Step by step solution to hard Sudoku level 4 game 5 explained
Solution to hard Sudoku level 4 game 5 explains finding a valid digit for each empty cell and the hard Sudoku technique used for the valid cell.
- A few words on hardness of Sudoku game puzzles
- Sudoku hard level 4 game 5
- Solving Sudoku hard level 4 game 5 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 hard technique of double digit scan.
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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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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.
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 the initial 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 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 green. Third stage valid cells are colored blue, fourth stage valid cells are colored pink and the fifth stage blue green.
This helps to identify the cells identified as valid in the present stage.
We'll now solve the Sudoku hard game 6 and after the solution we'll explain in more details, the Strategy and techniques of solving a Sudoku hard game.
A system of internal links to these techniques sections and the solution section help 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.
We'll show the puzzle board again for convenience of understanding.
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 hit: R4C4 1 by Scan 1 on R6,C5,C6 -- DSA on R9C9: DS in R9 and R9C9 [2,4,7,8]: but, [4,7] in C9 and 8 in home 9 cell larger square: R9C9 2.
DS [1,3,8,9] in C9 and R7C9: [3,8,9] in R7: R7C9 1 by DSA -- Reduced DS [3,8,9] in R8C9: [3,8] in bottom right 9 cell square: R8C9 9 by DSA -- R6C9 9 by Scan for 9 on R4,R5,C9 -- DSA on R9C3: DS [4,7,8] but 4 in C3 and 7 in home 9 cell square: R9C3 8 -- R9C6 4 -- R9C8 7.
Cycle (4,5) in R8C7, R8C8.
R6C4 4 by Scan on C5,C6 -- R7C4 2 by DSA [2,3,7,8] in C4 -- R7C2 4 by scan for 4 by Cycle (4,5} in R7 and 4 in C3 -- R4C1 4 by Scan C2,C3 -- By DS [1,2,5,9] in C1 Cycle [1,5,9] in R1C1, R1C2, R1C3: R8C1 2 by exclusion.
You may verify the actions taken till now from the following second stage status.
Notice that we have avoided the time consuming evaluation of possible digits or DSs for empty cells as far as possible. This strategic approach speeds up the process.
Solution to Hard Sudoku level 4 game 5 Stage 3
With 5 in bottom middle 9 cell square R7C5 6 by cancellation -- R7C3 5 by exclusion -- Three Cycles formed in R8 but these are of no great use.
Attention now turned to top left 9 cell square with only 3 empty cell left to be evaluated. At the moment this is the most promising zone that we'll evaluate.
It results in Cycle (7,8) in R1C2, R1C4: R1C7 9 by exclusion -- R1C1 5 by cancellation.
R4C6 6 by Scan on R6,C5 -- R5C8 6 by Scan R4,C7, C9.
With no immediate easy breakthroughs visible, 2 to 3 digit subsets or DSs are evaluated in promising empty cells conveniently.
Patient DS analysis of cells that are affected or LIGHTED UP by at least 6 digits revealed no breakthrough till the discovery of Cycle (3,7,8) in top middle 9 cell square.
This restricted the digits possible in cell R3C5 as [2,5,9].
But with [2,5] in R3, we finally get the breakthrough R3C5 9.
R3C5 cell is also colored turquoise blue to show its breakthrough status.
Rest of the valid cells are easy to get.
For reducing clutter and ease of understanding let's close at this point and show you the status below.
Solution to Sudoku hard Sudoku level 4 Game 5 Final solution Stage 4
R3C1 1 by cancellation -- R2C1 9 -- R3C8 3 -- R4C8 5 -- R8C8 4 -- R8C7 5 -- R5C7 2 -- R4C7 8 -- R6C9 3 -- R2C9 8 -- R3C7 7 -- R3C6 8 -- R2C7 4 -- R2C8 1 -- R1C4 7 -- R1C2 8 -- R2C4 3 -- R8C4 8 -- 8C6 3 -- R8C5 7 -- R5C3 1.
Cycle (3,5) in C2 and left middle 9 cell square.
With 2 in C2, R6C2 7 -- R6C3 2 -- R2C2 6 -- R2C3 7 -- R8C2 1 -- R8C3 6 -- R4C3 3 -- R5C2 5 -- R5C5 3 -- R4C5 2 by exclusion -- R2C5 5 -- R2C6 2 -- R6C6 5 -- R6C5 8 by exclusion.
After the crucial breakthrough in R3C5 9 at Stage 3, it has been all clear with easy pickings.
Of interest is the fact that in the whole hard Sudoku game, the only crucial breakthrough has been in R3C5 9. That's why finding this single point of weakness in the whole puzzle has been hard to find.
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.
A rarely encountered powerful 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. We'll not also elaborate on this further.
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 highlighted 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 highlighted 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 Sudou hard game.
To go through the solution of this Sudoku hard once more, clicck 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.
Though there has been no need to use the more advanced Sudoku hard techniques of Single digit lockdown or Double digit scan, the hard Sudoku game posed a fair challenge to find its single point of weakness by DS evaluation of most of the empty cells.
With early multiple breakthroughs, however much advanced techniques are needed for their discovery, the Sudoku game breaks open quickly. On such occasions, the time consuming DS evaluation for empty cells is avoided.
That way, this hard Sudoku took a good amount of time to solve.
Other Sudoku puzzles you may like to go through at leisure
Sudoku hard Expert level 5 puzzles
Hard Sudoku level 4 puzzles
Hard fourth level Sudoku puzzle 5, Strategies Techniques and Solution
New York Times Hard Sudoku puzzles
Hard Sudoku level 3 puzzles
You may access all hard Sudoku level 3 solutions at Third level hard Sudoku.
Medium level 2 puzzles
You may read through all medium level 2 solutions at Second level medium Sudoku.
For beginners, Sudoku beginner puzzle solutions are at Beginner level Sudoku.