Hidden Sets -- "Insider" Secrets from the Testing Department

In the process of creating the SBY program, our development team had to test our code by using it to solve hundreds of puzzles. We had to be sure that it could cope with the great variety of complex situations that can arise from time to time. One of the last and somewhat tricky situations that we had to understand was what we came to call "hidden sets".

A hidden set occurs when a block, a row or a column contains a number of digits in certain cells and where the number of digits and the number of cells is equal. These situations are not obvious because a number of other digits also occupy the same Options Grid. When this situation arises, all digits (except the digits in the set) can be removed from the applicable  cells in the block, row or column. Confused? I know we were. But a picture is worth a thousand words so let's examine five situations from real Sudoku puzzles.

Hidden and Exposed Sets: But before getting into this subject it is helpful to distinguish hidden sets from what we arbitrarily call "exposed" sets. But be warned -- sometimes even exposed sets can be tought to find. I just defined a hidden set in the preceding paragraph. An exposed set. on the other hand, occurs when 2 or more digits (and ONLY those 2 or more digits) occupy 2 or more cells in the same block, column or row. As you go through the situation below, you are going to learn about eliminations in both of these situations. To set the scene, here are how eliminations are handled in these two situations:

Hidden Sets: all digits other than the digits that make up the set can be eliminated from the identified holding cells in the applicable block, column or row

Exposed Sets: all digits in the defined set can be eliminated from the non-holding cells in the applicable block, column or row

As you examine the Hidden and Exposed Set situations below, you will begin to realize just how valuable the Options Grid can be. It provides you with a means by which patterns can be recognized. This is only possible by viewing the OG values for all cells -- not for isolated cells. It is impossible to see the patterns when OG cells are viewed in isolation.

Having figured out hidden and exposed sets, we started to notice that they come up quite frequently when simpler techniques are available. But, if you recognize these sets it is possible that other less complicated, but more numerous, situations can be avoided. Thus, we feel that learning to spot those sometimes elusive sets can really elevate your skill level AND decrease the time needed to solve a Sudoku puzzle.

-------------------- SITUATION 1 (Hidden Double) --------------------

Take a close look at the block on the right. Look for the digits 4 and 6. If you look carefully, you will see that they appear in only 2 cells -- the cell in the center of the block and the cell in the bottom right-hand corner. This is why we call them "hidden" doubles -- because they are difficult to spot when buried in with all those other digits. And this is an example of what we mean when we refer to a "pattern". Only by looking at ALL the OG values in the 9 block cells can we be sure that both the 4 and the 6 appear in JUST those 2 cells. And finally, you will also notice that we have satisfied our important "hidden sets" rule -- the number of digits (two) is identical to the number of cells (two) that contain those digits.

Having found those elusive digits, we can now eliminate the 7 in the center cell and the 3, 5 and 8 in the cell in the bottom right-hand corner. This is because WE KNOW that the center cell MUST contain a 4 or a 6, just as the bottom, right-hand cell MUST ALSO contain a 4 or a 6 -- all other digits are impossible. Notice that when using the SBY program, eliminated OG options are easily identified by the dark blue background color, where the original number is still "readable" but also easy to ignore as further solving proceeds.

So... what have you learned? Well, you've learned that solving a Sudoku puzzle is all about MAKING ELIMINATIONS and in this example of a "found" hidden set we were able to eliminate a total of 4 digits in 2 of the cells in the block. Hey, every little bit helps as we push towards the final solution.

------------- SITUATION 2 (Hidden Triple AND Exposed Triple) -------------

Now, let's move on to a second example. Take a close look at the block on the right. This situation is a bit more complicated than the previous one. Here's a clue. Look for the digits 1, 3 and 8. Notice that there is a 1, 3, and 8 in the cell at the top, left; a 1, 3 and 8 in the cell in the top row in the center and just a 1 and 3 in the cell at the bottom, left. That makes 3 digits and 3 cells. It doesn't matter that the 8 did not appear in the cell on the bottom row.

Thus, we can eliminate the 6 in the top, left cell; the 5 and 6 in the top, center cell and the 2 and 6 in the cell in the bottom row as shown in the block just below and on the right.

Actually this block contains a second set of triples, but this set is "exposed". Notice that the 3 cells in the middle row contain just a 2, 5 and 6 in that:

• the first has has a 2 and 6
• the second has a 5 and 6
• and the third had a 2 and 6

This makes three digits and 3 cells. Thus the 2, 5 and 6 can be eliminated from all other cells in the block as shown on the right.

Thus, using either one of these approaches gives us the information needed to make the displayed eliminations. The important thing is to become adept at finding a defined number of digits that appear in the same number of cells.

-------------------- SITUATION 3 (Hidden Double) --------------------

Examine the first block on the right. This block came from the same puzzle as the previous one, but was solved in a different way. Can you spot the hidden doubles in this block. Pretty easy - right! The center cell in the top row and the cell in the bottom row both contain a 3 and an 8 and no other cells contain these digits. Having found this situation, notice the eliminations that can be made in the block on the far right.

-------------------- SITUATION 4 (Hidden Quad) --------------------

Remember that columns and rows are just like blocks -- they must contain 9 unique digits in each of the 9 cells. Hence, just for a little variety, this example will deal with a column instead of a block. Keep in mind that we could just as easily have used a row.

From the first column on the right, look for the digits 1, 3, 8 and 9 and see if they appear in 4 cells. Sure enough, by starting at the top, we see:

• a 1, 3 and 8 in the first cell

• a 1 and 3 in the third cell

• an 8 and 9 in the fourth cell

• a 1, 3 and 9 in the sixth cell

That's 4 digits and 4 cells. Therefore all digits except the 1, 3, 8 and 9 can be eliminated from the 4 cells where the 4 digits were found.

Now, look at the column on the far right to see all the eliminations that are possible because of how cleverly we spotted those "hidden quads" (a name we invented). We can now do the following:

• eliminate the 6 from the first cell

• eliminate the 2 and 6 from the third cell

• eliminate the 2 and 6 from the fourth cell

• eliminate the 2 form the sixth cell

Go to the head of the class if you noticed that after the eliminations, we are left with an exposed triple of 2, 4, 6 in cells 2, 7 and 8.

-------------------- SITUATION 5 (Exposed Triple) --------------------

Let's deal with one last example of an exposed, but somewhat subtle, exposed triple. The trick here is to separate the digits that are a part of the applicable set of 3 digits that DON'T appear in any of the 3 cells that have been determined to HAVE TO contain one of the 3 digits and no other.

Check out the first column on the right. Look for the digits 4, 8 and 9. Notice the following:

• the 8 and 9 appear in the third cell

• the 4 and 9 appear in the fifth cell

• the 4, 8 and 9 appear in the sixth cell

That makes three digits and three cells. Therefore the three identified cells must contain one of these three digits and from that we know that all occurances of a 4, 8 or 9 can be eliminated from all other cells in the column.

Thus, we can make the eliminations that are seen in the column on the far right. To be specific, we can do the following:

• eliminate the 8 in the fourth cell

• eliminate the 4 in the seventh cell

• eliminate the 8 in the eighth cell

 Once again, it's all about MAKING ELIMINATIONS.

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Summing Up -- What Have We Learned?

By using the SBY program, we have promised you that you will get better at solving Sudoku puzzles. Was this just an idle boast or is there some substance to this claim? To answer this question, le't review what we have learned. Well, we have learned that the solving of intermediate and complex Sudoku puzzles requires 3 things:

• PATTERNS: you need to understand the concept of what we call a "pattern" and get good at recognizing them. The above description of hidden and exposed cells is just one of many patterns that can be identified. On other sites and in good reference books on Sudoku you can learn about other techniques with scary names like X-Wing, Sword-Fish, Jelly-Fish, etc. We encourage you to seek out this reference material.

• ELIMINATIONS: having identified a pattern, you have to take action and that action invariably requires eliminating possible digit(s) from the set of digits that have already been identified as being "non-conflicting" with the basic rules of Sudoku. The case studies that appear above clearly demonstrate how a typical pattern is identified and used to make the eliminations that appear with a dark blue background color.

• TOOL: OK, you've identified a pattern and from that you've identified some eliminations that must be made. Now you need a tool that will facilitate the elimination process. That's where the SBY Options Grid comes in. As you have already learned, the Options Grid lets you see the original permissable values that can be placed in each cell and then it provides a means by which eliminations can be "marked" but can still be visible so that your work can be checked. Bear in mind that the SBY Options Grid is nothing more than an automated approach to the "pencil markings" that are referred to in reference material. Accordingly, the Options Grid that is included on the printed SBY grids is just as useful as the electronic version on the home page of the actual SBY software offering.

Take what you have learned on this page and from using the SBY program to help raise your Sudoku puzzle solving skills to the next level!

Happy Sudoku Solving,

.... Brian Yager and the SBY Team