Tips and Techniques for Solving Sudoku Puzzles
It has the look of a Calculus final. A scattered group of numbers staring you in the face, seemingly tossed at random into the many squares of the puzzle. You swallow hard, sharpen your pencil and embark on your journey, not knowing if you will return victorious or completely insane. You are just starting a game of Sudoku, the wildly popular puzzle game that has established itself as one of the most addictive games in the world. We now proudly welcome you to the Chess and Poker Dot Com Sudoku strategy guide that will help you tackle this always entertaining puzzle. It assumes you've played through several puzzles and are familiar with the game.
This unique puzzle is based on a simple set of rules that interestingly enough also guide a player to solving each puzzle. A complete solution requires that each Block, Row and Column of the puzzle board must contain the digits 1-9 exactly one time each, but not more than once. So once the puzzle is correctly solved, you will be able to look across any Row, down any Column or in any Block and find each of the nine digits only once in each segment.
Segments of a Sudoku Puzzle
Blocks are composed of 9 squares inside of a slightly darkened border on the Sudoku board. We have the Blocks in our Sudoku strategy guide labeled A through I, displayed faintly in the background of each Block. Rows and Columns are also made up of 9 squares, with Rows running the length of the board horizontally and Columns standing vertically. Understanding this layout is very important as you will not be able to solve a Sudoku puzzle without this basic knowledge. We will refer to the Sudoku board using these notation terms throughout the Sudoku strategy guide. An example of a partially solved Sudoku board is provided in the next paragraph as a reference to these basic guidelines.
All of the segments interact with one another perfectly in a solved Sudoku board. This means that what is true for one segment must hold true for the other two. So if you correctly solve a number in a Row, it must also be in the correct spot for the Column and Block that it happens to intersect with. If for any reason this is not true, then the number has not been correctly solved. In the notation graphic to the left, we can see that the entire A-block has been solved, as well as one full row and column.
Notice where the solved row and column cross one another in the E-Block, with a "5" marked at the intersection. If this number is correctly solved for the puzzle, then it is also already in the correct position for all of the other segments that it affects. For example, there won't be another 5 anywhere in the same Row or Column as the solved Five and of course not repeated in the E-Block. As you continue to solve other locations of the board you'll find that 5's in other segments of the board will have prevented the 5 from being placed into any square other than the correctly solved square it currently resides in, since that would violate the simple concept of once per segment. Now that we know the Sudoku rules, let's learn how to solve them!
Sudoku Strategy: Squeezing Solutions
In a game of Sudoku, a player will typically begin their puzzle by working to solve as many numbers as they can by using several common techniques, inevitably getting stuck at some point when faced with the more difficult numbers to find. That's where our advanced strategy techniques will kick in. But let's look at the two most common methods first, Squeezing and Cross-Hatching.
Squeezing is the process of using two solved numbers in connected Blocks of the Sudoku board to solve the same number in the last connected Block. When there are already two of any number solved in connected blocks, we can reduce the total number of legal squares the same number can be located in for the target connected Block. In the graphic to the right, we can see that the number "2" (deuce) has already been solved in both the A and B Blocks. This means that we can use this information as leverage when trying to solve for the Deuce in the connected C Block. We have highlighted the "off-limits" squares in red across the board based on the location of the A and B-Block deuces. Of course, we know that there can only be one deuce per Block per the rules so we don't need to highlight any other squares in the A and B-Blocks (we already know they're off limits).
The top three squares of the C-Block are off limits for the C-Block deuce due to the rule that states each Row must contain each digit only once. In this case the B-Block deuce has already been solved. It sits in the top row of the puzzle board as well. Therefore, there cannot be another deuce in the entire row. Remember, since there is already a deuce in the top row, there can't be another deuce in the entire row based on the rules. This also holds true for the bottom three squares of the C Block, because of the deuce located on the same row already in the A Block. This leaves only the middle three squares of the C Block with the option of containing the deuce. Upon further inspection, we see that two of these squares already contain numbers, leaving only one available square open. Since the deuce can't be in any of the off-limits squares based on the rules (highlighted in red), we know that the correct location of the C-Block deuce must be in the last open square.
This process is known as Squeezing, and should be the first technique you use when starting out a new Sudoku puzzle. For some of the easier puzzles, using this simple technique alone will allow you to solve a large batch of numbers. When you move up to attempt the more difficult puzzles, however, Squeezing will play a much smaller role in solving the puzzle but will always be a necessary component in solving them. Using this basic technique, we can now build up to even stronger concepts used to solve the more difficult numbers on the board.
Sudoku Strategy: Cross-hatching Solutions
Cross-hatching is similar to the Squeezing technique except that with Cross-hatching you will be using more than two numbers in more than two connected Blocks to solve for a number. In the example above, we used the two deuces from the A-Block and B-Block to eliminate legal squares for the last remaining deuce in the C-Block. When there was only one square left, we knew it had to be the correct square as no other options were available based on the rules of the game. But what happens when attempting a Squeeze isn't enough?
In the first graphic, we have used the 7's found in the I-Block and F-Block to eliminate legal squares for the 7 in the connected C-Block. But once the off-limits squares were highlighted, there remained two available legal squares (shown with question marks) for the C-Block 7. When we did a Squeeze above, there was only one legal square available, which made things much easier for us. But we'll need a bit more information to solve the C-Block 7, so we begin to look at the other Blocks which are connected with the C-Block. When we scan the A-Block and B-Block, we find a 7 that has already been solved in the B-Block. We can now perform a Cross-hatch.
Using the 7 from the B-Block, as well as the 7's from the I-Block and F-Block we can now reduce the legal squares for the C-Block 7 even more. In the second graphic above, we see that using this technique has reduced the available legal squares for the C-Block 7 to only one square, which of course must be the correct placement based on the rules of the game. Cross-hatching might seem difficult at first, but with a little practice you will develop your Cross-hatching skills very quickly. This is a very important technique that will be put to use throughout all Sudoku puzzles of any difficulty quite frequently. It should become second nature to you in no time at all.
Sudoku Strategy: Reducing The Three-or-Less Segments
Squeezing and Crosshatching will have their best results in the opening stages of solving a Sudoku puzzle and should give you a solid foundation of solved numbers to work from, but at some point you will most likely run out of numbers to solve using those two basic methods. It will then be time for a slightly more advanced technique.
Reducing is the process of logically eliminating possible numbers in a Block, Row or Column using simple deductive reasoning. When any segment has been solved to the point that only three or less unsolved numbers remain in it, the segment is ready to be Reduced. When you spot these three-or-less segments (which can be Blocks, Columns or Rows), a helpful technique will be to pencil in the remaining unsolved numbers for each three-or-less segment in the margins of the Sudoku board. It is important to note that some players will begin using pencil marks in segments that have many more unsolved numbers than just three-or-less unsolved squares. Except in some more advanced techniques (described below), this is almost always unnecessary and can confuse the player by cluttering up the board. For our Sudoku strategy guide, we will only focus on segments containing three-or-less unsolved squares as described here.
In this mini-example, we have focused on a sample B-Block which only needs the 1-3-6 numbers solved. This is because the numbers 2, 4, 5, 7, 8 and 9 have already been solved for the segment. Note that we have made pencil marks of the remaining numbers for this segment in the margins of the puzzle. This makes it very simple to remember what numbers we are dealing with and focuses the attention to the appropriate segments. For Rows and Columns it is usually best to make the pencil-marks just outside of each segment. With Blocks it is usually easiest to label them at the bottom of the puzzle, with the Block name beside the pencil marks. This should avoid any confusion that might occur between the different segments. Once all of the remaining unsolved numbers are determined for each of these segments, we can begin Crosshatching to reduce them further.
Sudoku Strategy: Crosshatching The Pencil Marks
In the examples below, we'll focus on the B-Block of this new puzzle which we can see only needs the 3-4-7 numbers solved. To further illustrate this, we've made pencil marks inside of each of the unsolved squares. Many players find this helps them keep track of the remaining numbers, but once you've gained some experience you'll soon be able to do this step in your head. Since we know what numbers need to be solved but not where, we must use the technique of Cross-hatching that we learned above.
In the first graphic, we're ready to begin crosshatching the pencil mark options we've arrived at. We'll now take a look at the Blocks connected to the B-Block to see if we can use any already solved numbers there to assist in our reductions. In the second graphic, we see that the 3 in the E-Block has already been solved. Since the rules state that there can be only one digit, one-time each per segment, we know that there cannot be another three in that entire column. Therefore, we can crosshatch that 3 into the target B-Block and erase the 3 options from that column only. Notice that we didn't erase the 3 pencil mark from the upper-left hand square of the B-Block. This is because that the 3 in the E-Block only applied to that column. There is no other 3 visible on this sample board that can be used to eliminate the 3 as an option for this particular square, so the pencil mark must remain for it there. We can now see that for two of the unsolved squares, only the 4 or 7 can be the correct solution. Now that we've at least eliminated some options, let's try for more reductions!
Using the same logic as above, we continue our scan of the Blocks connected to the target B-Block to see if we can find any more numbers to use to our advantage. We find a 4 located in the A-Block. Using our crosshatching skills, we can now eliminate the 4 as an option in the bottom unsolved square of the B-Block. This is of course because since there can only be one 4 for each row, and it's already been solved for the row in the A-Block, we know it is not a legal option anywhere else in the entire row. In the graphic above we have eliminated the 4 pencil mark from the bottom unsolved square of the B-Block. Now we can see that only the 7 pencil mark remains as an option for that particular square. What does this tell us? Since we know all of our calculations are correct up to this point, then the 7 must be the correct number for that square. We've solved for the 7 in the B-Block! In the next graphic to the right, we have officially placed the 7 into its solved position after erasing the mini-pencil marks. Now that we've done that, we can then use this newly solved number to further reduce the remaining options in the last two unsolved squares.
Since we've solved the 7 for the B-Block (and the entire column where it resides), we can now eliminate any remaining 7 pencil marks that we had previously been using there. After doing so, we can see that now one of the squares only has a 4 pencil mark remaining. With no other options available, we know that this square must be the correct location for the 4. And once we've erased the pencil marks and written in the 4, we know that only one number remains, the 3. It must go in the last unsolved square in the upper left-hand corner of the B-Block. We've completely solved this B-Block!
Sudoku Strategy: Spotting the Lone Number
Using the above Sudoku strategies, you should be able to deeply solve most of the easier puzzles and discover a large batch of solved numbers for many of the moderate-to-hard puzzles as well. Sometimes it is even possible to further optimize the steps to help speed things along (and avoid getting burnt out in the process). One such optimization is know as "Spotting the Lone Number". In the last strategy tip we learned how to pencil mark the available options for segments with three-or-less unsolved squares. Furthermore, we learned to use the cross-hatching technique to further reduce the pencil mark options and in some cases even solve the entire segment. But there is often an even more efficient method when performing these types of reductions.
In the mini-example to the right, we've focused in on a D-Block segment. It is already solved down to only three numbers remaining, which are the 2-8-9. These have been penciled in already for us and have been reduced as much as possible by using the crosshatching methods above. Since there aren't any available solved numbers left to help reduce the pencil mark options any further, we'll have to use some more logical deductions to move forward. Looking at the D-Block, it can be shown that the 9 can be solved immediately. To do so, we must simply find the lone number and solve for it.
The Lone Number in any given segment can occur when one of the many available pencilmark options is only found in only one of the unsolved squares. Let's take a look at our example. The remaining unsolved numbers are 2-8-9, and two of the squares have already been reduced to only having the 2 and 8 as available options. This means that there is only a single 9 pencil mark in the entire D-Block. This means that even though hypothetically all the options are still available (and are all still penciled in the square as 2-8-9), the 9 is a lone-number. If there are no other 9 pencil marks except in this particular square, we now know that it must be the correct square to place the 9 in! The Lone Number theory holds true for all segments, not just Blocks, so constantly keep watch for Lone Number opportunities such as this to speed your solutions along and get more numbers solved.
Sudoku Strategy: Advanced Crosshatching
During your Sudoku solving career you may decide that you are ready to graduate to some of the more difficult Sudoku puzzles, often titled "Very Hard". You may find that attempting the basic and somewhat advanced techniques you learned above may not get you quite as far as before. However, keep in mind that with these basic concepts alone you can solve any Sudoku puzzle (with the possible exception of some super-advanced versions, sometimes deemed "Diabolical" or similar intimidating terms). You just need to expand your horizons a bit to put the techniques you've learned into better use. Let's look at our final example, which should help us see that even some seemingly impossible solutions can be arrived at by simple logic.
Although it might not jump out at you just yet, the A-Block 9 can be solved in only three steps from the first graphic above. Even though there are absolutely no numbers solved in the A-Block to this point, using combinations of the strategy we've already learned will guide us on how to accomplish this. In the second graphic we have started the solution by attempting a crosshatch with solved 9's in the B, D and G-Blocks. However, it looks like we're stuck with two unsolved squares that the A-Block 9 could go in (shown with question marks), if only for the simple fact that the C-Block 9 has not yet been solved and therefore cannot be used to eliminate either of these two options. But you'll start to notice when any given number has several solved locations throughout the board, you may be able to combine techniques to logically, and creatively, reduce options.
In the next graphic we have decided that since the C-Block 9 hasn't been solved yet, we'll give it a try! Using solved 9's in the B-Block and F-Block we attempt to crosshatch the C-Block 9 but are unsuccessful in completely solving it. However, in the process we have quarantined the remaining squares that the 9 will go into, even though we don't know which exact one yet. For instructional purposes, we've pencil-marked the two locations (typically, we save our pencil-marking for segments containing three unsolved squares or less). Now we can examine the location of the two squares that will contain the C-Block 9. They both reside on the topmost row of the C-Block. We have made a very important discovery. Since we know that the C-Block 9 must be in one of these two pencil-marked squares, based on the rules we can further understand that the 9 for this row has been effectively solved for the rest of the segment. We don't need to actually solve the C-Block 9 at this point, because as long as we know that it must go into one of those two squares, both on the same row, we can use this information to cancel out the top three squares of the A-Block as seen in the next graphic to the right. There is now only one remaining square left for the A-Block 9, and we can confidently place it there.
Keep in mind that this technique, as well as all of the others we have discussed, apply to all segments of the Sudoku board, not just one or the other. They can be used up and down Columns, in either direction across Rows and in any Block configurations. Using this final method will allow you to "crack" some of the tougher numbers in your Sudoku puzzles and breakthrough with a solution.
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