1st Sudoku puzzle at 4th level of hardness and its solution explained with strategies and techniques
This is the 1st puzzle session at fourth level of hardness that is supposed to be very hard.
As expected, this is found to be a difficult puzzle with multiple bottlenecks to overcome even at late stages.
Digit subset analysis, Cycles and digit lockdown techniques are the main advanced techniques and resources used for overcoming the bottlenecks. Solution of each step is 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 first 4th level puzzle has only 25 cells filled. At this advanced stage, if you remove even a single digit in the puzzle, not only it may increase the hardness greatly, it might even destroy the uniqueness of solution of the puzzle.
The 1st 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 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. 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 single digit lockdown and so is much more effective.
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.
The main strategy for solving a hard Sudoku puzzle is to use 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 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 7 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.
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 R8C1, look at the row R6 with six digits missing in it—1, 2, 4, 5, 6 and 7.
Now cross-scan column C1 to identify any digit these six appearing in the column C1.
As 5 and 7 are the two digits out of the six that appear in the column, cancel these two from the six digit subset of R8C1 to reduce it to [1,2,4,6].
Lastly check the third dimension of the home square, the 9 cell bottom left square, for any more possible digit cancellation.
With no additional digit cancellation, the valid digit subset or DS for the cell would be four digits [1,2,4,6].
None of these four digits appear in the home square, home column or the home row for the celll R8C1.
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 may fill the cell.
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 trying for a breakthrough and so reduce the full DS evaluation load.
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 in a 9 cell square, in a 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 the puzzle now.
The Sudoku 1st 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 still better—created in a spreadsheet.
The first valid cell identified is—R3C4 3, scan C5, C6—pretty simple. This first fill in a stage is colored turquoise blue as a convention.
Next few valid cells obtained by direct scans are—
R9C9 4, scan R7,C7,C8 -- R5C6 4, scan R4,R6,C4 -- R5C6 4, scan R4,R6,C4 -- R3C5 4, scan R1,C6 -- R2C3 4, scan R1,R3,C2 -- R8C1 4, scan R9,C2,C3.
Digit 4 is fully populated in all 9 cell squares, and good thing is Cycle (1,5) is automatically formed in cells R1C5 and R1C6 in R1 because of cancellation of 2 in R1 from DS [1,25] in both the cells. Next fill by exclusion is—
As a strategy, we don't delay in identifying a Cycle and reaping its benefits immediately.
We take up cross-scan again to get the next few fills as—
R4C5 8, scan R6,C4,C6 -- R5C2 8, scan R4,R6,C1 -- R3C3 8, scan R1,C1,C2 -- R1C1 9, scan R3,C2,C3 -- R6C2 5 scan C1, C3.
Now we force-create the 3 cell Cycle (1,2,6) in cells R3C1, R5C1 and R9C1 by DSA in these 3 cells—
For R5C1 DS [1,2,6] as, digit 5 is cancelled by scan in C1 from the DS [1,2,5,6] in R5 -- For R3C1 DS [1,2,6] as, digit 7 is cancelled by scan in C1 from DS [1,2,6,7] in R3 -- For R9C1 DS [1,2,6] as digits [3,7] are cancelled from DS [1,2,3,6,7] in R9C1 home square by cross-scan in C1 and R9.
This is a minor breakthrough that resulted in—
R6C1 3 by exclusion.
Before we close this stage
Cycle (1,6,7) formed in top right 9 cell square -- Cycle (3,8,9) formed in R2 in same square -- R2C2 1 by exclusion.
A lot has been achieved at this stage with formation of many Cycles and valid cell fills.
The results are shown below with valid cells found at this stage colored green. We'll color the valid cells in the next stage in a different color to distinguish the cells filled in various stages.
Stage 3 of Sudoku puzzle solution: 4th level game play 1
The immediate positive results are—
R2C9 9 as 8 cancelled by scan on C9 from DS [8,9] -- Cycle (2,6) in R4C9 and R4C9 in R4 by cancellation of 9 from DS [2,6,9] in R4C9 -- R4C6 5 by cancellation of 6 due to cycle (2,6) from DS [5,6] in R4C6 -- R1C6 1, DS cancel -- R1C5 5 by cancellation -- R7C6 6 exclusion -- Cycle (1,2) in R7C5, R9C5 -- R8C4 5, scan R9, C5 -- R5C4 2, DS cancel of 5 from DS [2,5] -- R9C4 7 exclusion -- digits [1,2] cancelled in DS [1,2,6] in R6C5 -- R6C5 6.
Cycle (1,6) in R5, R5C8 5 by exclusion -- R7C7 5, scan R8,R9,C8,C9 -- R7C8 9, scan R8,R9,C9 -- R4C7 9, scan R6,C8,C9 -- Cycle (2,6) in R4 -- R4C8 3 by exclusion -- R2C8 8, DS cancel -- R2C7 3, exclusion in R2 -- R9C8 8 scan R7, R8, C8.
We'll close actions of this stage and show you the results below.
Stage 4 of Sudoku puzzle solution: 4th level game play 1
Three digits 4, 8, and 9 fully filled, six digits are left with not many empty cells. But there is no easy valid fill even at this late stage. This is the hallmark of a hard puzzle.
The bottleneck is now broken through by the rare and powerful single digit lock discovery—
The single digit 6 is locked in C9 of 9 cell right middle square -- it follows, no 6 can occur in C9 outside this 9 cell square -- R1C9 7 by cancellation of locked 6 from its DS [6,7].
This is the much needed breakthrough. The immediate positive results are—
Cycle (3,6) in R1C2, R1C3 --R3C1 2 by cancellation of 6 from DS [2,6] -- R3C2 7 by exclusion-- Cycle (1,6) in R9, R9C5 2 by exclusion -- R7C5 1 exclusion.
The rest fills are straightforward as all bottlenecks are overcome. We'll show the rest of routine fills in the last stage. The crucial results of current stage is shown below.
Final Stage 5 of Sudoku puzzle solution: 4th level game play 1
The first fill at this stage is achieved by a Cycle—
Cycle (1,6) in R3C8 and R9C8 of C8 -- R8C8 7 by exclusion.
And subsequent fills—
R7C9 2 by cancellation -- R4C9 6 by cancellation -- R5C9 1 by exclusion -- R5C1 6 by exclusion -- R4C3 2 exclusion -- R9C1 1 exclusion -- R9C8 6 exclusion -- R3C8 1 exclusion -- R3C7 6 exclusion -- R6C3 1 exclusion in 9 cell square -- R6C7 2 exclusion -- R8C7 1 exclusion -- R7C2 3 cancellation of 2 from DS [2,3] -- R7C3 7 exclusion -- R1C2 6 cancellation of 3 from DS [3,6] -- R8C2 2 exclusion.
R8C3 6 exclusion -- R1C3 3 exclusion.
The final solved game 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 of the empty cells avoiding longer length DSs. Too much clutter makes it difficult to spot patterns.
DS evaluation was in stages, that's why problem steps were repeated from step 3.
The breakthroughs were by force-creating or normal formation of Cycles and single digit lockdown.
Digit subset analysis remained a necessary part practically for taking any action.
Level of hardness being high in this puzzle, careful attention to details was needed most of the time.
Checkout and tally these seven steps with the process of solving this puzzle.
This is what we call the life cycle of Problem Solving, an independent and enormously powerful overlay subject standing on its own.
Watch out for the next 4th level Sudoku puzzle solution.
Other Sudoku game plays at fourth level hardness
Hard fourth level Sudoku puzzle 1, 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.