Solving Sudoku – Four Group Intersection

Here we come to the most challenging technique yet for solving Sudoku. This one involves looking at the intersection of two rows and two columns. Look for a situation where you have the same possibility in four unsolved cells, where those four cells are the corners of a rectangle. Consider the following diagram:

Here we have 4 as a possibility in the fours cells marked in green. Examine the other marked cells. If 4 does not occur in any of the red cells, then it must not exist in any of the blue cells.

That is, if 4 is not in any red cells, then there are two possible arrangements for the 4‘s in the green cells: upper-left and lower-right; or upper-right and lower-left. Either way, with just the green cells, the columns will already have their cells with value 4. If there are any 4‘s in any of the blue cells, they can be eliminated as possibilities.

In practice, this situation is rather rare. Sometimes, you’ll find cases where you’ll find the same value in four corner cells. But more often than not, you find that value in both the rows and columns. In those cases, see if it’s possible to eliminate possibilities to get to a state where you can apply this technique.

Note that this technique can also scale up to three columns and three rows. But that’s an even more rare situation.

Well, those are all the techniques I’ve figured out. To summarize, continually use the basic techniques until you reach an impasse. If you’re lucky, and the puzzle author is kind, the basic techniques will be sufficient. But once you find yourself needing to make notes, you’re probably into the realm of the advanced techniques. First look at the groups with lots of solved values, and the cells with few possibilities. And keep plugging away at it. With experience, you’ll find it easier to recognize the situations where you can apply the more advanced techniques.


Solving Sudoku – Rectangle

This is the fourth posting in the series of techniques for solving Sudoku puzzles. First, I discussed the basic techniques. Then, I discussed single-group partitioning and two group intersection. Those techniques are enough for the vast majority of puzzles. Here, I discuss another technique that might be useful.

This technique takes advantage of an important characteristic of Sudoku puzzles. All puzzles have (or should have) one unique solution. Consider the cells at the four corners of a rectangle, for example:

Consider the cell at the lower right. If values 6 and 7 were eliminated, we would be left with an ambiguous state. That is, these four cells would all have the same two possibilities. There would be two possible solutions. But we can’t have that. We must have a unique solution.

Therefore, the possibilities 6 and 7 must remain, and we can eliminate the possibilities 4 and 5 from that cell:

And of course, once possibilities are eliminated, other opportunities to proceed will open up. In this particular example, if there were only three possibilities instead of four, we could write that in directly.

Next up, four group intersection.


Solving Sudoku – Two Group Intersection

Let’s continue with a study of the advanced Sudoku solving techniques. In my previous post, I looked at partitioning the cells of a single group. In this epistle, I look at what you can do with two intersecting groups. In this case, it’s a 3×3 square intersecting with either a row or a column. (If you consider the intersection of a row and a column, you can apply one of the basic techniques.)

Consider the groups marked in green in the following:

We have two groups: a row and a 3×3 square. Look at the three cells that belong to both groups, and consider the value 4. Within the row, the value 4 exists only in the intersecting cells. Within the 3×3 square, the value 4 must exist in one of those two cells that also belong to the row. Therefore, the value 4 can be eliminated from the other cells of the square.

With the techniques described up until now, you have the means at your disposal to solve the vast majority of puzzles. But there are still a few tricks left, which will have to wait until later postings. Next up, we look at the corners of a rectangle.


Solving Sudoku – Single-Group Partitioning

Okay, so you’ve mastered the basic techniques to solve Sudoku puzzles, and you’re still stuck on a puzzle. At this point, you’ll need to make notes, writing in small print the possible values at the top of each cell. There are a number of “advanced” techniques that can help you progress. At no time should you have to guess at a possibility. All puzzles should be solvable using analytical methods. However, note that most of the advanced techniques only help you to eliminate possibilities. You’ll still need to basic techniques to finish the puzzle. But eliminating possibilities should provide more opportunities to apply the basic techniques.

The technique I describe here involves looking at the cells of one group. Consider the following group with possibilities listed:

Before continuing, have a look to see if you can spot any possibilities can be eliminated.

Note that the values 1 and 2 exist in only two cells in that row, and not in any other cells. Since those two values can only be in those two cells, all other possibilities can be eliminated:

At this point, with those values eliminated, you may find opportunities for applying the basic techniques, allowing you to move forward.

One more note on this particular example: The full 3×3 square at the left is not shown. However, now that we know that two cells of the square only contain the values 1 and 2, we can now eliminate those values from the rest of the cells of that square.

Note that the technique can also be applied with three values. Look for a set of three values that exist in only three cells. All other values in those three cells can be ruled out. Likewise for four or more cells.

In the next installment, I look at an advanced technique involving two intersecting groups.


Solving Sudoku – The Basic Methods

Do you want to learn how to solve Sudoku puzzles but don’t know where to start? Read on! In this missive, I discuss the basic solving technique.

To start, consider the following puzzle:

Look at the cell marked in blue. An experienced player should be able to look at the puzzle and immediately know what the solution is for that cell. Scan the puzzle above that cell and also to the left. Consider that the value 1 cannot occur in any of the empty cells marked in green. We see that the blue cell is the only cell in the lower-left 3×3 group where the value 1 is possible.

(The keen observer will see another opportunity elsewhere in the puzzle where a 1 must be the solution for a cell.)

While solving a puzzle, as you’re entering values or eliminating possibilities, constantly look for these opportunities since these are the easiest to find.

Okay, so you applied this technique as much as you can and you’re stuck. What next? Consider the following diagram of the above puzzle but with a few values filled in:

Although we can still apply the first technique, let’s see if we can apply the second basic technique. Consider the cell marked in blue. That cell belongs to three groups: a column, a row, and a 3×3 square. Note that the value of the blue cell cannot be the same as any other value in those three groups. So, for that cell, list out its possible values. We see that eight possible values can be eliminated, leaving only one possible value for the cell: 1.

To find opportunities to apply this technique, look for groups with lots of values already filled in.

I’ll leave the rest of the puzzle to you. Click here to print out a clean copy of this puzzle. It’s the first puzzle in that document.

Using these two techniques, you can solve most puzzles published in newspapers and magazines. To practice, try out the Beginner Puzzles. In my opinion, the best way to solve Sudoku puzzles is by pen and paper, relaxing in an easy chair.

When starting out, it may be useful to write in the possible values in small print at the top of a cell, crossing them out as you progress through the puzzle. But with experience, you should be able to apply both of these techniques without making any notes.

In later columns, I’ll cover the more advanced solving techniques. Next up, single-group partitioning.