How to Solve Sudoku Hard level 4 game 9 quick and easy
Breakthroughs in Sudoku hard level 4 game 9 by identifying digit patterns Parallel scan, Single digit lock, Cycle, Naked singles and each step explained.
- A few words on hardness of Sudoku game puzzles
- Sudoku hard level 4 game 9
- Solving Sudoku hard level 4 game 9 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.
- How a single digit candidate valid cell is identified by Digit Subset Analysis (DSA) in solving a Sudoku hard puzzle.
- How digits possible for all empty cells (DSs) enumerated while solving a Sudoku hard puzzle.
- Single digit lockdown and its use in solving a Sudoku hard puzzle.
- Sudoku technique of double digit scan: For simplifying a hard Sudoku puzzle.
- Sudoku technique of parallel scan for a single digit on a row or a column: For a breakthrough in hard Sudoku puzzles.
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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.
In the same way, for the other techniques also you may jump to details of a technique and after refreshing your concept return to the point from where you jumped.
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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 thing 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.
But differentiating between hardness levels 3 and 4 is not always clear.
To be honest, we cannot say all 4th level Sudoku puzzles are harder than all the 3rd level puzzles we have solved. Both level 3 and level 4 Sudoku puzzles can be categorized simply as Hard Sudoku or Sudoku hard whichever way you like to say.
From our experience of solving hard Sudoku puzzles we have found the following general attributes of a Sudoku hard puzzle,
- Significantly less number of cells filled in a hard Sudoku puzzle compared to easier Sudoku. Generally, a hard Sudoku game comes with maximum 26 number of cells filled. And this number can be as low as 22 or 23.
- But along with cells filled, the specific arrangement of the cells filled also plays an important part in determining hardness of a Sudoku game. For example, a 26 cells filled hard Sudoku may be harder to solve than a 24 cells filled hard Sudoku.
- At the beginning of the game, how many easy row column scan hits you can get is an indicator of the hardness of the puzzle. If you can't get any easy opening valid cell by row column scan or DSA, you would have to look for possibility of applying advanced hard Sudoku techniques of single digit lock, double digit scan or parallel scan. In case you fail to identify such a positive result-bearing digit pattern, the only alternative remains for you is to take up time-consuming evaluation of possible digits or DSs of empty cells.
- Extra hard Sudoku puzzles have a tendency to pose hurdle after hurdle for you to cross by making multiple breakthroughs.
- Lastly a sure sign that the Sudoku puzzle is hard is in its tenacity and resistance to break open completely till the very last stage.
But, do not take our word to assess the hardness of a Sudoku hard. Assess it for yourself by solving more Sudoku hard puzzles and trying to solve them in less and lesser time.
Overall, you'll find that solving a Sudoku hard puzzle elegantly will give you great pleasure.
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.
A note on the Sudoku game playing mode in solving the Sudoku hard game
Following our solution steps, the Sudoku hard game may be solved either with paper and pencil or better on a spreadsheet.
Main point is: NO HELP IS TAKEN FROM A SOFTWARE FOR SOLVING THE GAME.
We prefer this approach because the pleasure in discovery of a breakthrough can be fully enjoyed this way.
The solution is broken up into stages for concentrating only on the present state of the game and easy backtracking in case of an error discovered at a later stage.
A note on cells colored with different colors conveying specific information
At any stage, the first cell found where a valid digit is entered, is colored TURQUOISE BLUE. All other Stage 2 valid cells are colored light banana leaf green. Third stage valid cells are colored light dull blue, fourth stage valid cells are colored light pink and the fifth stage sea blue.
In case of special breakthrough of a valid cell, it is also occasionally given TURQUOISE BLUE color.
Cells involved in a Cycle are colored yellow and also in orange if two Cycles are adjacent.
This helps to identify the cells identified as valid in the present stage as well as identifying valid cell hits during earlier stages.
We'll now solve the standard hard Sudoku game 9 and after the solution we'll explain the Strategy and techniques of solving a Sudoku hard game in details.
In this puzzle game, the breakthroughs have been forced at the earliest avoiding possible digit evaluation of all empty cells. The solution is reasonably quick and each of the breakthroughs and valid cell hits explained step by step.
A system of internal links to these techniques sections and the solution section helps you to navigate between the solution process and the detailed techniques used by clicking on the links.
Please spend your time fruitfully on the game trying to solve it before going through the solutions.
Let us solve our hard Sudoku puzzle now.
The puzzle is shown again so that you can follow easily.
Let's repeat, 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 valid cell hit is by row column scan, but this is only one,
R6C8 3 scan R4,C9.
But this single valid cell of 3 helps to form two Cycles consecutively ending in a a valid cell hit,
Cycle (1,3) in R5 -- Cycle (2,5,9) in R1C5, R4C5, R6C5 as 1 is eliminated by Cycle (1,3) -- R9C5 1.
It's time now to force in valid cells hits by possible digit analysis or DSA reduction of naked singles in promising cells,
DSA reduction [1,3,5] from DS [1,3,5,6] in bottom left major square and in R9C1 -- R9C1 6.
Going is hard and comparatively more powerful digit pattern of single digit lock needed for a breakthrough,
Single digit lock on 1 in R4C7, R4C8 by scan on 1 in C9 - Cycle (2,7) in R3C1, R4C1 by DSA reduction of [1,3,8] from DS [1,2,3,7,8] in C1 -- R6C1 1 reduction -- R7C1 3 reduction - R1C1 8 -- R2C8 8 scan R1, R3, C7.
And then last few valid cell hits at this stage,
R7C2 1 scan R8,C3 -- R8C2 5 reduction.
Cycle (2,4,7) in top left major square by DSA reduction -- R3C2 6 reduction -- R1C2 3.
Observe that I have altogether avoided evaluating unproductive DSs for empty cells and focused on discovering breakthrough digit patterns and advanced Sudoku hard techniques. This strategic approach speeds up the solution considerably.
More next stage. Results of actions taken shown below.
Solution to Sudoku hard level 4 game 9 Stage 3
Finding a valid cell is not yet easy and we spotted next an opportunity to use parallel scan for the nrxt breakthrough. A parallel scan for a specific digit on a row or column is always equivalent to formation of a Cycle.
We prefer a parallel scan if it can be spotted. And now not one but two parallel scan pattern could be spotted,
Parallel scan for 6 on empty cells of R1 -- 6 in top middle major square, and in C7 -- R1C9 6.
Parallel scan of 5 on C7, 5 in top left major square and 5 in R8 -- R4C7 5 -- Cycle (4,8) in R4C9, R6C9 by reduction of 1 in C9 from DS [1,4,8] in right middle major square -- R4C8 1 by reduction of [4,8].
With Cycle and DSA, single digit lock needed to be used for the next few valid cell hits,
Cycle (2,9) in C5 by DSA reduction -- R6C5 5 -- Single digit lock on 2 in R3C8, R3C9 by scan 2 in C7 -- R3C1 7 -- R4C1 2 by reduction.
With 2 in R4C1 and 5 in R4C7, R4C5 9 by reduction -- R1C5 2 by reduction.
R4C2 4 reduction -- R6C2 9 -- R6C3 7 -- R2C3 4 -- R2C2 2.
Results of steps taken till this point shown below.
Solution to Sudoku hard level 4 game 9 Stage 4
The game is not yet fully broken open and the next valid cell hits are by Cycle and DSA reduction,
With 4 in R4C2, R4C9 8 reduction -- R6C9 4 reduction -- R6C4 2 reduction of 8 -- R6C6 8 8 reduction.
With [6,7] in both R3, R9, Cycle (2,4) in R3C8, R3C9 by reduction of [6,7] from DS [2,4,6,7] in C8 -- R7C8 6 reduction of [2,7] -- R8C8 7.
R8C7 9 by reduction of [2,4,5] from DS [2,4,5,9] in bottom right major square -- R8C3 2 reduction -- R7C3 9 reduction -- R7C4 5 reduction.
No more difficulties left. Rest of the easy finds will be shown in next final stage.
Results of steps taken in this stage shown below.
Solution to Sudoku hard level 4 Game 9 Final Solution Stage 5
Among valid cell hits in this final stage by reduction, a Cycle and a Parallel scan also used for speeding up solution,
With 5 in R7C4, R7C9 2 reduction -- R9C9 5 -- R3C9 -- R3C8 2 -- R9C8 4.
R9C4 9 reduction of 2 -- R9C6 2.
Next a Cycle and a parallel scan are used,
Cycle (4,7) in R1C4, R1C7 -- R1C6 9 reduction.
Parallel scan on three empty cells of R2 for 5 -- with 5 in C4 and C7, R2C6 5.
All the rest are by reduction,
With 4 in C6, R3C6 1 reduction -- R3C7 4 reduction -- R1C7 7 reduction -- R2C7 1 -- R1C4 4 -- R2C4 -- R2C4 7 -- R4C4 6 -- R4C6 7 -- R8C4 3 -- R5C4 1 -- R5C6 3 -- R8C6 6.
The final solution of the hard Sudoku 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.
The new technique of parallel scan is preferred as it results in faster identification of a valid cell than equivalent formation of Cycle.
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.
Let's see two examples of forming Cycles of captive digits and their effects.
Examples of Cycles of Captive digits
The following shows the formation of the first Cycle.
With no scope of any easy valid cell hit by row column scan, we take up possible digit subset analysis or DSA in promising zones.
We'd be happy if we discover a cell for a unique digit or even a Cycle of captive digits in this process.
And see what we have got by DSA,
A valuable Cycle of (1,5,9) in cells R1C1, R2C1, R3C1.
DSA for creating this Cycle has not been easy:
Possible digit subset DS for six empty cells [1,5,6,7,8,9] in top left major square is reduced by [6,7,8] to [1,5,9] in R1C1, R2C1 and R3C1.
As a result, the three digits [1,5,9] become captive in these three cells Cycling between the three cells only.
Effect of the yellow colored Cycle is,
Reduction of the three digits in all other empty cell DSs in the parent major square and parent column C1 (creating two more Cycles).
In a way, a Cycle is a multi-cell multi-digit lock and always is a valuable asset to have.
At this point, the Cycle (1,5,9) doesn't produce a direct valid cell hit, but sure enough we'll get the breakthrough using the Cycle soon.
Follow the figure below.
Cross-Scan for 5 in R3 and C4 creates a single digit lock of 5 in cells R2C5, R2C5 in R2 and in top middle major square. Result is,
This single digit lock reduces 5 from DS [1,5,9] in R2C1 and creates in turn a smaller Cycle of (1,9) in R2C1, R3C1.
By standard reduction property of a Cycle,
Digit 9 is reduced from DS [5,9] of R1C1 and gives us the breakthrough R1C1 5.
This reduction by a shorter Cycle is sometimes called as hidden groups.
We consider these names of naked groups, naked subsets, hidden groups or hidden subsets too many and artificial as well.
Instead, we would always use the concept and name of Cycle for this valuable Sudoku digit pattern.
In solving any hard Sudoku game, 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.
The Sudoku technique of DSA, as we call it, is kind of bread and butter technique for solving medium level 2 Sudoku puzzles to Expert level 5 Sudoku puzzles.
We consider this as an operation that is not only used for unique digit valid cell identification, but also to create Cycles and other assets for solving harder Sudoku puzzles.
The hard Sudoku game in the following figure may have its first breakthrough valid cell by by DSA. Can you find how?
The Sudoku game is obviously hard with only 23 cells filled with digits.
Any possibility of easy valid cell by row column scan? There is none.
Without getting desperate, we start next with the Sudoku technique of possible digit analysis DSA.
We choose directly the top row R1 with 5 digits filled as the most promising zone, and land on cell R1C6 as seemingly the most promising cell lighted up by maximum number of unique digits.
Our hunch proved to be right. The possible digit subset in four empty cells of R1 and so in R1C6 is [1,6,7,9] and interconnecting column C6 reduces three of the digits [1,6,9] to give us the first breakthrough as R1C6 7.
We have delayed showing the result to give you a chance to make the breakthrough yourself. Now the result of the first breakthrough is shown below.
R1C6 7 by DSA reduction of [1,6,9] from DS [1,6,7,9] in R1 and R1C6.
It further creates three more valid cells,
R1C7 6 by DSA reduction of [1,9] in R7 -- R1C1 1 DSA reduction of 9 -- R1C2 9.
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.
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, sometimes 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 to be 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 lock - Conditions for single digit lock - 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 by a SINGLE COLUMN SCAN.
How a single digit lock is formed by a column scan and what is its effect in a hard Sudoku game
Digit 1 in column C9 disallows the two empty cells (marked as x) R4C9, R6C9 in right middle major square leaving only two cells R4C7, R4C8 for digit 1 in this major square.
This is how digit 1 is locked in row R4 as well as in the parent major square R4R5R6-C7C8C9.
Its effect is,
Digit 1 cannot appear in any of the empty cells of the row R4. The single digit lock acts as if digit 1 actually is filled in one of the two cells R4C7, R4C8.
Using this single digit lock of 1 in R4, and 1 in C3, apply a row column scan for 1 on left middle major square to get the breakthrough: R6C1 1.
This in turn results in a second valid cell, R7C2 1 by row column scan single digit lock in R4, C1 and C3.
We'll see now a second example of single digit lock.
How a single digit lock is formed by row column cross-scan and what is its effect in a hard Sudoku game
The following figure shows a second example of single digit lock, but this time formed by cross-scan of a row and a column.
In the above hard Sudoku game, digit 3 in row R1 and column C2 debars the three cells R1C3, R2C2, R2C3 for 3. This is same as scanning for a digit with the objective of getting a single valid cell for the digit scanned.
But in this case, two cells R2C1 and R2C3 are left in row R2 that can be occupied by digit 3. Though we don't have a valid cell, the result of this single digit 3 locked in the two cells in R2 and in top left major square is no less effective.
The single digit lock acts like the actual presence of the digit in R2.
So scan now for digit 3 on top right major square.
3 in R1, 3 in R2 lock and 3 in C9 gives you the valid cell R3C7 for 3.
This new hit in turn participates in a second scan for 3 in R7, C7, C9 giving R8C8 3.
Almost always a single digit lock provides an important breakthrough.
As a strategy, right from the word go we look for a single digit lock while carrying out row column scan. Even if we cannot use a single digit lock immediately, it is recorded for future use.
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.
A parallel scan is carried out for a specific digit on the empty cells of a promising row (or column). Because of presence of the specific digit in the interconnecting columns (or rows) for all empty cells of the scanned row (or column) except one, the valid cell for scanned digit can be identified as this cell.
The digit pattern and the technique to identify a breakthrough valid cell by parallel scan is shown in the figure below,
The parallel scan for digit 6 is done in this case on the empty cells of R1. Out of 4empty cells R1C4, R1C6, R1C7 and R1C9, digit 6 is disallowed in the first two by 6 in top middle major square and disallowed in R1C7 by 6 in C7.
This leaves only the single cell R1C9 where digit 6 can be placed. That becomes the valid cell for digit 6.
Observe that as a result a Cycle (4,7,9) is formed in the rest of the three empty cells in R1.
If you could have identified the Cycle before parallel scan, you could automatically have got the valid cell without parallel scan. That's the interesting property of parallel scan, if you can spot one, you would be sure to find an equivalent Cycle as a result.
To us, valid call by parallel scan is easier and faster.
To go through the solution of this Sudoku hard once more, click here.
The solution presented may seem to be quick and easy, but it is so only because of forcing the breakthroughs at the earliest using DSA, parallel scan and single digit lock to their fullest potential.
This way, time consuming long possible digit evaluation for all empty cells avoided as much as possible and thus solution speeds up.
Overall, this is a good hard Sudoku puzzle game that has been a pleasure to solve quick.
Other Sudoku puzzles you may like to go through at leisure
Sudoku hard Expert level 5 puzzles
Hard Sudoku level 4 puzzles
Sudoku hard level 4 game 9 quick solution explained step by step
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.