5th Sudoku puzzle at 4th level of hardness with its solution, strategies and techniques explained
This is the 5th puzzle session at fourth level of hardness of very hard category.
This is a very hard puzzle with multiple bottlenecks to overcome right from the start.
Cross-scan of row, column and 9 cell square, Digit subset analysis and advanced use of Cycles provide the main breakthroughs. Though single digit lockdown and an amazing 6 cell long Cycle are observed, these are not used for breakthrough.
DS evaluation for empty cells needs to be done early but in selected cells and limited to 3 digit long DSs.
For most Cycles force-creation approach is adopted and solution of each step 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. Even though its filled digit number is not very low, it should definitely be classified as a very hard puzzle by its combination of filled digits.
It is generally true that, at this level, if number of digits filled in the puzzle is reduced even by 1, its difficulty may increase significantly.
5th Sudoku puzzle at level 4 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 trying to solve it 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 or a single digit lockdown.
One of the most powerful patterns that we have to use while solving very hard puzzles 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. Though this was present, there was an easier path to break the bottleneck available. Nevertheless, 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 corresponding 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 valid DS for cell R1C1 in our puzzle, look at the row R1, column C1 and top left 9 cell square that have together a set of 7 unique digits 1, 2, 3, 4, 6, 7 and 8 with two digits missing in it—5 and 9.Valid digit subset or DS for R1C1 is then [5,9]. These are the only two candidates for filling up this cell.
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.
Three 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,
- 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, and
- When taking up DS evaluation of promising zones, limit evaluation of DS length to 3 so that clutter is reduced. More often than not, a breakthrough pattern emerges with 3 digit long DS evaluation itself.
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 5th puzzle at level 4 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—R4C4 1 cross-scan R6, C5, C6. This is the only direct cross-scan valid digit obtained.
R9C9 2 by DS evaluation scan R9, C9 and bottom-right 9 cell square; this is breakthrough -- R7C9 1 again by DS evaluation for the cell -- and then R8C9 9 by the same method of DS evaluation -- R6C8 9 cross-scan R4,R5,C9.
The first Cycle (1,5,9) in R1C1, R2C1 and R3C1 is formed by force-creation using DS analysis for the cells and this is an important element to use for valid cell finding.
The second Cycle (7,8) in R1C2, R1C4 is again force-created by scan R1, C2 and R1, C4 respectively. This gives the immediate breakthrough of—
R1C7 9 DS cancellation by Cycle (7,8) and of 5 by scan top right 9 cell square.
R9C3 8 home square DS cancellation.
To show further progress clearly we'll close the stage at this point and show the results below.
You may verify the actions taken till now from this second stage status.
Notice that we have evaluated the DS of a limited number of promising cells that too DS length restriced to 3. This strategic approach speeds up the process.
Stage 3 of Sudoku puzzle solution: 4th level game play 5
The first four valid cells are easy to find—
R1C1 5 cancellation scan R1 -- R9C6 4 DS cancellation --R9C8 7 exclusion -- R6C4 4 cross-scan C5, C6.
Next two involved opportunity spotting by DS analysis and subsequent 8 easy valid cell finds—
R7C4 2 DS cancellation of [3,7,8] from DS [2,3,7,8] in C4 scan R7 -- R7C5 6 DS cancellation [4,5] from DS [4,5,6] in R7 scan 9 cell middle bottom square -- R7C3 5 DS cancel 4 scan C3 -- R7C2 4 exclusion in R7 -- R8C2 2 cancel 4 -- R4C1 4 cancel 2 -- R4C6 6 scan R6,C5 -- R5C8 6 scan R4,R6,C7.
Three automatic Cycles in R8 formed for future ease of cancellation—
Cycle (4,5) in R8C7, R8C8 -- Cycle (1,6) in R8C2, R8C3 -- Cycle (3,7,8) in R8C4, R8C5, R8C6 in bottom middle 9 cell square.
And before closing the next important Cycle is force-created—
Cycle (3,7,8) in R1C4, R2C4, R3C6 in top middle 9 cell square by DS analysis.
For ease of understanding let's close at this point and show you the status below.
Stage 4 of Sudoku puzzle solution: 4th level game play 5
The immediate breakthrough is given by Cycle (3,7,8) in top middle 9 cell square—
R3C5 9 DS cancel [2,5] from [2,5,9] scan R3.
This followed by a series of easy valid cells—
R3C1 1 cancel -- R2C1 9 cancel -- R3C8 3 cancel -- R4C8 5 cancel -- R5C7 2 cancel -- R4C7 8 cancel -- R6C9 3 cancel -- R2C9 8 cancel -- R4C2 3 cancel -- R5C3 1 cancel -- R5C2 5 cancel.
Let's again close at this stage to keep clutter at a lower level and go over to the final stage for ease of explanation. The status results are shown below.
Stage 5 of Sudoku puzzle solution: 4th level game play 5
The first series of valid cells finds are just by cancellation and exclusion—
R8C3 6 cancel -- R8C2 1 cancel -- R2C3 7 cancel -- R6C3 2 exclusion -- R6C6 5 cancel -- R2C6 2 cancel -- R4C5 2 exclusion -- R5C5 3 exclusion -- R8C8 4 cancel -- R8C7 5 cancel -- R2C8 1 cancel -- R6C2 7 exclusion in left middle 9 cell square -- R1C2 8 cancel -- R2C2 6 cancel -- R1C4 7 cancel.
The also rest are straightforward—
R6C5 8 exclusion -- R8C5 7 cancel -- R2C5 5 exclusion -- R2C7 4 cancel -- R3C7 7 cancel in 9 cell home square -- R2C4 3 cancel - -R3C6 8 cancel -- R8C6 3 cancel -- R8C4 8 cancel.
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 a number of promising cells for force creation of Cycles. Later when required another set of empty cell DSs were evaluated for the next breakthrough.
Information collection, analysis and key pattern identification was in stages.
Other Sudoku puzzles you may like to go through at leisure
Hard Sudoku level 4 puzzles
Hard fourth level Sudoku puzzle 5, Strategies Techniques and Solution
New York Times Hard Sudoku puzzles
Hard Sudoku level 3 puzzles
You may access all hard Sudoku level 3 solutions at Third level hard Sudoku.
Medium level 2 puzzles
You may read through all medium level 2 solutions at Second level medium Sudoku.
For beginners, Sudoku beginner puzzle solutions are at Beginner level Sudoku.