in this case, a 9.
Find a row, column, or block where a number appears in only 1 of it's cells. That one cell becomes the number there was only one of. In the example above, the 4 is the hidden single which would actually uncover the 8 as another hidden single, uncovering the 6 as yet another. The logic being that it is the only possibility for that specific number within the line or block, so it has to be that number to satisfy the "have numbers in every column, row, and block". Naked doubles are a pair of cells in a block, column, or row that have the same 2 numbers and no others after pencil marks.
Also because of the placement, if there were any 4's or 6's in the top left block aside from this double, then they would be erased as well. The logic being, whatever one of the cells is, the other cell has to be the other, so on the same row, column, or block no other cell could share that value. All the other numbers but the doubles can be erased from the 2 cells. In the example above, the 2 and 9 are exclusive to the corner cells and so the other numbers in the cells got erased.
The logic being that since those 2 cells were the only ones with those 2 numbers, neither of them could have been any other number so they were erased.
A group of 3 cells with 3 different numbers which force each other to be the other of the 3 if any of their possibilities are chosen. Just like with the doubles erases all other of any of the 3 numbers from the rest of the cells not included. In the example above, 5, 6 and 9 monopolize those 3 spaces so all the others in the same row have to remove those 3 numbers. Within a block, row, or column 3 cells with 3 exclusive numbers between them and none of the other cells.
Erases all other numbers but the triples themselves from the 3 cells. In the example above the 4, 8, and 9 were exclusive to the 3 cells so all the other numbers were removed. Same deal as Naked triples, only its between 4 cells and numbers. You know the drill Removes all numbers but the 4 from the 4 cells.
An X-wing occurs when within 2 rows or columns there are only 2 instances of a number and they share the same column or row forming a rectangle. Erase all other instances of that number in line with any of the four numbers making the rectangle. In the example above, there is an X-wing formed with 4's and all the ones in line with them get erased. The logic being, in both rows, there are only two possibilities that can be a 4, and since they form a perfect rectangle, if you choose one on the first line, then you would have to choose the one to the diagonal of it to satisfy that line, and same deal if you chose the other to start.
No matter which one you start with, all cells in line with the 4 main corners of the rectangle would be proven to not possibly be the same number. The unique rectangle occurs when there is a rectangle made entirely of pairs where one of them happens to have an extra number along with the pair.
Remove the extra numbers from the cells circled in red. Do you think hidden triples are tough to find? Try quads. Naked quads are like naked triples with the exception that four cells contain only four distinct candidates in a row, column, or region. In the example at the left, the naked quads are circled. They are 3, 5, 6, and 8. Remove any instance of these four numbers from the other cells in this row.
The last of my Sudoku tips for this article is to look for hidden quads. As the name suggests, hidden quads are four cells containing only four distinct candidates in a row, column, or region. These four numbers are hidden by additional candidates. Hidden quads are very difficult to find. The good news is I have rarely seen them. Maybe because they are hidden so well! They occur only in a few of the more difficult puzzles.
In my example at the left, the hidden quads circled in red are 1, 5, 6, and 8. It is safe to remove the extra digits 3,4,7,9 from these four cells.
I hope these Sudoku tips will help you in your quest to become a professional Sudoku puzzle solver. For more advanced Sudoku Tips, read my pages below. Many solvers pencil in candidates and look for common…. Read More ». Download this free Sudoku worksheet to help you solve all of your Sudoku puzzles.
This handy Sudoku Excel template will allow you to enter and remove…. Sudoku Solver Review Use a Sudoku solver to solve your most difficult sudoku puzzle instantly! You have several Sudoku solvers from which to pick.
You must be logged in to post a comment. Sudoku Tips to Make you an Expert. Hidden Singles In the example at the left there are two hidden singles. Now for my next Sudoku tip.
Hidden Pairs In the example at the left there is a hidden pair 2 and 9. Hidden Triples Hidden triples are much harder to spot. Hidden Quads The last of my Sudoku tips for this article is to look for hidden quads.
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