Learn how to solve expert Sudoku level 5 game 9 quickly
The Sudoku expert level 5 game 9 solved using advanced Sudoku techniques and strategy step by step to make learning how to solve expert Sudoku easy.
- A few words on Expert Sudoku
- Expert Sudoku level 5 game 9
- Solving expert Sudoku level 5 game 9 by advanced Sudoku techniques and strategy
- Expert Sudoku Strategy for quick solution
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A few words on Expert Sudoku
This level of Sudoku we consider as equivalent to level 5 Sudoku, the hardest Sudoku.
An expert Sudoku will definitely be hard, but the most distinctive feature of an expert Sudoku is its numbers of digits filled initially. The number of filled digits in the 81 cells to start with will be in most cases 22 or 23, a very low figure.
But interestingly you may find even a 23 filled cell Sudoku easier to solve than a level 4 Sudoku with 25 filled cells. We won't assure that you will find any expert Sudoku solved here as extremely difficult for you.
But that won't be the norm. Generally an expert Sudoku will be really hard. As hardness of a Sudoku puzzle cannot be measured and is in reality a relative and changing attribute, you cannot point to a Sudoku classifying it with hardness level 5.
That's why we say at this point that distinctive feature of an expert Sudoku will be its number of filled digits as 22 or 23 in most cases. In a rare case, if we find a 24 cells filled Sudoku very very challenging we would include it here as an expert Sudoku.
To sum up, if you are after learning how to solve very challenging Sudoku puzzles, this should be the category for you.
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.
The detailed strategy for quickly solving expert Sudoku is placed after the solution.
Following is the solution of the puzzle explained step by step in details. The techniques used should be understood from the detailed explanatory.
Please spend your time fruitfully on the game trying to solve it before going through the solutions.
Solving Expert Sudoku level 5 game 9 Stage 1: Breakthrough by Double digit scan, Single digit lock, Parallel digit scan and Cycles of twins and triplets
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.
Strategy adopted in brief: start with row column scan coupled with aggressive breakthroughs by advanced Sudoku techniques.
To skip the rest of the brief strategy click here.
As a strategy we always start row column scan from digit 1 and continue till digit 9. In the process we may get valid cells, but otherwise our lookout is for identifying a single digit lock by cross scan.
If we get one, we should get usually a breakthrough in one step or two.
In addition, at this early stage while carrying on row column scan, we remain alert for the possibility of a double digit scan or even a parallel scan also.
When possible valid cell by a row column scan is exhausted, we take up the next two patterns of possible digit analysis DSA technique and forming Cycles of twins or triplets (or of even larger number of cells).
A little later when more cells are filled with unique digits, we raise our alertness for more powerful breakthrough digit patterns such as an X wing.
On the whole, we are always ready to force a breakthrough using even an advanced Sudoku technique at the early stages if feasible.
On the other hand, we are always ready to look for the simplest method of row column scan even when facing a bottleneck.
Main focus is to either find a valid cell or forming a breakthrough pattern such as Single digit lock, Cycles of twins or triplets or X wing at the earliest.
In the process, avoiding enumeration of time-consuming and unnecessary evaluation of possible digit subsets in empty cells speeds up the solution considerably.
While scanning for 1 and then 2, scan for the two digits together formed Cycle of twins (1,2) in R3C7, R3C8 by [1,2] in R1 and C9. It's a perfect example of breakthrough resource creation by double digit scan. We'll shortly see how valuable the Cycle of twins (1,2) will prove to be.
The first valid cell by row column scan is: R5C9 7 by scan for 7 in R4, C7, C8 -- Scan for 4 in R1 on empty cells of top right major square forms the breakthrough single digit lock on 4 in R2C9, R3C9. This has been possible because of the Cycle twins already existing in top right major square.
By the single digit lock, digit 4 cannot appear in any cell in C9 other than these two cells. Effectively it is as if digit 4 already exists in C9.
This gives us the major breakthrough R7C7 4 by scan for 4 in R8, C8 and C9 lock on 4.
It follows that R9C5 4 by scan for 4 in R7, R8 -- R5C4 4 by scan for 4 in R6, C5, C6 -- R4C1 4 by scan for 4 in R5, R6, C2 -- R3C3 4 by scan for 4 in C1, C2.
We'll leave R2C9 4 for next stage to keep the single digit lock undisturbed for ease of understanding.
Turning our attention to bottom three major squares valid cell, R8C6 7 by scan for 7 in R7 -- a Cycle of triplets (1,8,9) formed in R7C4, R7C5, R7C6 by leftover digits in bottom middle major square.
Time for an opportunistic valid cell in favorable zone, R9C8 2 by scan for 2 in R8, C9.
Another opportunistic breakthrough in R8C8 5 by scan for 5 in C7, C9.
An aggressive important breakthrough in R7C9 3 by parallel digit scan for 3 simultaneously on empty cells of C9 as the cells R2C9, R3C9 are eliminated by 3 in top right major square, R4C9 eliminated by 3 in R4 and R9C9 eliminated for 3 by 3 in R9. This leaves the single cell R7C9 in C9 for digit 3.
Parallel digit scan technique has the potential for creating major breakthroughs. To us, it's a very important advanced Sudoku technique.
With 6 in R8, R9C9 6 by scan for 6 in R8 followed by R8C7 8 by exception as the only digit left in bottom right major square.
A breakthrough naked single R9C3 1 by DSA reduction of [7,8] in C3 from possible digit subset [1,7,8] in R9 -- R9C2 7 by reduction of 8 because of 8 in C2 -- R9C1 8 by exception in R9.
Two pending Cycles formed in bottom left major square by leftover digits -- Cycle of twins (3,9) in R8C1, R8C3 and Cycle of triplets (2,5,6) in R7C1, R7C2, R7C3.
We'll take the pending valid cell, R3C8 1 and R2C9 4 next stage.
Observe how much ground has been covered in the first stage of solving this very hard expert Sudoku.
Result of the steps taken shown below.
Solving Expert Sudoku level 5 game 9 Stage 2: Breakthrough by Single digit lock, Double digit scan, DSA reduction and Cycles of twins, triplets and quadruplets
Highlight color of valid cells at this second stage is light blue.
First pending valid cell for 4 in R2C9 4 by scan for 4 in R1, R3, and next with 2 in R9C8, R3C8 1 by reduction of 2 -- R3C7 2 by reduction of 1.
With 8 in R7C7, R1C7 6 by reduction of 8.
Time to form the breakthrough advanced Sudoku digit pattern of single digit lock on 2 in R4C6, R5C6 by cross-scan for 2 in R6, C5. This lock contributes to the valid cell R2C4 2 by scan for 2 in R1, R3, C5 and by lock on 2 in C6. The lock disallows any other empty cell in C6 to have digit 2.
A special breakthrough next.
When we discover that digits [7,8] in left middle major square reduces [7,8] from possible digit subset [1,3,5,7,8,9] in R6, further check on C7, especially targeting R6C7, corroborated the expectation that it also has [7,8] thus forming two Cycles in R6.
First a Cycle of twins (7,8) in R6C4, R6C5 and next a larger Cycle of quadruplets [1,3,5,9] by leftover digits in the rest four cells in R6. This is equivalent to a double digit scan for [7,8] on empty cells of R6 to form the Cycle of twins.
The DS [7,8] helps to form the Cycle of triplets [7,8,9] in R1C4, R3C4 and R6C4 in C4 resulting in the valid cell R7C4 1 by reduction of [8,9]. This is equivalent to a parallel scan for 1 on the empty cells of C4.
For ease of understanding let's stop at this point and show you the status below.
Solving Expert Sudoku level 5 game 9 Stage 3: Breakthrough by Parallel digit scan, Cycle of twins
Highlight color of valid cells at this third stage, which is not the last stage, is light grayish yellow.
The first important breakthrough R3C1 7 by scan for 7 in R2, C2, C3.
It reduces the possible digit subset [7,8,9] to [8,9] in R3C4 to form a breakthrough Cycle of twins (8,9) in R3C4, R3C9 reducing [8,9] in DS [5,8,9] in R3C6 to R3C6 5.
An important breakthrough at this late stage in R5C5 5 by parallel digit scan for 5 in empty cells of C5: R1C5, R2C5, R3C5 eliminated for 5 by 5 in top middle major square, R4C5 eliminated by 5 in R4, R6C5 eliminated by Cycle of quadruplets (1,3,5,9) in R6 and R7C5 eliminated by Cycle of triplets in R7 leaving the single cell R5C5 for 5.
By the leftover digits [1,2,9] in central middle major square, DS in two cells R4C6, R5C6 are [1,2,9] and these two join with [1,9] in R2C6 to for a Cycle of triplets (1,2,9) in C6 -- breakthrough R7C6 8 by reduction of 9 -- R7C5 9 by reduction of 8 -- R4C51 by reduction of 9.
R2C6 1 by scan for 1 in C5 on empty cells of top middle major square.
Naked single R2C5 6 by DSA reduction of [3,7,8] from possible digit subset [3,6,7,8] in C5 -- R3C5 3 by reduction of 6.
R1C2 3 by scan for 3 in R2, R3 -- R3C2 6 by reduction of 3.
By DSA reduction breakthrough Cycle of twins (2,9) formed in R4C2 and R4C6, but we'll see its great effects next stage as it breaks down the puzzle completely.
The puzzle game at this stage is shown below.
Solving Expert Sudoku level 5 game 9 Final Stage 4: First breakthrough by already formed Cycle of twins and rest are easy valid cells
Highlight color of valid cells at this fourth and last stage is light cyan.
By the Cycle (2,9) in R4 debarring both digits in any other cell of R4, breakthrough valid cell R4C9 8 by reduction of 9 -- R3C9 9 by reduction of 8 -- R3C4 8 by reduction of 9 -- R6C4 7 by reduction of 8 -- R6C5 8 by reduction of 7 -- R1C5 7 by reduction of 8 -- R1C4 9 by reduction of [7,8] -- R1C8 8 by reduction of 9.
Cycle (2,9) in R4 causes a further reduction of 9 in possible digit subset in R4C8 to result in valid cell R4C8 6 -- R5C8 9 by reduction of 6 -- R5C6 2 by reduction of 9 -- R4C69 by reduction of 2 -- R4C2 2 by reduction of 9.
With 2 in R4C2, R7C2 5 by reduction of 2 -- R7C3 6 by reduction of 5 -- R7C1 2 by reduction of [5,6].
With 5 in R7C2, R2C2 9 by reduction of 5, R2C1 5 by reduction of 9.
R6C2 1 by exception in C2.
R5C1 6by scan for 6 in R6, C3.
R6C3 5 by scan for 5 in R5, C1.
With 1 in R6C2, R6C7 3 by reduction of 1 -- R5C7 1, R6C1 9 by reduction of 3 -- R8C1 3 by reduction of 9 -- R8C3 9 by reduction of 3 -- R5C3 3 by reduction of 9.
Final solution below.
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.
Valid cells by row column scan occasionally are also called as hidden singles, but to us process is row column scan and outcome is valid cells with unique digits.
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.
For a specific promising cell, considering digits existing in three parent zones, the subset of possible digits is enumerated. If the subset consists of a single digit, it becomes a naked single and we get a breakthrough by possible digit analysis or DSA technique.
And wherever possible, Cycles of twins or triplets are formed that in any situation are a treasure to have and Cycles play a key role in quick solution.
In a Cycle (1,2) of twins say in R8C5 and R9C5, the two digits may appear in any one of only these two cells in the zone column C5 and the bottom middle major square. By this twin digit Cycle, it becomes a certainty that as a group the two digits, locked in these two cells, cannot appear in any other cell in the parent zones of the Cycle and thus possible digits in the corresponding empty cells are reduced in C5 and bottom middle major square.
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.
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 lock on a single digit in a row or column inside a 9 cell major 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 lock technique indicates in a way the hardness of the puzzle.
In solving this Expert Sudoku, we have mentioned the use of a technique of Parallel scan for a single digit on the cells of a row or column. Occasionally this may provide a very critical difficult to get breakthrough.
Similarly an exceptional pattern and technique for breakthrough is double digit scan where instead of a single digit, a scan for two digits simultaneously is carried out on empty cells of a major square. This usually has a valuable outcome.
A powerful Sudoku digit pattern is 4 cell single digit lock that may be encountered in very hard or expert level Sudoku hard puzzles. Naturally, it is a superset of the more common single digit lock in 2 cells and so is much more effective.
This powerful digit pattern is the single digit rectangle or X wing. In solving this Expert level 5 Sudoku hard puzzle, X wing provided the most important breakthrough.
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 looking for key patterns all the time. Parallel digit scan, Double digit scan, Single Digit lock, Cycles of twins or triplets, Valid cell by DSA are some of the key patterns. Occasionally you may need to use X wing for a difficult breakthrough.
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.
Finding valid cells has been really tough at the start. But taking up the challenge, use of advanced techniques for breakthroughs at the earliest has been effective.
The aggressive approach in the beginning in fact makes the later stages of solution comparatively easier and overall solution faster.
This one is truly a very hard expert level Sudoku puzzle.
Other Sudoku puzzles you may like to go through at leisure
Sudoku hard Expert level 5 puzzles
How to Solve Expert Sudoku Level 5 Game 9
Hard Sudoku level 4 puzzles
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.