Upgrade Your Skills in Sudoku: Medium and Hard Sudoku Solving Tips With Sample Videos
Have you reached the point in your Sudoku solving where you are ready to try out some of the harder puzzles—expert Sudoku, hard Sudoku, medium Sudoku? Do you want some new techniques to make these harder puzzles more solvable? If so, you have come to the right place.
If you are already very confident with easy (one- or two-star) Sudoku, then you already know the rules of the puzzle and some of the simpler techniques for solving. You know that each row of nine cells must contain each digit 1–9 once; each column of nine cells must contain each digit 1–9 once; and each 3x3 block (or region) of cells must contain each digit 1–9 once.
You probably also know that letters or symbols can be used instead of the digits 1–9 to create (and then solve) a puzzle, and you may know that in some of the variety puzzles a region is made up of an irregular shape—still nine cells, but gerrymandered to create a different kind of solving challenge and enjoyment.
Examples of Simple TechniquesClick thumbnail to view full-size
The earliest techniques for most solvers are the Slice'n'Dice or cross-hatching technique (demonstrated in the videos below), the rule of necessity, and the technique of completion. Completion simply means that if only one space is empty in a row, column, or block, then you know what digit belongs there because there can be only one digit unused—assuming that the region has been completed correctly up to that point.
The rule of necessity tells you that you must find some place to put each digit in each row, column, or block. By comparing needed digits with those given or added in neighboring blocks, you can eliminate the impossible and finally select the correct digit. Slice 'n' Dice allows the solver to eliminate certain cells from consideration for a specific digit, by mentally crossing off the cells in the same row, the same column, and the same block where that digit is present.
As an example, if a 5 is present in cell 24 (= row 2, column 4), then no other 5 can be entered in row 2 or in column 4; and since this cell is in Block 2, no other 5 can be entered in Block 2. The solver visually scans these three regions (row 2, column 4, block 2) and eliminates those cells from consideration for placing a 5. When this is done using several different 5's (sometimes coming from different directions), in many instances only one cell will remain possible for the 5 in a given block.
Ghost Slice 'n' Dice and Column/Row Determination
A more complex form of Slice 'n' Dice is the technique of Double Scanning (or Ghost Slice 'n' Dice). In this case, the digit you wish to place may not actually appear in a certain row or column - and yet, because you know which row or column it will occupy, you can eliminate that region from consideration in the block where you are working. (This also is demonstrated in the videos.) Sometimes you will encounter a situation that points to a specific row or column for a given digit, because of where that digit must be entered in the adjacent blocks.
For example, let's say that in Block 1 you have discovered that the 3 cannot be placed in the top row; and in Block 3, you notice the same thing—the 3 cannot be placed in the top row. That tells you that the 3 must go in the top row in Block 2. You may even be fortunate enough to determine exactly which cell will take the 3. The same technique can be used with columns, rather than rows. The picture here shows how we came to place the "1" digit in cell 25.
One Choice or "What Fits?"
A technique that you may have learned with easier sudoku puzzles is One Choice or "What Fits?" That is, you may find a cell where you can count off all the digits that cannot belong in that cell, leaving only one possibility. If you wish to go looking for a cell where that technique might be used, remember that it is not the total number of digits appearing that matters—it is the number of different digits that counts.
So, rather than looking for a cell in a block with 4 digits, in a row with 6 digits, and in a column with 5 digits (all of which may use the same digits), you may have better luck checking out a cell at the intersection of a row with 3 digits, a column with 3 other digits, and in a block that has 2 still unused digits, as long as those digits are different from one another.
Twinning in Row 8Click thumbnail to view full-size
Triplets in Row 5 and Another (Neglected) in Block 5Click thumbnail to view full-size
Twins and Triplets
Twinning (or triplets or quads) is another technique that is used often in the next levels of sudoku. This concept works in two ways. One is known as a "naked pair." That is an instance when in a given region there are two cells that contain only two candidate digits, the same two digits in both cells. (For example, 1 and 9). Since there are two cells and only two digits (the same two), then one of the digits must belong to one of the cells and the other digit must belong to the other cell.
But even before you determine which cell takes the 1 and which one takes the 9, you already know that the 1 and 9 cannot go anywhere else in that region. So if these twin cells are in the same block, then the 1 and the 9 cannot go in any other cell within that block. If the twin cells are in the same row, then the 1 and 9 cannot go in any other cell within that row; the same is true if the twin cells are in the same column.
If you have penciled in any candidates, then you can use this principle of twinning to eliminate the twin digits from other cells in the same region, if they have shown up elsewhere. Or you can avoid placing them in there in the first place, as you notice in the picture here. In that case, placing the candidates in Block 9 shows that only the 5 and 9 can be used in the empty cells.
Since they are both in the same row, then neither 5 nor 9 can appear as a candidate in any other cell within that row. If you first filled in the candidates for Row 8 of Block 8, you could include the 9 in cells 84 and 86 - initially. But then, upon completing Block 9 or completing Row 9 of Block 8, you would see a Twin Pair (a Naked Pair) containing a 9; that tells you that the 9 could not be used either in cell 84 or 86.
The second way that twinning works (a "Hidden Pair"—not shown here) happens in a situation when other digits occur in the same cells as the twin digits, but those two digits appear only in two cells in that region (row, column, or block). In that case, all the other digits can be eliminated from those two cells.
As an example, let's say that the candidates in cell 57 are 1, 4, 6, 7, and 8, and the candidates in cell 59 are 1, 2, 5, 8, and 9. You see that 1 and 8 appear in both of these cells, which are in row 5. As you check across that row, you see that no other cells offer 1 or 8 as a candidate. In other words, even though other candidates appear to be possible in cells 57 and 59, the 1 and the 8 have no other possible homes in row 5. Therefore, you can eliminate all other digits as candidates for those two cells. When you do that, even though you still may not know where the 1 and the 8 go, you will eliminate theoretical placement for those other digits, and that may lead to the certainty of where to place them.
Triplets (and beyond) work in the same two ways, but with a slight variation. In those cases, it is not necessary that all three digits appear in all three cells. For example, let's say you see a row that contains a cell with 6 and 7 as the only candidates; two other cells in the same row contain only 6, 7, and 8. That makes up a triplet. The 6, 7, and 8 must go in those three cells (but the 8 cannot go in the first one mentioned). That also tells you that those three digits cannot be used in any other cells in that row.
But it could also be true that one of the cells contains (only) 6 and 7; a second one in the same row contains (only) 7 and 8; and a third one (still in the same row) contains (only) 6 and 8. That is also a triplet. Those three digits must be used (in that row) only in those three cells, but limited as indicated.
Finally, one last technique to mention is that of Forced Choice (aka a Forced Chain). In this situation, you have completed all the cells that you can determine with certainty; then you have penciled in the candidates in the remaining cells, keeping aware of Twinning, etc., so that you don't place a digit in a cell where it is not a viable candidate. You want to pencil in all candidates, but only the ones that are truly possible.
After using the penciled-in candidates to solve additional entries with certainty, you can use the Forced Choice technique. With this, you choose one cell which contains only two candidates, and you select one of them as your "choice." (Tip: I mark this cell with an asterisk so that I can remember where I started.)
Since you don't know yet whether that choice is correct, use some method of "choosing" that will alert you to the choice without erasing the other candidate. You may decide to underline your choice, draw a circle around it, or lightly pencil-slash through the unselected one, for example. When you have chosen one of the candidates, check other cells in the same row, column, and block, to see which candidates are forced because of the choice you made, and then mark them similarly.
Remember that you don't know yet whether these choices are correct; you are essentially following a hypothesis to its inevitable conclusion. You will either come to some point where a choice contradicts another one (by selecting a candidate that had previously been selected in another cell in the same region), or you might solve the entire puzzle. A third, unpleasant, possibility is that the puzzle is so complex that you can neither solve it nor find a contradiction. In that case, you will need even more advanced techniques.
Step-by-Step to Solve an Expert Sudoku Example
Outcome of Forced Choice
There have been times when I have followed Forced Choice for many selections, only to find the contradiction in the last possible cell. It would be so nice if a long string of non-contradictions proved that you were on the right track, but that is not always true. There have also been some times when the Forced Choice was so difficult that I realized it's time for me to look into the next level of solving techniques. When I do that, I will come back and share them with you!
Forced Choice works best when you find a good cell as your starting point. You want to start in a cell where one choice will lead to another choice in another cell within the same region; that choice will lead to another, then another, and another. It is possible to use some of the same techniques in Forced Choice as in actual solving of a puzzle. That is, sometimes one choice will eliminate candidates in such a way that a specific digit appears in only one cell in a given row or column; in that case, that is the Choice that was forced. Sometimes twinning may result in such a way that some digits can be eliminated within a block. While using Forced Choice, all of the choices are hypotheses, and they must derive from some previous action in the chain.
Once you encounter a contradiction, you know that the original choice you made was incorrect - if you have derived all of your choices through sudoku logic; and so the other candidate in the original pair must be correct and can be entered as an actual digit (as shown in the Expert Sudoku Example video). It can be very enjoyable to have a whole string of correct entries come from one contradiction. In some cases, it may be necessary to start the Forced Choice over once again, after completing just a few entries.
Step-by-Step to Solve a Hard Sudoku Example
Other Tips and Puzzle Strategies
Good solvers have suggested all kinds of practical tips that have helped them. Some say to use only pen; some say to use only pencil; one commenter suggested using pen for the obvious, definite entries in the puzzle, then using pencil when you get to the entries that you feel less certain about.
Some like to go in numerical order to check and solve for digits; after finding all the definites possible in one round, they then go back through the digits in order again. But some solvers prefer to start with the digits that appear most frequently among the givens; solve those as far as possible, then go back and pick up the ones that did not appear as frequently. It is a good idea to count often; as you go through a row, column, or block, check to see where each digit will go or could go, and count off the digits as you check for them.
My own method is to start with what is obvious - those digits that appear the most often. I start by using Slice 'n' Dice as much as I possibly can with all of the obvious digits, then with the ones that are less apparent. Then I turn to these other methods: "What Fits?", the rule of necessity, completion, twinning and beyond. I return to Slice'n'Dice frequently, perhaps immediately after I have placed a few digits by any other method, simply because it is so easy to use.
A Last Suggestion That Helps Me
When I play Sudoku on Google-Plus, I have one more helpful little trick that works for me, although it may not for others. That is, since this is on a computer, I keep a blank sudoku grid in a file on my desktop. When I have solved all of the obvious entries possible in the Google-Plus Sudoku Game itself, I open my blank grid and fill in candidates in each empty cell. That makes it possible to see examples of "What Fits?" or Twinning, etc.
I always fill in candidates in the region that has the fewest empty cells. For example, if I see that a certain row has only two empty cells, then I fill in the two candidates in those cells (in my grid); in some cases, the fewest empty cells may be in the block or in the column.
When I have done all I can with those candidates, then I turn to the Forced Choice technique. On my computer Sudoku Grid, I just select the candidate I wish to choose, highlight it, and change its color. That makes it easy to see which candidates I have selected in all of the cells. (And I do remember to mark the first cell with an asterisk so that I know which pair of candidates contained either a correct or an incorrect choice.) I admit that I don't really enjoy using Forced Choice, but it definitely does make it possible to solve some more difficult puzzles.
Every solver will find the methods that work best for them, and those methods may be as individual as the people who work the puzzles. Maybe you will add some of these described techniques to your bag of tricks so that you can start to tackle harder sudoku puzzles. Good luck, and happy puzzling!
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