3rd Sudoku puzzle at 4th level of hardness with its solution, strategies and techniques explained
This is the 3rd puzzle session at fourth level of hardness that is placed in very hard category.
This is found to be really a very hard puzzle with multiple bottlenecks to overcome right from the start. Valid Digit subset evaluation to a large extent needed to be done very early that indicates the hardness of the puzzle.
Digit subset analysis, Cycles and digit lockdown techniques are the primary techniques and resources used for overcoming the bottlenecks.
A new structure of chained single digit locks is discovered and converted to a new technique first time. Solution of each step with strategies and techniques adopted explained fully.
We thought it time to say a few words on hardness of Sudoku puzzles entirely learned through solving the puzzles that we published here. If you so decide, you may very well skip this following section.
A few words on hardness of Sudoku puzzles
First to state is—
There is no simple generally accepted criterion that determines the hardness level of a Sudoku puzzle as there are no well-defined hardness levels in the first place.
Then what about the hardness levels of our puzzles that we have solved and explained till now? One aspect we may say is—surely the 2nd level puzzles are more difficult than the 1st beginner level ones. In the same way, our 3rd level puzzles solved surely are more difficult than the 2nd level puzzles.
In fact, by solving the third levels puzzles we have discovered a host of new structures that gave rise to powerful techniques. Mind you, that is all by self-learning—by patient search for new digit patterns and creating a new technique. This lies at the heart of problem solving.
Self-learning by pattern discovery is a key to solving harder problems.
To be honest, we cannot say this first 4th level Sudoku puzzle is harder than all the 3rd level puzzles we have solved. No new technique needed to be used in solving this puzzle.
Having said that, it cannot be denied that this puzzle needed more concentration till final solution. This is a certain indicator of hardness of the puzzle. We think that this general difficulty level arises from the single criterion of NUMBER of CELLS FILLED in the puzzle game.
As you can perceive—the less is this number the more cells you have to fill with valid digits yourself and at least it would be more tedious, if not more difficult.
The minimum number of filled cells reached by the 3rd level puzzles is 26, whereas this 4th level puzzle has just 26 cells filled. At least from this parameter it can be perceived that the puzzle is hard to solve. Its special filled digit combination has made it really a very hard puzzle to solve.
The 3rd Sudoku puzzle at 4th level of hardness
The following is the Sudoku puzzle that should engage your mind for some time. The Rs are the row labels, Cs are the column labels and this we define as the stage 1 marked on top left corner.
You may go through the next five sections for learning strategies and techniques for solving hard Sudoku puzzles in brief. Or, you may skip.
Following these five concept sections, the solution of the puzzle is explained step by step in details.
Please spend your time fruitfully on the game before going through the solutions.
Overall strategy adopted and techniques used
As a strategy we always try first—the row-column scan to find the valid cell at any stage because that is the most basic and easiest of all techniques.
When easy breaks by row-column scan becomes hard to come by, the next technique is used.
Next easy to use technique used is—identification of single valid digit for a cell by Digit Subset Analysis or DSA in short. This technique is explained in a following concept section.
And wherever possible, Cycles are formed that in any situation are a treasure to have and Cycles play a key role in quick solution. Concept and use of Cycles are explained in a following section.
You may wait for Cycles to form automatically in a column or row, but a proactive approach of forming a Cycle by DS analysis speeds up the solution process considerably.
The last resort of filling each empty cell with valid digit subsets is to be taken when it is absolutely necessary. Only with all empty cells filled with valid digit subsets, the possible breakthrough points in a hard puzzle can be discovered. Strategically for faster solution, it is better to delay this time consuming task as much as possible.
Full DS population process is explained in a following section, but any experienced Sudoku player would be doing it as a routine.
In hybrid strategy, a few of the cells of interest are filled with DS of shorter length and analyzed for a breakthrough such as forming a Cycle.
One of the most powerful patterns that we have used for highly positive result each time is the lockdown of a single digit in a row or column inside a 9 cell square so that the digit is eliminated from all other DSs in the locked row or column outside the 9 cell square. The necessity of use of this digit lockdown technique indicates in a way the hardness of the puzzle. This technique is also explained in a following section.
A still rarer pattern is 4 cell single digit lockdown in a rectangular formation that we have found only once. Naturally, it is a superset of the more common single digit lockdown in 2 cells and so is much more effective.
A basic part of overall strategy is,
Whether we search for a breakthrough of a bottleneck or a valid cell identification, our focus usually is on the promising zones, the zones (row, column and 9 cell square combined) that contain larger number of filled digits including Cycles.
The main strategy should always be to adopt the easier and faster technique and path to the solution by looking for key patterns all the time. Digit lockdown, Cycles, Valid cell by DSA are some of the key patterns.
Focus when solving a hard Sudoku puzzle should be on using the technique that would produce best results fastest. Easy to say, not so easy to do—comes with practice.
Structure and use of a Cycle
Form of a Cycle:
In a Cycle the digits involved are locked within the few cells forming the cycles—they can't appear in any other cell in the corresponding zone (row, column or 9 cell square) outside the few cells forming the cycle.
For example, if a 3 digit cycle (4,7,8) in column C2 is formed with a breakup of, (4,7) in R1C2, (4,7,8) in R5C2 and (7,8) in R6C2, the digits 4, 7 and 8 can't appear in any of the vacant cells in column C2 further.
If we assume 4 in R1C2, you will find R5C2 and R6C2 both to have DSs (7,8) implying either digit 7, or 8 and no other digit to occupy the two cells. This in fact is a two digit cycle in the two cells. Together with 4 in R1C2, the situation conforms to only digits 4, 7 and 8 occupying the set of three cells involved in the cycle.
Alternately if we assume 7 in R1C2 (this cell has only these two possible digit occupancy), by Digit Subset cancellation we get, digit 8 in R6C2 and digit 4 in R5C2 in that order repeating the same situation of only the digits 4,7 and 8 to occupy the set of three cells.
Effectively, the three digits involved cycle within the three cells and can't appear outside this set of three cells. This property of a cycle limits the occupancy the cycled digits in other cells of the zone involved (which may be a row, a column or a 9 cell square) generally simplifying the situation and occasionally providing a breakthrough.
Use of a cycle:
In the example of Cycle above, if a vacant cell R8C2 in column C2 has a possible DS of (1,4), as digit 4 has already been consumed in the cycle (4,7,8) in the column, only digit 1 can now be placed in R8C2.
This is how a new valid cell is broken through which otherwise we were not able to find out in any other way.
In any hard Sudoku game solution, creating, analyzing and using the structure of Cycles play a very important role.
How a valid cell is identified by Digit Subset Analysis or DSA in short
Sometimes when we analyze the DSs in a cell, especially in highly occupied zones with small number of vacant cells, we find only one digit possible for placement in the cell. We call valid cell identification in this way as Digit Subset Analysis.
For example, if in row R4 we have four empty cells, R4C1, R4C3, R4C6 and R4C9 with digits left to be filled up [1,3,5,9] we say, the row R4 has a DS of [1,3,5,9] that can be analyzed for validity in each of the four empty cells.
By the occurrence of digits in other cells if we find in only cell R4C1 all the other three digits 3,5 and 9 eliminated as these are already present in the interacting zones of middle left 9 cell square and the column C1, we can say with confidence that only the left out digit 1 of the DS [1,3,5,9] can occupy the cell R4C1.
While evaluating the valid digit subset or DS of an empty cell, you would analyze not only the digits that are already filled in corrsponding row, column and 9 cell square, you must include the Cycles present in the three interest zones also.
This is how we identify a valid cell by Digit Subset Analysis.
You may also refer to our first and second game play sessions at level 2 where we first explained use of a Cycle and DSA.
On filling up of every empty cell DS or full DS evaluation
We have not yet discussed the filling up of every empty cell with their valid digit subsets or DSs.
Let us see this in a little detail.
For example, to evaluate the DS for cell R1C1 in our puzzle, look at the row R1 and column C1 that have together a set of 6 unique digits 1, 3, 4, 6, 7 and 8 with three digits missing in it—2, 5 and 9. At this point, valid digit subset or DS for R1C1 is [2,5,9].
Lastly check the third dimension of the home square of R1C1, the 9 cell top left square, for any more possible digit cancellation. One more unique digit 9 being there, this digit is cancelled from the set [2,5,9] and the final DS for cell R1C1 is just [2,5]. These two are the only candidates that can occupy R1C1 at this stage.
Basically for evaluating the valid DS for a cell,
You have to cross-scan the row and column as well as check against the home square digits to identify the missing digits that are the only candidates for filling the cell.
For full DS evaluation of all empty cells, this process is to be carried out for each empty cell carefully. Slightest error at this stage will land you into grave trouble later on.
This is a tedious and error-prone process.
In solving a hard Sudoku puzzle, there may be no option than to go through the full empty cell DS evaluation. But it should be done when it has to be done and as late as possible.
Two strategic approaches are adopted to minimize the overall work load in this process—
- First try to find valid digits and fill the cells as much as possible using any technique so that the number of possible valid digits in empty cells as well as number of empty cells are reduced, and,
- Identify promising zones to evaluate the DS of a few cells locally trying for a breakthrough and so reduce the full DS evaluation load later on.
The second is a dynamic approach that depends on your experience and skill in identifying promising zones.
Single digit lockdown and its use
Occasionally, after evaluating valid DSs for a large number of empty cells, you may find if you look closely, that,
A single digit appears only in the DSs of two or three cells inside a 9 cell square—in a single column or a row, and in no other DSs in the 9 cell square.
This is what we call as single digit lockdown.
If it happens in a row inside a 9 cell square, in no cell in the row outside the square the digit can appear.
And so you can eliminate all occurrences of the locked digit from the DSs in the row outside the 9 cell square. If you can do that, usually it would give you the much needed breakthrough. It is a very powerful structure. And same for single digit lockdown in a column inside a 9 cell square.
As an example, if DSs in R9C7 and R9C9 in row R9 and in the bottom right 9 cell square, are [1,4,8] and [1,4] and digit 4 appears only in these two DSs in the 9 cell square, you know that the digit 4 is locked in R9 inside the bottom right 9 cell square.
Then if the DS in R9C1 is [3,4,7], happily delete the locked out 4 from this DS to reduce it to just [3,7].
You may think, what is the point of it, what would it achieve after all!
Well, in a similar situation in the process of solving a hard Sudoku puzzle game, the reduced DS in R9C1 formed a cycle (3,7) in column C1 and helped to pinpoint a valid digit 4 in cell R2C1 and that started a deluge of valid cell finds. This proved to be the key turning point in the whole game.
Let us solve our hard Sudoku puzzle now.
Sudoku 3rd puzzle at fourth level of hardness
We'll show the puzzle board again for convenience of understanding.
To follow the details accurately, you should better have the game actually with you written on paper, or better still—created in a spreadsheet.
The first valid cell identified is—R3C3 6, scan R1,R2,C2. This first fill in a stage is colored turquoise blue as a convention.
Next few valid cells and Cycles obtained are—
R7C7 1, scan R8,R9,C8 -- R5C9 1, scan C7,C8 -- R4C2 1, scan R5,C3 -- R3C4 1, scan R1,R2,C6 -- R6C5 1, scan C4,R4,R5 digit 1 over -- R1C5 9, scan R2,C6.
Shifting focus to force-creation of Cycles and resulting valid digit finds, results in—
Cycle (2,57) in R1 cells R1C1 (2,5), R1C2 (2,5,7) and R1C9 (5,7) -- At least Digits [3,8] get cancelled from DS [2,3,5,7,8] in R1 by scanning each crossing column C1,C9 for R1C1, R1C9 and scan top left square and C2 for R1C2 -- Formation of Cycle (3,8) in R1C7, R2C8 by exclusion in R1 -- R2C3 3 cancellation of 8 scan C3 -- R1C7 8 exclusion -- R2C8 3 scan R1,R3,C9 -- R6C7 3 scan R4,C8 -- R3C8 2 scan C7,C9.
With no more easy scan visible, excluding 5 digit lengths (of little use in cycle formation as 5 digit long cycle are nearly non-existent), most of the empty cells are evaluated at this stage itself for Cycle formation or other pattern discoveries.
Immediately new Cycles and valid cells discovered—
[3,6,7] eliminated from DS [3,4,5,6,7,9] of each of R5C1, R5C2 and R5C8 Cycle (4,5,9) formed in these three cells of R5 -- R5C5 6 as [3,7] eliminated from DS [3,6,7] scan C5 -- R7C6 6, scan R8,C4,C5 -- R9C1 6, scan R7,R8,C3 6 over.
Now we'll identify a relatively rare 4 cell Cycle (4,7,8,9) in R9C3 (as 2 is locked in C3 left middle square), R9C7, R9C8 and R9C9. The immediate positive results are—
R9C5 2 exclusion -- R9C8 8 again by exclusion as this is the only cell out of the four that can have 8.
All the results achieved are shown in the graphic below. Cells R5C5 6 and R9C5 2 also are colored turquoise blue to indicate that these two provided breakthroughs.
You may verify the actions from the second stage status.
Stage 3 of Sudoku puzzle solution: 4th level game play 3
As digit 8 is locked in R8C5, R8C6 inside bottom middle 9 cell square, 8 gets eliminated from DS [4,5,7,8] of R8C2. Result of this small action is promising—
R7C2 8 scan C1, R8 digit lock -- R7C1 2 scan R8, C3 with 2 locked -- R1C1 5 cancellation -- R1C9 7 cancellation -- R1C2 2 exclusion.
But now even at this late stage, in spite of many Cycles, this point turned out to be a bottleneck.
So we took to the last resort—evaluating valid digit subsets for all the remaining empty cells. It is quick now as most of this job has already been done.
This is the opportunity to discover the single key pattern that would break the bottleneck.
Can you spot it?
We would just indicate that the key pattern is the single digit 5 locked in Row R4 in cells R4C5 and R4C6.
The stage is closed and results shown below so that you can examine its results and try to push forward on your own.
Stage 4 of Sudoku puzzle solution: 4th level game play 3
Digit 5 being locked in R4C5 and R4C6, 5 cannot appear in DS [4,5,9] of cell R4C7 outside this locked 9 cell square. As DS of R4C7 is changed to [4,9] a second lock of digit 5 appears inside the right middle 9 cell square in R5C8 and R6C8.
This is very rare case of a chain of single digit locks-first lock causing a second one.
The second lock helps reduce DS of R7C8 to (4,9) that gves a Cycle (4,9) immediately with R7C4.
Final breakthrough -- R7C3 5.
Next valid cell are easy and routine—
R5C2 5 scan C1, C3 -- Cycle (4,9) in R5C8, R7C8 -- R6C8 5 exclusion -- R8C7 5 scan R9,C8 -- Cycle (4,9) in R7C8, R9C9 -- R9C7 7 exclusion.
Note: You could have obtained this Cycle (4,7) in right bottom 9 cell square as R7C3 5 was identified. It would have been easier route to the solution.
Cycle (4,9) in R8C1, R9C3 -- R8C2 7 exclusion -- R3C2 4 exclusion -- R2C3 7 cancellation -- R2C4 2 cancellation -- R6C4 4 cancellation -- R6C3 2 cancellation -- R7C4 9 cancellation -- R8C4 3 cancellation -- R8C6 8 cancellation -- R8C5 4 cancellation -- R8C1 9 cancellation -- R5C1 4 cancellation -- R5C8 9 cancellation -- R7C8 4 cancellation -- R4C5 5 cancellation -- R2C5 8 cancellation -- R2C6 5 cancellation -- R2C9 4 cancellation -- R9C9 9 cancellation -- R9C3 4 cancellation -- R3C9 5 cancellation -- R3C7 9 cancellation -- R4C7 4 cancellation -- R4C6 2 cancellation -- R4C3 9 cancellation -- R3C6 7 cancellation -- R5C6 3 cancellation -- R5C4 7 cancellation -- over.
The last cell is colored yellow.
The final solved puzzle board is shown below.
Check for the validity of the solution if you need.
End note on Problem solving in Sudoku
Any puzzle solving involves essentially problem solving. The general steps are,
- First stage analysis and breaking it down into smaller chunks if possible as well as adapt the strategy of solving this type of problem,
- Solving the easier component problems so that the main problem size and complexity is reduced,
- Detailed information collection, that is, defining the problem in more details as far as possible,
- Second stage analysis of structure of problem (in this case of Sudoku) and information content,
- Key pattern identification,
- Use of the key pattern to create the breakthrough,
- Repeating the last five steps (steps 3, 4, 5, 6 and 7) for finally solving the problem.
As this Sudoku problem is large and complex, we had to stop at step 3 to collect detailed DS information of most the cells and afterwards facing a bottleneck gain the full DS evaluation at step 3 had to taken up again. Effectively then the last 5 steps are repeated.
Key pattern identification had occurred in many instances primarily by Cycles and single digit lockdown.
For the first time a new structure of chained two pairs of single digit lockdown discovered and used for the main breakthrough, though a second easier path to the solution also discovered.
Watch out for the next 4th level Sudoku puzzle solution.
Other Sudoku game plays at fourth level hardness
Hard fourth level Sudoku puzzle 3, Strategies Techniques and Solution
Assorted Interesting Sudoku game plays
These Sudoku game solutions are collected from various sources and are found to be interesting. You can get these Sudoku solutions at Interesting Sudoku not classified at any hardness difficulty level.
First and second level Sudoku games
Third level game plays
List of third level hard Sudoku game plays are available at Third level Sudoku.