Sudoku hard level 4 game 3 solution by Sudoku techniques
How to Solve Sudoku hard level 4 game 3 explains Sudoku techniques DSA, Cycle, Single digit lock, Possible digit enumeration and more.
The game is extra hard with multiple bottlenecks right from the start.
Digit subset analysis, Cycles and digit lockdown techniques are the primary techniques 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 used explained.
- A few words on Sudoku hard puzzles
- Sudoku very hard level 4 game 3
- Sudoku techniques -basic and advanced
- Sudoku basic technique - Row column scan to find single digit candidate for a cell
- 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
- Sudoku advanced technique:Single digit lock - how to identify and use for a breakthrough
- 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.
- Sudoku strategies for quick solution
- Solution to Sudoku very hard level 4 game 3 - Sudoku techniques and steps explained
To skip the next section on a few words on hard Sudoku puzzles click here.
A few words on Sudoku hard puzzles
First to say 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 digit patterns that helped to create 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.
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.
Though the number of digit-filled cells in the puzzle is a rough indicator on how hard the puzzle is, it is fully reliable. Usually, you may find Sudoku puzzles with digits below 26 to be hard.
Below is the Sudoku hard puzzle to be solved. The Rs are the row labels, Cs are the column labels and this we define as the stage 1 marked on top left corner. Give it a try before going on to the solution.
You may go through the next five sections for learning Sudoku techniques and strategies for solving Sudoku hard puzzles in brief.
To skip these sections and move on to the solution directly click here.
The first technique of row column scan is the most basic - like staple diet. It is included for completeness.
Basic Sudoku rule disallows any digit repeated in a row, a column or a 9 cell square such as R1R2R3-C4C5C6. The puzzle below shows how row column scan for finding a single digit candidate for a valid cell works.
With digit 1 in three zones, row R1, row R2 and column C6, Row column scan on digit 1 in 9 cell major square R1R2R3-C4C5C6 leaves only the cell R3C4 for digit 1.
By the digit 1 scan, all 5 other empty cells of the 9 cell square are COVERED by the influence of digit 1 in the three zones with three cells already occupied.
So, the SINGLE CELL R3C4 is left in the major square where only digit 1 can be placed.
Cells not allowed for digit 1 are colored blue. Only one cell in the zone is left for 1.
This is identification of a single digit candidate for a valid cell. 1 is placed in the identified cell and the CELL OCCUPANCY of the Sudoku hard puzzle increases by 1.
This is the most basic and the easiest to use Sudoku technique. No opportunity of a valid cell find by row column scan should be lost.
At a point in solving a puzzle, you may find no opportunity for a valid cell find by row column scan. But when a new valid cell is found by other means, the Sudoku digit position should be checked immediately because a new opportunity of row column scan might have arisen.
Scan for a digit may not always involve row and column, it may be scan of even a single row or a single column for a digit.
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.
Important to note
- Unless digits possible (DS) for a number of cells are already enumerated, you won't be able to find a Cycle. That's obvious. But don't enumerate DSs for all the empty cells at one go. Enumerate only for promising cells in 9 cell squares that have more clustering of digits considering three parent zones together. And don't enumerate DS length of 4 digits. Limit number of possible digits to be enumerated to maximum 3.
- Having said that, it IS to be mentioned that occasionally three digit DS enumeration is not enough for a breakthrough. Then you must go for 4 digit long DS enumeration and look for Cycles. It is rare to get a 4 digit 4 cell Cycle. But again we have one in this game itself.
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 4 empty 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.
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.
Let us solve our hard Sudoku puzzle now.
We'll show the puzzle game 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: Solution to Sudoku hard level 4 game 3 by Sudoku techniques
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: Solution to Sudoku hard level 4 game 3 by Sudoku techniques
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 gives 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.
More Sudoku hard puzzles you may like to solve and learn how to solve
The updated list of Solutions to level 3, level 4 and NYTimes Sudoku hard puzzle games:
Enjoy solving Sudoku hard.
By the way, Sudoku hard solution techniques are included with many of the solutions.
Enjoy also learning how to solve Sudoku hard in easy steps.