## Let's learn to play Sudoku

With a very brief introduction about the game we will start playing a Sudoku game in detailed steps, explaining why and how certain actions are taken.

### What is Sudoku

Sudoku is a single player game where you have to fill up the empty cells of the given 9 by 9, that is, 81 cell large square. In a specific Sudoku game some of the 81 cells will have already filled digits. The game we'll use to learn Sudoku is shown on the right.

As you can guess, there can be innumerable Sudoku games possible depending on which of the 81 cells are filled. Also, there can be very easy to extremely hard Sudoku puzzle games.

Sudoku has the treasure to engage minds of young or old alike for all time, in all countries. Not for nothing it is considered as the brain game of the century.

We'll start learning to play by solving an easy Sudoku puzzle by explaining each step of filling an empty cell in details.

**You will find the links of the beginner level easy Sudoku game plays given at the end.**

### The game

We will play the following Sudoku game board.

We will use the **labels C1, C2....C9** at the top **for identifying the columns** of the **large square with thick borders**. This square with exactly **81 tiny squares** many of which are vacant but some are filled with numbers, is a specimen of a Sudoku game board. There are millions of such different Sudoku game boards with the tiny cells filled up differently. You would perhaps never be able to exhaust this treasure even if you play hard.

Similarly we will use the **9 labels R1, R2....., R9** to identify **the 9 row**s of the square game board.

**To identify a specific tiny cell**, say the **cell with 6** at the bottom-most row, we will **mention its row label first** and **then its column label**. By this convention, **we will call this cell as R9C5**. With this system we will be able to identify exactly each of the $9\times{9}=81$ tiny cells of the large square Sudoku game board.

At a higher level, these **81 tiny cells are grouped** into $3\times{3}=9$ **medium sized squares** each **consisting of 9 tiny square cells**. To point out one for you, we have highlighted the bottom left medium group of **9 tiny cells in green color**. Notice that these 9 medium sized squares **also have thick borders**. These borders **help you to distinguish between adjacent such medium sized squares**.

For convenience we will call the tiny square cells as simply **CELLS**, the medium sized 9 cell 9 squares as simply **SQUARES** or 9-cell squares and the 81 cell game board as just **BOARD or a game board**.

At this moment you need to forget about the labels and the colors. Those are all our doings for ease of explanation. The **actual game board is the large square with 81 cells some of which already are filled up with digits in them.**

### Rules of the game

As you perhaps have understood, the very first rule of standard 3 degree $(3^2\times{3^2})=9\times{9}$ Sudoku game is,

**First rule:** *Each cell of the 81 cell board can have any of the nine digits 1 to 9 only, and nothing else*. In the beginning, some of the 81 cell will come filled up with digits and the rest empty. The player has to fill up all the cells gradually, following a few more stringent rules.

**Second rule:** Each row and each column can have the digits 1 to 9 without any repetition at all. **Ultimately** when you fill up all the empty cells of this game board, **each row and each column will have exactly nine digits 1 to 9 with no digit repeating**. That **will be the correct solution** of the specific Sudoku problem you had started with in the beginning.

But this is not all. We have a third very important rule.

**Third rule:** Each of the nine 9-cell squares have their own little world in which they won't allow you to put anything other than the digits 1 to 9 again without any repetition. Though this rule apparently makes the game more complex, this specific property helps greatly in solving a Sudoku game to its final legally filled up form.

*To follow the game play, you should draw the game board with the filled in digits on paper, or better still, create it in a spreadsheet program.*

At this point, you know what is Sudoku, and also the rules of the game.

You should now try to complete filling up of all the empty cells of the game given with digits 1 to 9 without violating any rule, before going ahead further. If you are really stuck, then refer to the solution. This is the best way to learn.

### The play begins

With this brief introduction, if you are playing the game for the first time you may feel quite confused about where to start. In fact usually there may be many starting cells that you might be able to **fill up with a VALID digit at this point** of starting the game. It is up to you where you start. **At this level of the game, the start won't be important**. You need just to put a VALID digit in an empty cell.

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

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

Unless this convention is followed you won't be able to proceed with the game at all. You will always be under confusion regarding the correctness of the position of the digit in the row or column.

Question is, *how to find out the cell in which you can put one and only one digit out of the digits 1 to 9*. It is understood that when you place a digit in an empty cell you do not violate any of the rules of the game, naturally.

### Row-column sweep

At this point stop for a moment and try to find that special cell in which you will be able to put one and only one digit.

#### Favorable zone

Here we will introduce the first basic concept of identifying the elusive cell that you can fill up with a valid digit.

A **favorable zone may be a medium sized 9-cell square** with **maximum digit occupancy or a 9-cell square filled up with maximum number of digits**. As you can perceive, **possible digits left to be filled in that high occupancy 9-cell square being small**, *cross-checking with connecting columns and rows*, **chances of finding the right cell where you can put your valid digit will be high**.

A favorable zone **may also be a highly occupied row or column**, where again number of cells (or digits) left to be filled up being small, *cross-checking with connecting column, row or the 9-cell square itself,* might hand over to you easily the elusive right cell where you can put your valid digit. We will see examples of favorable zones as we proceed further.

#### Favorable digit

On the other hand, you may perceive that, the **digit that is appearing maximum number of times** in the whole game board (a digit can appear 9 times maximum) will have a high chance of being a valid digit.

#### Basic scanning

The **very basic technique of finding the right cell** we use at this beginning stage of play is **to quickly scan rows and columns of 9-cell squares** and try to identify a 9-cell square where a cell is **locked out** or **boxed in** by the **appearance of a spcific digit on all sides except that particular cell**. That will be your valid cell where you will be able put the valid digit you were examining.

### Second stage

To understand how we do it, carefully examine the following figure where we have been able to put **1** in cell ** R2C8**.

As you have perhaps guessed, we went after the digit **1** and **scanned the 9-cell third column C7C8C9** and **9-cell first row R1R2R3**. Our interest is on the **intersecting 9-cell square R1R2R3-C7C8C9.**

**Scan the columns C7 and C9** to find that in each of these two columns **a 1 is present**. These are the **invalid columns for the digit 1**. That is the reason we colored the column **C8 in blue color indicating that this column is the favorable zone for 1** and further, **in the two empty cells R2C8 or R3C8** we may be **able to put a 1**. As a visual aid we have colored the C8 column. **In actual play you have to do** this * sweeping* or

**scanning****mentally**all the time.

Now to decide between the two favorable cells **we change direction from vertical to horizontal** and scan the first row of 9-cell squares searching for 1s in the three rows R1, R2 and R3.

**As both R1 and R3 have a 1 in them, the middle row R2 is a favorable row for 1.**

At last **we have found our first valid cell**. This is the **intersection of the two blue colored column and row, cell R2C8**. This cell **can have one and only one digit** **1** in it.

As soon as we put a 1 in it, all the first three rows get a 1 in each as well as the last three columns get a 1 in each. We won't consider these zones for placing a 1 any more.

Now we have to find the next valid cell.

This way the game proceeds.

### Third stage

This third stage board you would find very colorful.

We have not left pursuing the digit 1, and coming down, colored the last two rows R8 and R9 and the column C4 red indicating **these are invalid zones for digit 1** as each of these column and rows has a 1 in it.

As a result you would find that **the cell R7C5 is totally boxed in from all sides** and **is the only possible cell to have a 1 in it**. To highlight we have colored it dark yellow.

At this point only two 9-cell squares are left without a 1, but we won't be able to decide with certainty which two cells in these two 9-square cells must have a 1. To highlight this uncertainty and the fact that we have examined this possibility, we have colored the probable four more cell with dark yellow. We won't be able to find any positive result with digit 1 any more at this point of time.

A new thinking is required now in a new direction.

### Fourth stage

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

Again we went up to the first three rows and last three columns intersecting 9-cell square, but this time **searching for placement cell for the digit 2**.

**We got the R3C8 cell quickly** as **rows R1 and R2 are invalid zones for a 2** and only in the intersection of row R3 and column C8 we find **an empty cell R3C8** waiting for receiving a 2 and nothing else. After placing a valid 2 we colored it yellow to mark that this is the starting point at this fourth stage.

#### A new approach

Now instead of continuing our search for digit 2 placement, we notice that the 9-cell square R1R2R3-C7C8C9 itself has been **transformed to a very favorable zone** because of filling up of two empty cells. Now **only two empty cells R2C9 and R1C9 remain to be filled up**. *This change of course of action is the hallmark of any problem solving.*

You will always be alert for favorable zones or situation at this current moment, and whenever you find such a one, you must start examining and exploiting the new situation to its fullest extent before taking a break and review or proceeding along a conventional path.

By row scanning, we are able to fill up the rest two **cells with digit 8 first** and then **the only remaining digit 4** in the 9-cell square easily. This again is another tactic -** that is to examine what digits are left in the favorable zone.**

Turning our **attention to the column C9 we notice it to be a very favorable zone** and quickly fill up the rest three empty cells **with 5 first by scanning the effect of row R8** on the bottom right 9-cell square for 5.

**As the column C8 has a 5 in it**, the valid cell **for 5 turns out to be R8C9.** Now it has become easy to fill up the valid cells for 6 and 9, the remaining two digits, as cells **R7C9 and R6C9 by scanning the two rows R6 and R7. These rows are alternately occupied by 9 and 6. **

At this stage the column C9 is completely filled up.

Being aware that in the bottom 9-cell row we have got now two placements of 6 in two rows R7 and R9, we cross check with column C1 and **box in the cell R8C2 for a valid place of 6**.

**Can you say how we got the 4 in the cell R3C1?**

Again we changed direction and swept the first 9-cell square column for 4. R2C1 and R3C1 could have had a 4 but row R2 already has a 4. So the **valid cell for a 4 is R3C1.**

### Fifth stage

In the fifth stage we proceed at a bit leisurely pace.

In this stage **we first place 4 in cell R7C4 by boxing it in** and then at last put another 4 as the first digit in the central 9-cell completely unoccupied square by **cross scanning on both sides by two rows and two columns containing a 4.**

Notice that **we are employing apparently new tactics** as we are proceeding towards the end.

### Sixth stage

In this stage you would find we have made a lot of progress, but **to mark where we have started, we have colored the cell R9C7 containing a 4 in blue.**

we got this valid cell for 4 by first scanning the adjacent rows R7 and R8 in the bottom 9-square row and finally eliminating the cell R9C8 from consideration for 4 by scanning the column C8. We found a 4 in the C8 column as well. That was the start at this stage.

Next we put a 2 in valid cell R8C4 as it is boxed in by the presence of 2 in column C5 and row R9. As a chain effect still examining the bottom 9-cell row, we find the last two rows R8 and R9 have a 2 in each and thus **in cell R7C2 we find valid 2 boxed in or trapped. Similarly in cell R8C2 we get 6 trapped by a 6 in adjacent rows R7 and R9 each and also a 6 in the cross column C1.**

Now **still examining the 9-cell square R7R8R9-C1C2C3 for 7**, we find **7 trapped in cell R8C1** as both the adjacent rows R7 and R9 have a 7 in them.

**Putting a 3 in the remaining cell R9C1 is just a formality.**

At the end of this fruitful stage, **we find a valid cell in ****R7C7 for 3** by row scanning of R8 and R9. We take a halt at this stage to regroup our thoughts and strategies.

### Seventh stage

In this stage we fill up just two cells, **a 2 in R4C6 by scanning columns C4 and C5 and rows R5 and R6.** We needed to scan four columns and rows for 2 in this case.

Boxing in **a 3 in cell R5C8 is easy** as both the adjacent columns C7 and C9 have a 3 in them.

Now we are very near to our final solution.

### Eighth and final stage

At this last stage let's analyze the position of various zones containing still to be filled up cells.

The **very first thing** we noticed is the inherent problem associated with the two pairs of cells, R8C5, R8C6 and R8C8, R9C8. These four cells will finally have only the two digits 8 and 9 that we can see plainly. But apart from that we can't make any headway with these four cells, even though the associated 9-cell squares were quite healthily occupied.

Next we look at the central 9-cell square and there also draw a blank. It is still too sparsely populated for finding a breakthrough valid cell.

We decide, **the valid cell we will find in any of the three 9-cell squares**, R1R2R3-C1C2C3, R4R5R6-C1C2C2 or R1R2R3-C4C5C6. This analysis of prospective course of action is important just like in a real life problem situation.

We **got our valid cell in R1C3** by examining **the digit 3**, trapping or boxing it in horizontally by the row R3 and vertically by the columns C1 and C2, each containing a 3.

Exercise:We could have found a different valid cell in a different zone. Can you find it? This is an exercise for you. You have to analyze the seventh stage game board position for finding the alternative starting cell at this eighth stage.

This new valid cell for 3 in R1C3 sealed the fate of the **next valid cell R3C3 for 6**, as a 6 each in row R2 and columns C1 and C2 trapped it for 6.

Gradually things are getting easier, as we quickly **identify valid cell R2C2 for 7**, leaving the two remaining cells in the first top right corner 9-cell square, R1C2 and R2C1 to be eligible for the remaining two digits 9 and 5.

#### A useful tactic

Under this situation, we just look at cell R1C2 and its containing row R1 searching for the presence of either 5 or 9. We find a 5, and so obviously this cell is meant to accept only 9. Remaining cell R2C1 takes in 5 and completes the 9-cell square.

#### Column occupancy

Column C1 is so heavily occupied now that only one cell is left and that is allotted for 9. The cell is R4C1.

At an advanced stage of play you should always be alert for this state to appear to get an easily available valid cell.

Looking up we find a similar situation with the first row and fill up the lone empty cell R1C4 with 6.

#### The usual process again

We trap the cell R2C5 for 3 by the row R3 and column C6 each containing a 3. Now cell R2C6 is trapped for 9. The remaining lone cell R3C5 receives a 5.

This incidence of a 9 in column C6 seals the fate of the two pairs of cells at the bottom we had identified fit only for 8 and 9. Quickly we fill them up suitably.

### Now it is just a formality

At this position the board is so heavily occupied by digits you should be able to fill all the remaining cells easily by yourself even if this is your first Sudoku game.

### A few important points to remember

Before we end let us jot down a few important points for your convenience in playing Sudoku.

#### Playing medium

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

If possible always play Sudoku in a spreadsheet program. 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 the hardest Sudoku game in the world.

The advantage will simply be in the ease of creating a good looking Sudoku game board from a book or from mobile, delete the mistakes easily, undoing mistakes in series till you return back to a correct previous position with certainty. Errors are frequent as strong concentration sometimes waver. It is human.

A good habit is to copy the whole board of a correct position to a new position alongside the old board and play on the new board. This way, later you would be able to analyze your progress through the game very conveniently. It is a great help in learning.

**You can use color to mark special cells.** You have perhaps noticed that **we have colored all our newly filled up cells as red.** This always keeps you informed about the old supplied filled up cells and the cells you have filled up.

In a single worksheet you can play and preserve many Sudoku boards along with your own notes if any.

The advantages are many.

#### Build up your own strategies

In this page we have led you by hand. That is not good for learning. Unless you venture into a jungle on your own you won't know its ways. In other words, if you are serious about playing Sudoku, you have to play lots of games yourself and always should be in a learning mode, identifying and noting the special strategies and techniques that you have found useful. This is as true in Sudoku as in learning Maths.

#### Checking correctness of a game board

Always try to check correctness of a game you have finished without looking at the answers. This will increase your feel about the playing board.

#### Don't try to skip difficulty levels

By this we mean, when you feel confident about a difficulty level of Sudoku, don't be rash enough to skip one higher level of difficulty to try a problem at two higher level of difficulty. By mistake we have done this once resulting in great hardships.

You should move up one level of difficulty when you are completely confident of the present level of games you are playing.

#### Don't spend too much time on Sudoku

This warning is important as we know for sure that this game is great but it is also very addictive. It might encroach into your valuable work or study time.

### 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 a brief 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.