How to find Hidden Singles faster
What is Hidden Single
Hidden Single is one of the most basic yet powerful technique for solving Sudoku. Mastering this technique alone would allow you to solve any Easy puzzle on Sudoku Mood. The idea is simple: choose a digit and a row, column, or 3x3 block. Then, check how many cells in the selected row, column, or block can contain that digit. If there's just one such cell, you can place the digit in that cell.
Anyone who has solved Sudoku at least once has almost certainly discovered and used Hidden Single technique intuitively, even if they didn't know its name. However, finding a Hidden Single can sometimes take a lot of time. Let's look into ways to speed up the search.
Order of combinations
Before we proceed further, let's introduce some terminology. We refer to a specific row, column, or block as a group. The type of group - whether it's a row, column, or block - is called the group type. For instance, when we talk about column 7, the group type is "column" and the group is 7.
To find a Hidden Single or to conclude that there are no Hidden Singles, we need to check how many cells in each row, column, and block can contain each digit. Using the terminology introduced above, we need to check all combinations of group type, group, and digit.
The key difference between various approaches to searching for Hidden Singles lies in the order in which you check these combinations. Understanding this fact alone can help you develop your own approach - simply try different orders and choose the one that works best for you.
There are countless ways to order all possible combinations, but only two of them make sense. The first one is this: first, you choose the digit, then the group type, and finally the group. The second one is: first, you choose the group type, then the group, and then the digit. Let's call the first way digits first, and the second one digits last.
More choices to make
Once you've chosen either digits first or digits last, there are a few more decisions to make. First is the order of digits. This is simple - it rarely makes sense to choose anything other than 1, 2, 3, ... 9. The same goes for the order of groups within a chosen group type. For example, if you need to scan all rows, the most natural order is from row 1 to row 9.
Where it gets less obvious is the order of group types. Usually, it doesn't matter whether you choose rows first or columns first. However, there is a difference in what you look at first - blocks or rows with columns. For humans, it's a bit easier to spot a Hidden Single in a block than in a row or column. Since in Sudoku, you should always examine the easiest option first, it makes sense to look at blocks first. And because there's no difference between rows and columns, you can always stick to the same order. For example, blocks first, then rows, and then columns.
So far, we can conclude that among all the choices considered, the only significant one is whether to use digits first or digits last. That's good news because it greatly reduces the number of approaches to choose from.
Once you find a Hidden Single
If you've checked all possible combinations and haven't found any Hidden Singles, it's time to move on to other techniques. But what should you do if you do find a Hidden Single? You can't just check the remaining combinations and move on to other techniques. Any change on Sudoku grid could lead to new Hidden Singles appearing. So it may happen that if you check again one of those combinations you've just looked at without success, you'll find a Hidden Single that wasn't there before. Moreover, placing a new digit might lead to a Last Digit appearing somewhere on the grid.
So, what should you do after finding a Hidden Single? Well, there is no single right answer. What you should definitely do is to look for Last Digits - they are easy to spot. But after that you have several options. Let's say you chose digits first and found a hidden 7 in row 3. Then you can do this:
- You can check digit 7 and the remaining rows 4 to 9. Then digit 7 and all columns. Then digits 8, 9 and 1 to 6 with all blocks, rows and columns. Then digit 7 with all blocks and only those rows you checked before you found a Hidden Single - 1 and 2. In other words, you go in circle until you come again to the combination that resulted in finding Hidden Single.
- You start checking all combinations from the beginning - digit 1 and all blocks, rows, and columns, then digit 2 and so on. Until you check the last combination - digit 9 and column 9.
There is another, more sophisticated option. You can try to think of what Hidden Singles could have appeared after you placed a new digit on the grid. It could be only:
- The same digit in any block, row, or column on the grid. See examples 1a and 1b.
- Any digit in the same row, column and block where you placed the new digit. See examples 2a and 2b.
This trick can help you to reduce the number of combinations that need to be rechecked. However, be careful when using it - you might overlook some Hidden Single and jump to more difficult techniques. Use this trick only after you solved enough Sudoku puzzles and feel confident to try out more advanced approaches.
Example 1a shows a Sudoku puzzle where there's only one Hidden Single on the grid - digit 5 in row 7. After you place it, a new Hidden Single appears: digit 5 in block 5. For detailed explanation and visualisation see solution path of this Sudoku.
Example 2a shows a Sudoku puzzle where there's only one Hidden Single on the grid - digit 8 in column 6. After you place it, a new Hidden Single appears: digit 7 in block 5. For detailed explanation and visualisation see solution path of this Sudoku.
To sum up
The key to faster search for Hidden Singles is to use a well defined and clear approach.
The approach to finding Hidden Singles is mainly determined by two factors: whether you use digits first or digits last; and what you do after finding a Hidden Single. However, understanding the logical path we've taken to reach this conclusion allows you to add your own twists and tricks to speed up the process.
If you are a beginner Sudoku player, it might be helpful for you to read about finding Hidden Singles on the Hidden Single technique page.
Keep playing Sudoku and get better at it!