## Learn how to solve very hard expert Sudoku level 5 game 28 quickly simple way

Very hard expert Sudoku level 5 game 28 solved simple way using Sudoku solving techniques of Single digit lock, Asymmetric X Wing, Cycles and parallel scan.

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**Very Hard Expert Sudoku level 5 game 28****How to solve very hard expert Sudoku level 5 game 28 simple way by Sudoku solving techniques****Expert Sudoku Solving Techniques and How to Use Special Digit Patterns****.**.**Expert Sudoku solving techniques of single digit scan and double digit scan****Expert Sudoku Solving Technique of Possible Digit Subset Analysis (DSA) and how to find a naked single.**.**Special digit pattern of Cycle of twins or triplets and how to use it in solving an Expert Sudoku puzzle****Expert Sudoku solving technique of parallel scan for a single digit on a row or a column.**.**Single digit lock and how to use it in solving an Expert Sudoku puzzle****Expert Sudoku solving technique of Asymmetric X wing diagonal test**

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### Very Hard Expert Sudoku level 5 game 28

We have found the following Expert Sudoku game 28 very hard to solve. Major Sudoku Solving Techniques including double digit scan, parallel scan and asymmetric X wing diagonal test have been used for solving the puzzle. The techniques used are separately explained after the solution, but you must first try your best to solve the puzzle that will enrich your mind.

The Rs are the row labels, Cs are the column labels.

Following is the **solution of the puzzle explained in simple way.**

*Please spend your time fruitfully on the game trying to solve it before going through the solutions.*

### How to Solve Very Hard Expert Sudoku level 5 game 28 Simple Way by Sudoku Solving Techniques Stage 1: Single digit scan, Cycles, Single digit lock, double digit scan

R5C2 8 scan 8 in C1, C3, R6. R2C2 9 scan 9 in C1, C3.

**Double digit scan for [4,9] in R4, C5** on the vacant cells of central major squares create **Cycle (4,9) in C4** in the major square.

R9C6 9 scan 9 in **single digit lock in C4 and 9 in C5, R7.**

**Single digit breakthrough lock on 4 in R7C1, R9C1 by cross-scan for 4 in C2, R8** -- breakthrough R6C3 4 scan 4 in lock in C1, 4 in C2, R4.

Status Shown.

### How to Solve Very Hard Expert Sudoku level 5 game 28 Stage 2: DSA, Cycles, Single digit lock, double digit scan, parallel scan

With 4 in R6, R6C4 9 DS reduction -- R5C4 4.

**Double digit scan for [4,9] in R1, R2** on vacant cells of top right major square creates breakthrough Cycle (4,9) in R3C8, R3C9 in R3 and in the major square.

**Breakthrough R1C7** 8 scan 8 in R2, C8 -- **Cycle (2,5,7)** in top right major square.

**Critical breakthrough R5C7 9 by parallel scan for 9** on vacant cells of C7: 9 in R2, R6, R7, R9 eliminate four C7 cells R2C7, R6C7, R7C7 and R9C7 for 9 leaving the single cell R5C9 for 9. Always look for double digit scan, triple digit scan and parallel scan for quick breakthrough instead of time consuming possible digit formation in all major cells.

R5C9 5 by DSA elimination by [1,3,6] in C9 from Row R5 DS [1,3,5,6].

**Single digit lock on 6** in R4C4, R4C6 in R4 by scan 6 in C5 -- **breakthrough Cycle (2,7)** formed in R4C8, R4C9 by reduction of [1,6] in R4 in the cells from DS [1,2,6,7] in the right middle major square. R6C1 2 scan 2 in lock in R4, 2 in R5, C2 -- R8C3 2 scan 2 in C1, C2.

**Single digit lock on 1** in R5C5, R6C5 in C5 by scan 1 in R4.

Status shown.

### How to Solve Very Hard Expert Sudoku level 5 game 28 Stage 3: DSA, Naked single, parallel scan, Cycles, single digit lock, Asymmetric X wing

**Naked DS [7,8]** (instead of a naked single digit cell) in R8C5 and R3C5 by DSA forms **Cycle (7,8)** in C5 reducing DS in C5 to [1,2,3,5]. **Cycles (3,5)** and **(6,8)** formed in R4.

**Critical breakthrough R7C6 5 by parallel scan for 5** on vacant cells of C6: R1C6, R2C6, R3C6 eliminated by 5 in top middle major square and R4C6 eliminated by Cycle (3,5) in R4 -- R9C5 2.

**Single digit lock on 1 in R7C4, R9C4** reduces DS in R2C4 to [2,7] from C4 DS [1,2,6,7,8]. This cell joins with R2C7 to form **Cycle (2,7)** in R2 -- **Breakthrough R2C8 5 by DS reduction** of [2,7].

Short length DSs of 2 or 3 digits for the vacant cells are formed with ease for most of the vacant cells. Many single digit locks and Cycles are also identified. But no valid cell for a unique digit could be identified by DS reduction.

This is the situation where we will look for the more complex digit patterns, especially, the X wing digit pattern. But we don't have a normal X wing in the game stage. Instead, the less powerful **asymmetric X wing formed by single digit locks on 6 in R5C1, R6C2 and R5C8, R6C7 sharing rows R5, R6 has still the advantage of one of the two DIAGONAL VALUE PAIR VALIDITY.**

In other words, **diagonal test of an asymmetric X wing** establishes validity (or invalidity) of a specific digit placement in TWO CELLS AT ONE GO. Digit 6 at the ends of one of the diagonals will be valid and the other invalid. In most cases quick resolution of the invalid diagonal values clears the way for resolving all unique digit valid cell values easily with the valid diagonal values.

This **diagonal test for 6 is favored here because**,

- all four cells have two digit DSs,
- digit 6 is involved in a large number of other linked two or three digit cell DSs, and also,
- the X wing is
**spread out to its maximum width of three major squares**. As a rule, the more spread out the X wing, the more will be its effectiveness.

For these reasons we should get quick results, positive or negative, from the diagonal test.

We will test first the validity of the diagonal R5C8 6, R6C2 6 next stage. The chosen diagonal cell values for digit 6 are in fact invalid and it can easily be checked mentally in a few steps. As a rule, we first choose the invalid diagonal of an X wing for test. Once the invalidity is established quickly, rest of the larger number of cell results can be obtained easily.

Status shown.

### How to Solve Very Hard Expert Sudoku level 5 game 28 Stage 3A: Asymmetric X wing diagonal test

R5C8 6, R6C2 6 diagonal values will now be tested.

R5C8 6 -- R6C2 6 -- R5C1 3 -- R4C3 5 -- Cycle (1,3,6) in C3 -- R3C1 7 by reduction of 6.

With 6 in R6C2, R8C2 7 -- R8C5 8 -- **R3C5 7 clashing with R3C1 7 in same row R3.**

This proves **X wing diagonal choice of R5C8 6, R6C2 6 invalid. **

**Solution will come from the other valid diagonal values of R5C1 6 and R6C7 6 easily.**

Status shown.

### How to Solve Very Hard Expert Sudoku level 5 game 28 final stage 4: Resolution of unique digit valid cells with X wing diagonal valid values

**With valid diagonal choices of R5C1 6 and R6C7 6**, *all other vacant cells resolved easily into unique digit valid cells of the solution.*

R6C2 5, R4C3 3, R3C1 7, R3C5 8, R8C5 7, R4C5 5, R6C5 1, R5C5 3, R2C3 1, R3C3 6, R3C6 1, R2C6 3, R1C6 6, R1C3 5, R1C1 3, R4C6 8, R4C4 6, R9C4 1, R7C4 8, R8C1 1, R8C2 6, R8C8 9, R8C9 8, R3C8 4, R3C9 9.

R6C7 6, R6C8 1, R9C7 7, R2C7 2, R1C8 7, R4C8 2, R4C9 7, R1C4 2, R2C4 7.

R7C7 1, R7C8 3, R9C8 6, R7C2 7, R9C2 3, R9C1 5, R7C1 4, R7C9 1, R9C9 4.

All by simple DS reductions.

**Final solution** shown.

Check for the validity of the solution if you need.

### Expert Sudoku Solving Techniques and How to Use Special Digit Patterns

As a strategy * always try first—the row-column single digit scan to find the valid cell *at any stage, because that is the

**most basic and easiest of all techniques.**

While doing the single digit scan, look out for possible breakthroughs by double digit scan and even triple digit scan. Wherever possible, **Cycles** are formed that in any situation are valuable digit patterns to have and Cycles play a key role in quick solution.

Possible **Digit Subset Analysis** or **DSA** is a general technique that is the basis of finding a unique valid digit for a cell by Reduction, a Cycle or even the valuable digit pattern of a single digit lock. Whenever possible, short length possible digit subsets of 2 or 3 digits are to be formed in vacant cells by DSA.

A **Single digit lock** and an **X wing** are comparatively more powerful digit patterns that usually create important breakthroughs.

The **last resort of filling EACH EMPTY CELL with valid possible digit subsets** by DSA is to be taken when it is absolutely necessary. But,

Strategically for faster solution, it is better to delay this time consuming task as much as possible.

A **basic part of overall strategy** is,

Whether we search for a

breakthrough of a bottleneckor avalid cell identification, our focus usually is on thepromising zones,the zones (row, column or a major square) 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 lock, Cycles, Valid cell by DSA are some of the key patterns.

The six Sudoku solving techniques and special digit patterns are explained now.

#### Expert Sudoku solving techniques of single digit scan and double digit scan

Let us use the following Sudoku game as the starting point to explain the two techniques of single digit and double digit scan.

This is an initial stage of the Sudoku puzzle solution.

The result of breakthroughs by single digit scan and double digit scan are shown.

Digit scans are done on the cells of a major square. Single digit scan for 8 in R9C9 in C9 and in R5C3 in R5 blocks all other cells in the target right middle major square except R4C8. This is then the only cell in the major square where digit 8 can be placed. Single digit scan blocks then all other cells in the target major square except ONE cell to be occupied by the scanned digit.

This is the most used and most basic technique to find a unique cell for a digit.

In the same way, digits [1,9] appear in R6 and the pair of digits blocks R6C1, R6C2 and R6C3 for both the digits [1,9] in the target left middle major square. This is the **double digit scan**.

This leaves ONLY TWO VACANT CELLS for the two scanned digits [1,9] in the major square as well as in column C3.

This digit structure of two possible digits for only two cells in a column, row or a major square is termed as a two-cell Cycle.

Every opportunity of forming Cycles are utilized because, Cycles play a very important role in solving very hard to extremely hard Sudoku Puzzles.

Just as single digit scan on the cells of a major square is done searching for suitably affecting digit in multiple rows and columns, double digit scan can also be done with same two digits appearing in more than one row or column affecting the cells of a specific major square.

Double digit scan invariably provides an important breakthrough. Always look out for an opportunity for a double digit scan.

The concept of double digit scan can be extended to triple digit scan as well though it is rare. In this case a three digit long Cycle is formed.

#### Expert Sudoku Solving Technique of Possible Digit Subset Analysis (DSA) and how to find a naked single

The following is an initial stage of the Sudoku puzzle solution.

The result of finding a unique valid cell or naked cell by digit subset analysis technique applied on the above game stage shown now.

Digit Subset Analysis or DSA is a concept as well as a technique. By DSA, digits that can occupy a particular cell are identified.

This is an essential and very important function for identifying all other digit patterns possible in a target cell. When no easy unique possible digit in any cell can be identified, the only way to move ahead in solving the puzzle is to carry out DSA for PROMISING CELLS. The simplest type of promising cell is the cell with smallest number of possible digits DS of 2 or 3 digits (not 4 digits at first).

To identify a promising cell, identify first a row, column or major square with maximum number of already occupying digits. In the example above, row R4 is such a row with possible digit subset in the four empty cells [1,2,7,9].

Next, the cell R4C4 is easily identified as a promising cell as, [2,9] in Column C4 affecting the cell REDUCES the possible digit subset or DS for the cell to just [1,7]. Moving ahead, the third cell R4C6 gets DS [1,7,9].

And finally, for the fourth empty cell in the row R4C9, [1,9] in its parent major square and 7 in parent column C9 combine to form [1,7,9] to be reduced from the DS [1,2,7,9]. Result is, **R4C9 2**, a valid unique digit for a cell.

REDUCTION is a fundamental process in solving Sudoku puzzles.

**Naked Single:** By definition, a naked single is a digit that only can occupy a specific cell. If you analyze possible digits in R4C9 ignoring the earlier process of reduction from DS [1,2,7,9] in row R4, you will find only digit 2 can occupy the cell.

You may adopt this process of identifying a naked single **WITHOUT taking help of smaller possible digit subsets in vacant cells**, but this process is easier only occasionally.

#### Special digit pattern of Cycle of twins or triplets and how to use it in solving an Expert Sudoku puzzle

The following is an initial stage of the Sudoku puzzle solution.

The result of a breakthrough unique valid digit by forming 2 digit (twin) and 3 digit (triplet) Cycles is shown.

The digits [1,9] in R6 affect the possible digit subsets or DSs of vacant cells of left middle major square leaving only two cells of the square R4C2, R5C2 for the two digits [1,9]. It is not certain which of these two cells will finally be occupied by 1 or 9 but we can confidently say none other than these two digits are the **only eligible candidates** for occupying these two cells.

Thus a **Cycle (1,9)** is formed in these two cells restricting any other cell in the parent column and major square from having any of these two digits.

If we place 1 in R4C2, automatically R5C2 must have 9 and if we place 9 in R4C2 the cell R5C2 must have 1. Potentially these two digits Cycle between these two cells till their final positions are determined. That is why we can place the DS [1,2] in both the cells blocking any other cell of parent major square and column C2 from having these two digits.

This is a two digit Cycle and is the most frequently occurring one.

The direct positive result is formation of a second Cycle (3,5,7) in the three remaining vacant cells by exactly three remaining digits in the major square. This is a three-digit Cycle debarring all other vacant cells of parent row R6 to have these three digits. Result is formation of a third Cycle (4,8) in R6C5, R6C6 and a unique valid digit 6 in R6C9 as the REDUCED DS [4,6,8] in R6C9 is further reduced by [4,8] in C9 and right middle major square combined.

This breakthrough won't have been possible without the Cycle (3,5,7) in R6.

The main function of a Cycle is to REDUCE the length of possible digit subsets or DSs in affected parent zones and with each DS length reduction, certainty of getting a unique valid cell in the whole set of 81 squares increases.

#### Expert Sudoku solving technique of parallel scan for a single digit on a row or a column

The following is an initial stage of the Sudoku puzzle solution.

Result of carrying out parallel scan for digit 2 on the vacant cells of row R8 is shown.

In the relatively empty 81 square puzzle, digit 2 in left bottom major square and right bottom major square debar every vacant cell in the squares from having digit 2. With keen interest we observe, out of six vacant cells in row R8, four cells cannot have digit 2. If we can debar any of the two other remaining cells R8C5, R8C6 from having digit 2, we will get a unique valid digit breakthrough.

This actually happens, as 2 in R1C6 in C6 eliminates the fifth vacant cell R8C6 from having digit 2. We have achieved the breakthrough of unique valid digit 2 in R8C5 as if out of thin air.

In essence, a PARALLEL DIGIT SCAN for digit 2 is done on the empty cells of row R8. Even if digit 2 in R9C3 were in R2C3 or R3C3, the result would have been the same.

A parallel digit scan is done on vacant cells of a row or column (and NOT on vacant cells of a major square). Fortuitous presence of the digit scanned in other cells, all except one cell of the target row or column scanned are debarred from having the digit scanned for. This is an advanced and powerful Sudoku Solving Technique often providing a major unexpected breakthrough.

Observe, you could also have achieved the breakthrough by forming the five-cell long Cycle (1,3,4,6,8) in R8. But that would have been laborious. If you are aware of the possibility, a parallel scan will give you a quick and clean breakthrough.

#### Single digit lock and how to use it in solving an Expert Sudoku puzzle

The following starting position is an advanced stage of the solution. Because of Cycle (1,9) in C2 the DS in C2 is reduced to [2,3,4,5,8] which again is reduced by [2,4,8] in R1 to form DS [3,5] in R1C2 and forms the Cycles (3,5) and (2,4,8) in C2. Important result of these formations is DS [4,8] in R7C2.

The result of forming a single digit lock and breakthrough achieved by the lock is shown.

Shift your attention now to row R8 with DS [1,3,4,6,8] and especially the DSs of cells R8C1 and R8C3. By [1,6] in the parent major square, 8 in C3 and 4 in C1 the DSs in these two cells are reduced to [3,4] and [3,8] respectively. Recall that DS in R7C2 was [4,8]. These three promptly join together to form a breakthrough Cycle (3,4,8) which eliminates any other possibility of digit 3 in the two remaining cells R7C3 and R9C1 in the major square.

The digit 3 is now locked in only two cells R8C1 and R8C3 in R8 and bottom left major square. The two digits are highlighted by larger size and different color. This powerful digit pattern is what we call a **single digit lock.**

As this lock on 3 eliminates possibility of 3 in all other vacant cells of the row R8, especially 3 in DS [1,3] of R8C9, it provides the **breakthrough of R8C9 1**. Now double digit scan for [4,6] in R9, C9 produces Cycle (4,6) in bottom right major square and DS [3,7,9] in R9C7 reduces by [3,9] in C7 to R9C7 7. This is the **second major breakthrough** as it causes easily obtained valid cell chain starting from R5C7 4.

A Single digit lock is a very important digit pattern that may not only provide a major breakthrough but also may join suitably with a second single digit lock to form the more powerful and advanced digit pattern of X wing.

As a rule, while solving a hard Sudoku puzzle, the Cycles and single digit locks are identified and marked for immediate or later use.

#### Expert Sudoku solving technique of Asymmetric X wing diagonal test

We will use the following advanced stage of solving a very hard Sudoku puzzle to explain how the most critical breakthrough can be achieved by asymmetric X wing diagonal test.

Short length DSs of 2 or 3 digits for the vacant cells are formed with ease for most of the vacant cells. Many single digit locks and Cycles are also identified. But no valid cell for a unique digit could be identified by DS reduction. This is the situation where we will look for the more complex digit patterns, especially, the X wing digit pattern.

But we don't have a normal X wing in the game stage. Instead, the less powerful **asymmetric X wing** formed by **single digit locks on 6** in R5C1, R6C2 and R5C8, R6C7 sharing rows R5, R6 has still the advantage of one of the two DIAGONAL VALUE PAIR VALIDITY.

In other words, **diagonal test of an asymmetric X wing establishes validity (or invalidity)** of a specific digit placement in TWO CELLS AT ONE GO. *Digit 6 at the ends of one of the diagonals will be valid and the other invalid.*

We will test first the **validity of the diagonal R5C8 6, R6C2 6.**

R5C8 6, R6C2 6 diagonal values will now be tested.

R5C8 6 -- R6C2 6 -- R5C1 3 -- R4C3 5 -- Cycle (1,3,6) in C3 -- R3C1 7 by reduction of 6.

With 6 in R6C2, R8C2 7 -- R8C5 8 -- **R3C5 7 clashing with R3C1 7 in same row as shown in the following figure.**

This proves **X wing diagonal choice of R5C8 6, R6C2 6 invalid.**

Solution will come from the **other diagonal valid values of R5C1 6 and R6C7 6 easily.**

This **diagonal test for 6 is favored here because**,

- all four cells have two digit DSs,
- digit 6 is involved in a large number of other linked two or three digit cell DSs, and also,
- the X wing is spread out to its maximum width of three major squares. As a rule, the more spread out the X wing, the more will be its effectiveness.

For these reasons we should get quick results, positive or negative, from the diagonal test.

The chosen diagonal cell values for digit 6 are in fact invalid and it can easily be checked mentally in a few steps.

As a rule, *we first choose the invalid diagonal of an X wing for test.* Once the invalidity is established quickly, rest of the larger number of cell results can be obtained easily. In most cases quick resolution of the invalid diagonal values clears the way for resolving all unique digit valid cell values easily with the valid diagonal values.

*To go through the solution of this Expert Sudoku puzzle once more, click here.*

### More Expert Sudoku puzzles you may like to solve and learn how to solve

The **updated list** of **Solutions to Expert Sudoku **puzzle games:

**How to solve Expert Sudoku puzzle games full list****.**

Expert Sudoku solving strategy and techniques are included in many of the solutions.

*Enjoy solving and learning to solve Expert Sudoku puzzles.*