## Let's play Sudoku - Game 4

In the last session, we took lesser time to solve our exercise game given in session before that. This is because we have gained more familiarity with the game by now.

In this fourth session we will solve the Sudoku game we gave as exercise in the last session.

In this session we will spend little time on what is Sudoku and what are its rules of play and so on. If you want to know more please refer to any of our * session 1*,

*or the*

**session 2***.*

**session 3**The following is the game board we would solve in this session.

This is the game we left you to solve in the last session. We hope some of you have solved it already. In today's session we will move a bit faster than before as you have already experienced Sudoku game playing to some extent.

If you feel any difficulty, please refer back to our earlier sessions, * session 1, session 2,* or

*.*

**session 3**The 81 cell board above is a specific Sudoku game board with some cells filled up with digits from 1 to 9. These are the valid digits that you can use to fill up any empty cell.

While filling up any empty cell you must follow the **Sudoku game rule:**

You

must not repeat any digitin any column, in any row or in any 9-cell mediumsized square bordered by thick lines.

The whole board has 9 such 9-cell squares with thick borders.

*Ultimately your job is to fill up all the empty cells with digits 1 to 9 but without breaking the ** Sudoku game rule not even once.* Then only we would say you have successfully completed the game.

For convenience, we will use the **labels C1, C2, C3......C9** for** identifying the 9 columns** and **labels R1, R2, R3....R9 **for** 9 rows**.

We will refer to a cell by its row label suffixed with column label. For example * top left corner cell is R1C1* and

*bottom right corner cell is R9C9*.Each of the 9-cell groups we will call a 9-cell square and specify by the three row labels suffixed by three column labels. For example the top right **9-cell square is R1R2R3-C7C8C9.**

### Primary objective at each step

The **only objective at each step is to find a cell in which one and only one digit can be placed. **This is what we call** a valid cell.**

### Valid cell and Valid digit

We would repeat the definition of valid digit and the valid cell.

We **define a VALID digit as** the **digit you put in an empty cell** so that,

It is the only possible digit which you can put in the cell following the rules of the game.

As before, the **most important objective** is, *how to find out the cell in which you can put one and only one digit out of the digits 1 to 9*.

We have used the mot basic technique of **row column sweep or horizontal-vertical cross-scanning** and also the more powerful **technique of digit subset analysis**. We have used * Exclusion technique *wherever we found it possible.

We used the concept of a* favorable zone* as a column or a row or a 9-cell square with

*so the number of digits possible to fill up the empty cells in the zone is much less and so by eliminating the few other contenders getting only a single digit for a single cell is that much higher. We have also used digits with high occurrence in the board, that is,*

**high digit occupancy***to our advantage.*

**the favorable digits***For a favorable digit* that has appeared, say, in *7 cells out of maximum 9 possible all over the board*, finding the rest of the two valid cells for this favorable digit *by row-column sweep* is generally very easy because of elimination of a large part of the board by its occurrence in most zones.

### Let's find the first valid cell - first stage

We get our **first valid cell in R6C9 for digit 2** by scanning rows R4, R5 and column C8. We have colored it green and the rest of the valid cells that we would find we would color differently to distinguish it from the those cells that we fill up at this stage. We would always color the valid digits red to keep them separately recognizable from the digits supplied with the initial game board.

We get one more **2 in R9C5** by row R8 and columns C4, C6 before we turn our attention to digit 7.

Next we detect cell **R1C5 for 7** * boxed-in* by rows R2, R3. This fixes the cell

**R4C4 for 7**by rows R5, R6 and column C5. Notice that without 7 in R1C5 first we won't have been able to fix this cell for 7. This is what happens - once a cell is fixed for a digit, putting the digit in the cell opens up further possibilities for the same digit.

*This is why we keep on searching for valid cells for the same digit once we get one for the digit.*

**It becomes the favored digit**.Each time we put a valid digit in a valid cell we color it red to distinguish it from the digits supplied with original game.

Next **7 we get in cell R7C6** by columns C4, C5. This fixes in turn the cell **R9C7 for 7** by columns C8, C9 and row R7. This in turn boxes in **cell R8C3 for 7** by rows R7, R9 and column C2. Notice that we could have done the same by scanning first columns C1, C2, but then we would have had to scan both the rows R7, R9 in any case. * This is a more inefficient approach* of executing a row-column sweep compared to what we have done.

At this stage we have *finished filling up all 9 numbers of digit 7 possible.*

Turning our attention now to digit 1, the next valid cell we get is **R6C5 for 1** by scanning rows R4, R5 and column C6. Without leaving our interest in 1 we get the next **1 in R3C4** by scanning columns C5, C6 only. Similarly **1 in R2C1** is easy to identify by scanning rows R1, R3. We end our search for 1 by getting the valid unique **1 in R9C2** by scanning rows R7, R8 and column C3.

Freezing the board at this stage, as usual we copy it to a new place in our spreadsheet, remove the cell shadings and start analyzing fresh for finding the next valid cells.

### Second stage

At this stage we get the first valid cell **R3C6 for digit 6** by easy scanning of only two columns C4 and C5 each containing a 6. We are lucky in this case as out of the the three cells left by two column scan, two have already been occupied leaving only one cell for 6.

In the same way by two column scan we get the valid cell **R4C5 for 3**.

Now we apply the **special technique of digit subset analysis** on two remaining cells of column C5. The two remaining digits in this column are 5 and 8. We examine the first empty cell R2C5 for placing either 5 or 8 and find a 8 in this row already existing. Thus, **cell R2C5 becomes the valid cell for 5**.

The **only remaining digit 8** goes into the **only remaining** **cell R8C5**. This technique we call as **the exclusion technique**. Next we get digit **5 in cell R5C4** by sweeping columns C5, C6 and the **9 in R7C4** as the only remaining cell and digit in the column.

Lastly we apply the **digit subset analysis technique** on remaining two empty cells of row R2 for digits 6 and 9. As examining R2C8 we find a 6 in column C8, we fix this cell** R2C8 for the remaining digit 9 in the subset. 6 in R2C7 **follows automatically. If you analyze, you will find that this breakthrough you couldn't have achieved without applying the **digit subset analysis technique** and just by row-column cross scanning in any way.

We will freeze the board at this stage and go over to the next stage.

### Third stage

At this third stage, we have removed all the colors and started examining with a fresh mind.

The first valid cell we get in **R9C9 for 9** by scanning rows R7, R8 and a single column C8. This fixes the cell **R4C7 for 9** by scanning two columns C8, C9.

Next we get **R5C9 for 3** by rows R4, R6 and **5 in R6C8** by exclusion being the only digit left in the 9-cell square. Still going after digit 3 we get **R7C7 for 3** by column C9 and row R9, a special kind of scan.

You always should use cross-scan or exclusion to find a valid digit first. These are the basic techniques.

Now a single column scan of column C9 fixes cell **R9C8 for 8**. Same happens for **R8C9 for 6** by row R7 this time and finally **R7C9 for 4** by exclusion in this 9-cell square.

This last 4 finds the valid cell **R8C2 for 4** by rows R7, R9 and then **R4C1 for 4** by rows R5, R6 and column C2.

We freeze the board here and go over to the next stage.

**Fourth and final stage**

We start with **6 in R9C3**. It was the only remaining cell in row R9 that escaped our attention. Then **6 in R1C1** by columns C2, C3 and row R3. This finishes 6.

Next we get **8 in R3C7** by row R1 and column C8. This fixes cell **R3C8 for 3** by row R1, 4 for cell **R1C8** by column C7 and **2 in cell R1C7** by exclusion. This completes this 9-cell square.

Looking elsewhere we get now **R1C2 for 9** by exclusion in row R1. This fixes **R6R3 for 9** by columns C1, C2, **8 in R6C6** by exclusion in row R6 and then on to the** last 9 in the board in R5C6 by exclusion in column C6**. By exclusion we get **8 in R5C1**, then **5 in R4C2** by exclusion in row R4, **2 in R7C2** by exclusion in column C2, **5 in R7C1** by scan of column C3, **2 in R3C1** by exclusion in column C1, **8 in R7C3** by exclusion in row R7 and the last cell **R3C3 for 4** by exclusion all around. Notice that in this last stage we have used the easiest of all techniques, that is, the **technique of exclusion** - filling the last cell in a column, row or 9-cell square. This is the **fastest methd** of getting a valid cell and you should never fail to exploit this very favorable condition.

We would repeat now our recommendation regarding playing medium.

#### Playing medium

Should you play Sudoku using pen and paper, in a mobile or using something else? **Our strong recommendation is**,

Always play Sudoku in a spreadsheet program, if possible, at least at the beginning stage. We are not aware of any better medium of Sudoku game solving, be it an easy game like we have solved just now or the reportedly hardest Sudoku game in the world.

Lastly we leave a game for you to solve.

### A game for you to solve

We leave you here with a new game for you to solve. In our next session we will present its solution and another new game.

Enjoy.

### Other Sudoku game plays at absolute beginner level

**Sudoku beginner level game play 11**

**Sudoku beginner level game play 10**

**Sudoku beginner level game play 9**

**Sudoku beginner level game play 8**

**Sudoku beginner level game play 7**

**Sudoku beginner level game play 6**

**Sudoku beginner level game play 5**

**Sudoku beginner level game play 4**

**Sudoku beginner level game play 3**

**Sudoku beginner level game play 2**

**Sudoku beginner level game play 1**

### Assorted Interesting Sudoku game plays

These Sudoku game solutions are collected from various sources and are found to be interesting. You can get these Sudoku solutions at * Interesting Sudoku *not classified at any hardness difficulty level.

### Second and Third level Sudoku games

You will get links to all the 2nd level Sudoku game solutions at **Second level Sudoku.**

Links to third level Sudoku you will get first at 2nd level game solutions and links to fourth level Sudoku you will get in the 3rd level solutions.

It is **recommended** that without jumping over any of the hardness levels, one should progress through solving higher level Sudoku games strictly step by one step up. For example, you shouldn't play a 3rd level Sudoku game without being comfortable in solving 2nd level games.