## Sudoku hard Strategy and Techniques: Sudoku Hard level 4 game 4 solution

Learn Sudoku hard strategy and techniques while solving Sudoku hard level 4 game 4. Breakthroughs by DSA, Cycle, Single digit lock explained clearly.

Sections are,

**Sudoku hard level 4 game 4****Solving Sudoku hard level 4 game 4 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****Frequently used technique.****How a single digit candidate valid cell is identified by Digit Subset Analysis (DSA) in solving a Sudoku hard puzzle. Heavily used technique.**.**How digits possible for all empty cells (DSs) enumerated while solving a Sudoku hard puzzle****To be used judiciously, better to use partially.**.**Single digit lockdown and its use in solving a Sudoku hard puzzle****Helps to make a breakthrough in a Sudoku hard puzzle.**

*You may move directly to any of the above sections by clicking its link and return to previous position by clicking on browser back button.*

**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.*

### 4th Sudoku puzzle at level 4 of hardness

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.

We'll **first solve the Sudoku hard** using **strategies and techniques for solving Sudoku hard puzzles.**

The strategy and techniques for quickly solving Sudoku hard are explained with examples in five sections after the solution.

You may decide to go through the solution directly and take a look at the specific strategy or technique when you feel like.

The Sudoku hard techniques with internal links to the concept sections appear in the solution. You may just click on any such link, say for example * Cycle,* go through the concept and return by clicking on browser back button.

Or, you may decide first to go through the concepts and then go through the solution. To jump to the Strategy and techniques concept sections click **here.**

Following is the **solution of the puzzle explained step by step in details.**

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

### Solution to the Sudoku hard level 4 game 4

Let us solve our hard Sudoku puzzle now.

We'll show the puzzle board 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****—**R5C8 5 scan C7, C9. This is the only direct cross-scan valid digit obtained.

Subsequent results—

R4C9 7 scan C7 -- R6C1 8 scan C2, C3 -- R6C7 1 cancel [6,8] in DS [1,6,8] of R6C7 in right middle 9 cell square by scan R6 -- Cycle (6,8) in R7C7, R7C8 by exclusion in 9 cell square.

The Cycle (6,8) is formed automatically and we mark the Cycle for future use with no opportunity to use it now. With no more direct cross-scan opportunity visible three **Cycles are force-created** consecutively, resulting in a much sought after breakthrough—

Cycle (5,7) in R6C3, R6C5 cancel [2,3] from DS [2,3,5,7] in both R6C3 and R6C5 -- Cycle (2,3) in R6C4, R6C6 exclusion -- Cycle (1,4,8) in R3C1 DS [1,4], in R3C5 DS [1,8], and in R3C8 [1,4,8] --

R3C6 3home square cancel of [2,9] in DS [2,3,9].

This **breakthrough in R3C6 3** results in a series of valid cells by easy cross-scans or cancellations—

R6C6 2 cancellation -- R6C4 3 cancellation -- R1C7 3 scan R3, C8, C9 -- R2C3 3 scan R1,R3,C1 -- R7C2 3 scan R8,R9,C3 -- R3C2 2 scan R1,C1 -- R3C7 9 cancellation.

To show further progress clearly we'll close the stage at this point and show the results below.

You may verify the actions taken till now from this second stage status.

Notice that we have evaluated the DS of only the cells that we needed. This strategic approach speeds up the process.

#### Stage 3 of Sudoku hard puzzle solution: 4th level game play 4

The first two valid cells are easy to find—

R2C9 2 scan R1,C8 -- R8C2 7 by

DSA, cancel [1,9] from DS [1,7,9] in C2.

But it is not any more easy to find positive results. So looking for the **advanced structure of single digit lock**, we get immediate success.

The last find 7 in R8C2 eliminated 7 from DS[1,7,9] in R5C2 and with 7 in R4, created **digit 7 locked in R5C3 and R6C3 in C3 and locked inside left middle square****—**

R1C1 7scan R2, R3, C2, C3 7 locked.The digit 7 locked in C3 eliminated possibility of 7 in R1C3 acting as scan on C3.

And again the next positive result by a **second single digit lock** of 5 in R4C3, R6C3 in C3—

R2C1 5 scan R3, C2, C3 with 5 locked.

Next is a **bottleneck breakthrough that can be achieved in two ways**—either **parallel scan** for 2 in four columns C1, C5, C6 and C9 against R8 or **DS evaluation** of all the remaining empty cells of R8 to **find 2 alone in the DS of R8C7**—

R8C7 2 as lone cell DS with 2 -- R9C7 4 cancellation.

Identification and use of the **only cell DS with a specific digit in a zone** (row, column or 9 cell square) forms *an important additional technique for valid cell find.*

In addition to the first valid cell R1C9 for digit 2, three other cells are colored turquoise blue to mark each of these as a bottleneck breakthrough.

For ease of understanding let's close at this point and show you the status below.

#### Stage 4 of Sudoku hard puzzle solution: 4th level game play 4

**Cycle** (1,2) formed in C3 cells R7C3, R9C3 cancelling 1 in R9C1and then scan R9 cancelling 4 again in R9C1 gives first breakthrough at this stage followed by a series of easy valid cells.

This is a good example of a short 2** digit Cycle effecting a breakthrough** in a series of positive steps.

The valid cells now are,

R9C1 6 scan R8, scan Cycle (1,2) in 9 cell square -- R8C1 4 cancel -- R3C1 1 cancel -- R1C2 9 cancel -- R1C3 4 cancel -- R5C2 1 cancel -- R3C5 8 cancel -- R3C8 4 cancel -- R2C8 1 cancel -- R1C9 8 cancel -- R1C6 1 scan R2,R3,C4 -- R1C4 5 exclusion -- R2C5 6 scan C5, DS [4,6] -- R2C4 4 exclusion -- R4C5 1 scan R6,C4,C6.

Let's again close at this stage and go over to the final stage for ease of explanation. The status results are shown below.

#### Stage 5 of Sudoku hard puzzle solution: 4th level game play 4

Even at this stage direct scan was not visible.

So using next advanced technique, **Digit Subset Analysis or DSA** on cell R7C5 is done. The Digit Subset or DS in column C5 is [5,7,9] and with [5,9] in R7 we get the valid digit 7 in R7C5 -- **R7C5 7.**

The rest are straightforward—

R9C6 8 cancellation -- R9C8 7 cancellation -- R7C8 8 cancellation -- R6C5 5 cancellation -- R6C3 7 cancellation -- R8C5 9 cancellation -- R8C9 6 cancellation -- R8C6 5 cancellation -- R7C9 1 cancellation -- R9C9 9 cancellation -- R9C4 2 cancellation -- R9C3 1 cancellation -- R7C4 6 cancellation -- R7C3 2 cancellation -- R5C3 9 cancellation -- R5C4 8 cancellation -- R5C7 6 cancellation -- R4C7 8 cancellation -- R4C3 5 cancellation -- R4C4 9 exclusion in C4 -- R4C6 6 exclusion in R4 -- R5C6 7 exclusion in both R5 and C6 -- 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.

### Overall strategy adopted and techniques used for solving a Sudoku hard puzzle

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.

In solving this Sudoku hard, we have mentioned the use of an additional technique of **Parallel scan**for a single digit on the cells of a row or column.

In the case of this sudoku hard, parallel scan turns out to be one of the two possible techniques that can be used and so because of its minor role is not explained in more details.

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 bottleneckor avalid cell identification, our focus usually is on thepromising 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.**

### What is a Cycle and how to use it in solving a Sudoku hard puzzle

#### 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.

### How a valid cell is identified by Digit Subset Analysis or DSA in short

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

**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**

*3,5 and 9 eliminated*

**only the left out digit 1 of the DS [1,3,5,9] can occupy the cell R4C1.**Remember that,

While

evaluating the valid digit subset or DS of an empty cell,you would analyze not only thein corresponding row, column and 9 cell square, youdigits that are already filledmust 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.

### How digits possible for all empty cells (DSs) enumerated while solving a Sudoku hard puzzle

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 digitsthat 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.**

### Single digit lockdown and its use in solving a Sudoku hard puzzle

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.*

*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 this is a hard Sudoku, the hybrid strategy enabled avoiding the full DS evaluation for all empty cells which is a very tedious and time-consuming process. As a result solution process speeds up.

Key pattern identification had occurred in many instances by Cycles, DSA and single digit lockdown.

But the main breakthrough has been provided by single digit lockdown.

Overall, this has been a balanced Sudoku hard that needed careful strategic approach for quick solution.

### 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:

**How to solve Sudoku hard puzzle games full list (includes very hard Sudoku).**

*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.*