Sudoku hard Strategy and Techniques: Expert level Sudoku hard puzzle game 3 Quick solution
Can you solve the Expert level Sudoku hard puzzle game 3? In any case, you might like to have a look into the quick solution with each step explained.
- Expert level Sudoku hard level 5 puzzle game 3
- Solving Expert level Sudoku hard level 5 puzzle game 3 by Sudoku hard strategy and techniques
- Sudoku hard Strategy and techniques for easy solution
- What is a Cycle and how to use it in solving a Sudoku hard puzzle. Frequently used technique.
- How a single digit candidate valid cell is identified by Digit Subset Analysis (DSA) in solving a Sudoku hard puzzle. Heavily used technique.
- How digits possible for all empty cells (DSs) enumerated while solving a Sudoku hard puzzle. To be used judiciously, better to use partially.
- Single digit lockdown and its use in solving a Sudoku hard puzzle. Helps to make a breakthrough in a Sudoku hard puzzle.
- Sudoku hard technique of double digit scan. Helps to make a breakthrough in a Sudoku hard puzzle by creating new digit pattern like a Cycle.
- Expert level Sudoku hard breakthrough technique of X wing or single digit rectangle. Helps to make a critical breakthrough in a Expert level Sudoku hard puzzle.
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While going through the solution you may click on say Cycle whenever it appears, to know how to form and use a Cycle and then return to the previous position to continue through the solution.
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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.
We'll first solve the Sudoku hard using strategies and techniques for solving Sudoku hard puzzles.
The strategy and techniques for quickly solving Sudoku hard are explained with examples in five sections after the solution.
You may decide to go through the solution directly and take a look at the specific strategy or technique when you feel like.
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Following is the solution of the puzzle explained step by step in details.
Please spend your time fruitfully on the game trying to solve it before going through the solutions.
Let us solve our Expert level 5 Sudoku hard puzzle now.
We'll show the puzzle again for ease 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.
First few are easy valid cell hits,
R5C4 8 scan R4, R6 -- R6C4 1 DSA reduction of [2,3,6] from DS [1,2,3,6] in C4 -- R4C4 3.
R6C6 9 scan R4 -- Cycle (4,7) in R4C5, R4C6.
Next a minor breakthrough by Single digit lock on 7 in R7C2, R8C2 by scan 7 in R9, C3,
R5C1 7 scan for 7 in R4 Cycle (4,7), 7 in C2 single digit lock, 7 in C3.
Cycle (1,2) in R5C7, R5C9 by reduction provided another minor breakthrough but nevertheless a series of valid cell hits,
R5C3 9 reduction. R4C2 1 scan for 1 in R6, C1, C3. R4C3 2 scan R6, C1.
R8C3 6 DSA reduction of [3,4,5] from DS [3,4,5,6] in C3.
DSA reduction is a powerful technique that may not necessarily result in a single valid cell hit. Often, Cycles or other useful digit patterns are formed by DSA reduction. In conventional method of solution of a hard Sudoku puzzle, possible digit subsets for all empty cells are evaluated by DSA reduction only.
You may verify the actions taken till now from this second stage status.
Detailed evaluation of possible digit subsets in empty sets avoided as much as possible.
Stage 3 of Sudoku hard puzzle solution: Expert level Sudoku hard level 5 puzzle game 3
First a few quick valid cells by DSA reduction,
R8C2 7 DSA reduction of [3,5] from DS [3,5,7] -- Cycle (3,5) in R7C2, R7C3.
R9C5 4 DSA reduction of [2,6] from DS [2,4,6] in R9 -- Cycle (2,6) in R9C7, R9C9.
R4C5 7 reduction -- R4C6 4.
Next valid cell hits are by formation if Cycles and row column scan,
Cycles (1,8,9) and (2,6) in bottom middle major square by DSA reduction.
Cycle (1,4,9) by DSA reduction in bottom right major square -- R7C8 7 reduction -- R7C7 8.
R6C7 7 scan for 7 in R4, R5, C8, C9.
R3C6 7 scan for 7 in R1, R2, C5.
With fairly large number of empty cells filled up with valid digits, it was easy to fill the empty cells with possible digit subsets quickly.
For ease of understanding let's close at this point and show you the status below.
Stage 4 of Sudoku hard puzzle solution: Expert level Sudoku hard level 5 puzzle game 3: Breakthrough by X wing digit pattern
With no more easier valid cell hits by Cycles, DSA or single digit scan directly, it is the right time to look for single digit sharing over more cells. This would create more complex digit patterns of X wing, XY wing or Sword fish which would give use the critical breakthrough we need so much.
Search for single digit shared over more than 2 cells
With knowledge of what to look for, it doesn't take much time to spot the X wing digit formation,
4 in R1 removes 4 from all possible digit subsets in C7 except R2C7, R3C7. This is indicated by red colored digit 4.
Similarly single digit lock on 4 is created in R2C3, R3C3 by scan C1, C2.
These four numbers of digit 4s are shared between two columns C3, C7 and two rows, R2, R3. It creates the powerful digit pattern of X wing.
EEFECT OF X WING is to,
Remove all other incidence of digit 4 from these two columns and rows.
To know more on the advanced digit pattern and technique of X wing click here.
Specifically in this Sudoku hard game, CRITICAL BREAKTHROUGH started with formation of a Cycle and thereby creating a valid cell by reduction of digits because of the Cycle,
Reduction of 4 from DS [2,4,6] in R2C6 would create Cycle (2,6) in R2.
Cycle (2,6) in R2 reduces [2,6] from DS [2,6,8] in R2C6 giving us the CRITICAL BREAKTHROUGH of R2C6 8.
Its effect will be shown in next stage. We'll not disturb the status here.
Observe the stringent condition that in each of the two columns C3 and C7, digit 4 could be placed only in the two cells shared also among their two rpws. The pairs of digit 4 in each column are in single digit lock condition.
If the single digit locks are considered separately, the locks couldn't affect incidence of digit 4 in any of the cells in their rows, but as the two rows of the two pairs of cells also are common, lock on digit 4 becomes active on the two rows also.
Among these four numbers of digit 4s only two diagonally opposite digit 4s can occur in actual case thus locking both pairs of columns and rows for digit 4. Check yourself.
Interestingly, though the 4 cell lock of X wing started with single digit locks on columns, breakthrough happened by reduction on commonly shared cells in ROWS.
The status results are shown below.
Stage 5 of Sudoku hard puzzle solution: Expert level Sudoku hard level 5 puzzle game 3: Solution by X wing
Though rest of the large number of cells not yet hit could be filled with valid digit in a series of valid cells, not all hits are by routine reduction. Care needs to be taken even at this late stage.
Because of Cycle (2,6) in C2,
R2C6 8 by reduction of [2,6] -- R8C6 1 reduction -- R8C8 4 reduction -- R7C5 9 -- R8C5 8 reduction -- R6C8 6 reduction of 4.
R4C7 5 reduction of 6 -- R6C9 4 reduction of [5,6].
R8C9 9 reduction [1,4] in R8 -- R7C9 1 -- R5C9 2 reduction -- R5C7 1 reduction.
With 2 in R5C9, R9C9 6 reduction -- R9C7 2.
With 6 in R6C8, R6C2 5 -- R7C2 3 -- R7C3 5.
DSA analysis of Possible digit subset in C8,
DS in C8 is [1,2] -- with 1 in R2, R2C8 2 by reduction -- R1C8 1 -- R3C5 1 scan R1, R2 -- R2C4 6 reduction of 2 in the breakthrough Cycle -- R7C4 2 reduction -- R7C6 6 -- R1C6 2 as only cell left for 2 in C6.
R3C2 2 scan R1, R2, C1, C3.
With 5 in R4C7, R4C1 6 reduction -- R3C1 5 -- R1C9 5 scan for 5 in R3C7 -- R3C9 3 reduction.
R2C5 5 scan R1, R3 -- R1C5 3.
Last few valid cells naturally are by row column scan and reduction,
R2C3 3 scan R1, R3, C2 -- R3C3 4 only cell left for 4 in C3 -- R2C7 4 as the only diagonally opposite digit 4 left in X wing formation -- R3C7 6 sas the only cell left in R3 for 6 -- R1C7 9 as the only cell left in C7 for 9 -- R1C1 8 reduction -- R9C1 9 reduction -- R9C2 8 -- R1C2 6 -- R2C2 9.
The final solved puzzle is shown below.
Check for the validity of the solution if you need.
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. This is what we call forced creation of Cycles.
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 enumeration 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 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.
In solving this Sudoku hard, we have mentioned the use of an additional technique of Parallel scan for a single digit on the cells of a row or column. But it has a minor and unimportant role in discovering the concerned valid cell. We'll not elaborate it here.
A rarely encountered powerful pattern is 4 cell single digit lockdown in a rectangular formation that may be encountered in very hard or expert level Sudoku hard puzzles. 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.
Form of a Cycle:
In a Cycle, the digits involved are locked within the few cells forming the cycle. The locked digits 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). This generally simplifies the situation and occasionally provides a breakthrough by reducing the number of possible digits in the affected cells.
A number of Cycles are shown below from a Sudoku hard solution stage:
Cycle (1,2,6) in column C1 is over all three 9 cell squares on the left. It affects only the column C1.
Cycle (3,8,9) in top right 9 cell square is also in row R2, so it should affect both the 9 cell square and R2.
But Cycle (3,6,7) in top right 9 cell square is formed only in the 9 cell squares, it affects only the cells in the 9 cell square.
Can you see another Cycle in row R1 apart from Cycle (1,6)? The second Cycle (3,6,7) is formed by the cells R1C2, R1C3 and the far away cell R1C9. This Cycle affects only the row R1.
Can you say which are the affected areas for Cycle (1,6) in R1?
Two cells of this Cycle belong to row R1 as well as to the top middle 9 cell square. So the Cycle affects two areas, the row and the 9 cell square. This will be true for any two digit Cycle.
Use of a cycle:
In the example of cycle described 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. You get a single valid digit 1 for R8C2.
This is how a new valid cell is obtained using a Cycle that was not visible otherwise.
In any hard Sudoku game solution, creating, analyzing and using the pattern of Cycles play a very important role.
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.
Identifying a valid digit in a cell by DSA is like a bread and butter technique. It is possibly the most heavily used technique after the simplest row-column scan.
Though DSA may not be considered as an advanced technique it often provides a much required breakthrough. So always look for a valid cell by DSA.
An example of a breakthrough at the late stage of Sudoku hard puzzle solution by DSA is shown below.
We'll do DSA on cell R7C5. The possible digit subset or DS in column C5 and hence in cell R7C5 is [5,7,9], but the two digits [5,9] both are present in row R7.
So eliminating these two from the three digit DS for R7C5, we get the single valid digit 7 for R7C5 --- R7C5 7.
This is a breakthrough even at this late stage.
We have not yet discussed the enumeration of every empty cell with their valid digit subsets or DSs.
Let us see this in a little detail. We'll enumerate the possible digit subset or DS for empty cell R8C1 in the following Sudoku game.
Target cell R8C1 is colored green. Unique set of digits in the three zones—bottom left 9 cell square, row R8 and column C1 colored yellow—will determine the DS for empty cell R8C1.
To enumerate the DS for cell R8C1, look at the row R8 with six digits missing in it—1, 2, 4, 5, 6 and 7.
Now cross-scan column C1 to identify any of these six appearing in column C1.
As 5 and 7 are the two digits out of six that are missing in the intersecting row R8, cancel these two from the six digit subset for R8C1 to reduce it to [1,2,4,6]. Considering row R8 and column C1, possible digits that can occupy R8C1 till now are the DS [1,2,4,6].
But R8C1 also belongs to a 9 cell square and filled digits in it will affect the DS for the cell.
So 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 cell 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 small DSs 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.
Occasionally, after evaluating valid DSs for a number of empty cells, you may find 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 (or a column) inside a 9 cell square, the digit cannot appear in any other cell in the row (or the column) outside the square.
This eliminates all occurrences of the locked digit from the DSs in the row (or the column) outside the 9 cell square. Usually it creates a much needed breakthrough. It is a very powerful pattern.
Single digit lockdown - Conditions for single digit lockdown - how to identify it
Two conditions for single digit lockdown pattern,
- the digit can be placed in only two or three cells of a column or a row, AND,
- the locking cells must also be in SAME 9 cell square.
The third desired condition is,
- The lockdown to be effective, the locked digit should not be present as a single cell candidate in both the adjacent two 9 cell squares through which the locked column or row passes.
The following shows an example of single digit lockdown of 5 in cells R7C1 and R9C1.
How a single digit lockdown is formed
Look at columns C1, C2 and C3 in the bottom left 9 cell square R7R8R9-C1C2C3. Out of 3 empty cells, the cell R7C3 is debarred for placing digit 5 as column C3 has a 5 and it lights up the cell for digit 5.
5 can appear only in two cells in column C1, R7C1 and R9C1 and in no other cell in the 9 cell square or the column C1.
It is locked inside these two cells in C1 and 9 cell parent square.
How a Sudoku single digit lockdown is used - What it does
The locked digit 5 eliminates itself from the DSs of the other two empty cells R5C1 and R6C1 and a new Cycle (2,3) is created in C1.
Focus again on the bottom left 9 cell square. With Cycle (2,3) in C1, another Cycle (5,9) is formed in the two cells of the 9 cell square. As a result, digit 1 becomes the only digit left and cell R7C3 only cell left for it in the 9 cell square.
Still more happens. With 1 in C3 now, digit 9 now must occupy the cell R6C3.
These two single digit candidates obtained by the single digit lockdown of 5 affects other cells and breaks the bottleneck.
As a strategy, always form a single digit lock as soon as it is discovered.
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.
This technique sounds simple, but being aware of its existence and identifying it would always result in an important breakthrough. This digit pattern usually occurs in very hard Sudoku.
We will explain this advanced Sudoku hard technique on the following situation in a Sudoku hard game,
Notice the two digits [1,6] appearing in both row R4 and C5. Together these two result in DIRECT FORMATION OF CYCLE (1,6) in central middle 9 cell square.
This is a double digit scan simultaneously on a row and a column.
Now observe a second set of double digits [3,9] in C5 which DIRECTLY FORMS TWO CYCLES (4,7,8) AND (3,9) IN CENTRAL MIDDLE 9 CELL SQUARE.
This is a double digit scan on a single zone of C5.
Finally, with 3 in C4, R4C4 9 and R4C6 3.
Together these two double digit scans have produced two valid cells and two Cycles. It is a major breakthrough early in the Sudoku hard game.
This special and not often occurring digit pattern involves A SINGLE DIGIT SHARING SAME TWO ROWS AND TWO COLUMNS.
This can be imagined as an advanced form of single digit lock over one pair of rows and one pair of columns in the formation of a rectangle. That's why we call it a single digit rectangle.
This digit formation is commonly known as X wing.
The following figure shows an X wing formation on digit 4 at an advanced stage of solution.
At this advanced stage, possible digit subsets for all empty cells are evaluated after the valid cells and easily formed Cycles are created. By this approach, time waste is minimal.
Interestingly, if you examine closely you will find that all the possible digit subsets are involved in one or more than one Cycle.
The second more important aspect of this puzzle status is,
You cannot find any more valid cell breakthrough by the more frequently used techniques of row column scan (hidden singles), DSA (naked singles), Cycles (naked and hidden groups), single digit lock (direct interaction), double digit scan or parallel scan.
In this situation especially, you need to identify the more complex breakthrough digit patterns of X wing, XY wing or Swordfish. Specialty of all these next advanced level digit patterns is,
Usually all of X wing, XY wing or Swordfish digit patterns usually involve 3, 4 or more cells sharing one common digit.
Observe that we have indeed such a digit pattern formation of 9 in four cells R1C1, R7C1, R1C4 and R7C4.
This single digit of 9 is shared between two columns C1, C4 and two rows R1, R7.
What's so special about this formation?
The special effect of the X wing formation is simply,
The commonly shared digit cannot appear in any cell of these rows and columns except these four cells.
As a result all instances of commonly shared digit in these other cells are eliminated from the possible digit subsets, and this REDUCTION INVARIABLY PRODUCES A CRITICAL BREAKTHROUGH.
In our example game, the X wing cause reduction of 9 in possible digit subset [3,9] in cell R1C2 and causes the most critical breakthrough of the puzzle in the form of,
In fact, with this valid cell breakthrough rest of the cells are all reduced to valid cells just by reduction.
Notice that we have joined the diagonally opposite pairs of digit 9 by two lines forming a large X. One of the two diagonals will actually occur blocking all appearances of 9 in the two columns and rows.
Let's see a second example in the following figure.
Here also a stage is reached when no easy breakthrough could be seen. This is a sure sign of breakthrough by more advanced techniques of X wing, XY wing, Sword fish or likes that involve a single digit shared between more than 2 cells.
Though spotting such a digit pattern spanning at least 4 cells is not easy, if you know what to look for, you can find the pattern quickly. Also you need to practice.
In the above figure digit 4 is shared between two columns C3, C7 and two rows R2, R3. The property that makes this digit configuration a valuable and proper X wing is:
In each column, digit 4 can occupy only these two cells and no other cell. That is, digit 4 is locked in the two cells in both C3 and C7.
No doubt that this is a case of two numbers of SINGLE DIGIT LOCKS OVER TWO COLUMNS SHARING THE ROWS ALSO.
Effect is two-fold. First, digit 4 will have to be removed from all possible digit subsets in the cells of R2, R3, C3 and C7 as well as in cells of their home major squares, top right and top left.
Result is the CRITICAL BREAKTHROUGH OF FIRST FORMING CYCLE (2,6) IN R2C4, R2C8 AND THEN THE VALID CELL R2C6 8 BY REDUCTION OF (2,6) BY THE CYCLE IN R2.
That single valid cell resulted in ALL THE REST OF THE VALID CELLS TILL THE FINAL SOLUTION.
To go through the solution of this Sudoku hard once more, click here.
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.
Most of the initial valid cell finds have been forced by identification of special digit patterns and techniques and only then the rest of the empty cell possible digit subsets are evaluated quickly. This speeds up solution.
Key pattern identification had occurred in many instances by Cycles, DSA, and single digit lockdown.
Critical breakthrough has been provided by X wing formation of a single digit shared between four cells in two columns and two rows. Use of this powerful but complex technique for the ultimate breakthrough places the Sudoku hard game surely at Expert level 5.
More Expert Sudoku puzzles you may like to solve and learn how to solve
The updated list of Solutions to Expert Sudoku puzzle games:
Expert Sudoku solution strategy and techniques are included in many of the solutions.
Enjoy solving and learning to solve Expert Sudoku puzzles.