Let's play Sudoku - Game 2
First we'll briefly introduce Sudoku and then start solving the second Sudoku game for absolute beginners.
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.
Last time together we solved our first Sudoku game board successfully while explaining what the game is all about and how to play it step by step. We will start today's session with a recap of Sudoku game playing.
Recap of Sudoku in brief
Today we will play the following Sudoku game board.
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 little step by each little step leaving no scope of any doubts about why and how we have filled up a particular square.
As a recap, this 81 cell board has some of the cells filled up with digits ranging from 1 to 9. In fact these are the only digits with which you can fill up an empty cell.
The binding compound rule is:
You must not repeat any digit in any column, in any row or in any 9-cell medium sized square bordered by thick lines.
The whole board has 9 such 9 cell squares. Each such 9 cell square is bordered with thick lines to show them as separate regions.
Ultimately your job is to fill up all the empty cells with digits 1 to 9 but without breaking the compound rule not even once. Then only we would say you have successfully completed the game.
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. Before going ahead further, you should now try to complete filling up of all the empty cells of the given game with digits 1 to 9 without violating any rule. If you are really stuck, then refer to the solution. This is the best way to learn.
For convenience, we will use the labels C1, C2, C3......C9 at the top for identifying the 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 9-cell square in the middle of the board is R4R5R6C4C5C6.
Primary objective at each step
The final objective at each step is to find a cell in which one and only one digit can be placed. That is what we call a valid cell.
With beginner level game boards such as this one, you will always be able to find a valid cell at each step.
With your filling up of every empty cell by a single possible digit following the stringent rules, the complexity of the game gets reduced by just a little bit.
In other words, if a game board comes with many cells filled up already, its difficulty level generally is not high. Conversely if you find a game board with very few cells filled up, finding the first valid cell itself may become a very difficult task.
We should mention here that this is a general criterion to measure the difficulty level of the game. It gives just an idea. It is not a metric or scientific measure of difficulty of a Sudoku game board. And at this stage we should be aware of this general rule only.
If you think a bit you will be able to appreciate that the chances will be more in your easily finding a valid cell where no digit except a single digit out of the 9 is possible, if the number of cells already filled up is more. As a particular digit may be placed only once in a row or column or a 9-cell square, the more the number of filled up digits, the less are the possibilities of placing a digit in a particular cell.
In fact that is the whole objective - to reduce the possibility of filling up a cell to exactly one digit.
Valid cell and Valid digit
I would repeat the definition of valid digit and the valid 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 put in the cell following the rules of the game.
It means, firstly in the containing row, column and the 9-cell square, there is no occurrence of the digit you are going to put in a cell, and also, with all other digits, each of those digits appears in either the row or the column or the 9-cell square containing the particular cell being examined for filling up. That leaves only one digit as valid for placement in the particular cell.
As before, the most important 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.
The first basic technique that we will have to apply constantly is the row column sweep or horizontal-vertical cross-scanning.
Row-column sweep or Cross-scanning
We look at contiguous (adjacent) three rows and columns of a 9-cell square zone and if in any such three rows or columns we find a digit to have appeared twice then these two columns or rows become invalid zones for the particular digit leaving only one row or column for it to be placed in, and that too in only one 9-cell square.
There will now be at most three empty such cells for placement of the digit. Now we change direction of scanning by 90 degrees and scan those three columns or rows for occurrence of the digit. If you are lucky you will eliminate by this process two out of three cells for placement of the digit being considered leaving a single valid cell where without any shred of doubt you can put the digit under consideration.
By this cross scanning you have not only been able to find the valid cell, but also ensured that the row, column or the 9-cell square containing the valid cell does not have any single occurrence of the digit you are considering to put in the valid cell.
A favorable zone may be a column or a row or a 9-cell square with high digit occupancy so that possibilities of filling up the empty cells in the zone is much less and so the possibility of getting a valid cell is high.
On the other hand, as a particular digit can appear in the whole board only 9 times, a digit with high occurrence in the board has a higher chance of helping you to find its rest of the valid cells (each digit finally has a particular valid cell in a row or a column or a 9-cell square). We call a digit with a high number of occurrence as a favorable digit.
Let's find the first valid cell - first stage
We observed that 7 has appeared in the top two rows making this zone as invalid for 7. Further examining we see that 7 occurs in the first two 9-cell squares at the top. So it must have a valid cell in the right top 9-cell square. Examining still further, we find the valid cell possibility to be further reduced to the two cells R3C7 and R3C9.
Now we cross-scan changing direction of scanning for digit 7 by 90 degrees to scan the column C7 and C9. As C9 has a 7 in it and C7 does not have a digit 7, the favorable cell R3C9 has now turned invalid leaving only R3C7 valid for 7. In no other cell you can put the digit 7 in the 9-cell square R1R2R3C7C8C9.
This cell colored light green is our first valid cell. We have colored the rows and columns invalid for 7 in light grey shade for ease of analysis.
We write 7 in the valid cell R3C7 and color it red to distinguish it from the digits that came with the game board.
Now we have to find the next valid cell.
This way the game proceeds.
Continuing with digit 7, we turn our attention to bottom three rows, shade the rows R7 and R9, and column C4 in grey identifying them invalid for 7. In this process we have been able to box-in or lock the cell R8C5 from four sides leaving no other option than to put 7 in it. This is cross-scanning again.
At the third stage, we have removed all the colors and started examining with a fresh mind.
In the second stage we have already seen that though only two more valid cells for 7 remains we cannot find one such with certainty now. So we turn our attention to the top three rows again and easily box-in the cell R2C7 for digit 1.
After getting a valid cell in R2C7 for 1, we find this 9-cell square to be heavily filled up and so is favorable.
We get digit 9 first in R1C9 by row scanning only (rows R2 and R3 invalid for 9 and so shaded). And lastly we apply the exclusion rule for placing 8 in R3C9.
Exclusion rule says,
Put the last 9th digit left in a column or row or a 9-cell square with the rest eight cells already filled up with eight digits.
We must always be aware of sudden appearance of such a most favorable zone for a valid cell.
Noticing that the column C9 is quite a bit filled up we focus our attention on this column only.
In this stage we apply an interesting and powerful technique that we call, Digit subset analysis technique.
We observe the column C9 to be favorably filled up by 6 digits leaving only the digits 2, 5 and 6 left to be filled up in the cells R6C9, R7C9 or R8C9. We mentally put each of these three digits in each of these cells and see in the containing row how many of this subset of 3 digits 2, 5 and 6 appear.
In row R8 we find 2 and 5 occurring leaving only 6 for the valid cell R8C9. This is what we call Digit subset analysis technique.
In advanced Sudoku games this technique is a frequently employed one.
We continue working in the bottom three rows.
In this stage we box-in cell R7C3 for 6 by invalid shading of rows R8, R9 and column C1.
Our tendency this session is to remain in the vicinity of the last cell filled up.
We get valid cell R8C3 for 1 by simple invalid shading or scanning of columns C1 and C2.
Continuing in the same 9-cell square we get R7C1 for 3 by simple row scanning of rows R8 and R9 and then R9C1 for 4 by exclusion. This 9-cell square is now filled up completely.
In this stage we got R7C4 for 1 by cross-scanning of column C5 and row R8 and R9. We get 5 in R7R9 by scanning rows R8R9 and column C7. R6R9 gets 2 by exclusion in column C9.
In this stage, first we get 4 in row R8 by noting that only digits 4 and 8 are left in the row and for 4 only the cell R8C8 is left (first two 9-cell squares already have a 4). In R8C4 the digit 8 goes in by exclusion. In both cases we have applied basically exclusion technique.
Remaining in the vicinity again, we detect R9C8 for 8 by simple scanning of only column C7 boxing-in the cell completely.
In this stage we go after digit 4 with a vengeance. First we trap cell R5C7 for 4 by scanning only column C8.
Then we get 4 in R1C2 by scanning columns C1 and C3 and row R2.
Next we get a 4 in R3C5 by scanning rows R1, R2 and column C6.
Lastly we locked cell R4C4 for 4 from by scanning R5, R6 and columns C5, C6.
In this stage we get 8 in R2C1, 8 in R5C5 and 1 in R4C6 all by cross-scanning, our most basic technique.
In this stage we could pin down 6 cells.
First, 6 in R2C2 by scanning R3 and C3. Second, 6 in R1C4 by scanning R2 and R3. Third, 2 in R1C3 by exclusion in row R1. Fourth, 6 in R5C6 by scanning R4, R6 and C4, fifth, 9 in R5C8 by scanning R4 and sixth, 5 in R4C8 by exclusion in right-middle 9 cell square.
R3C6 gets 2 first by scanning R2, then R2C5 gets 5 by scanning C6, next R3C3 gets 3 by scanning C1, next R3C1 gets 5 by exclusion in 9 cell square, then R5C3 gets 5 by scanning by scanning C1 and C2 and lastly R6C4 gets 5 by scanning R4, R5, C5 and C6.
This last stage is always easy and we won't explain it.
We would repeat our recommendation regarding 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 hardest Sudoku game in the world.
We love spreadsheet because of its abilities:
- to form perfect looking 81 cell Sudoku game board with conveniently put digits in cells,
- to write and erase content in any cell painlessly,
- in undoing a series of steps, should you detect an error at any step, We assure you errors do happen. Erasing with pencil and erasure is painful, undoing is so much better
- in copying and pasting a whole game situation to a new place just by the side of the previous stage. Analysis of progress through the game is possible in doing this,
- write special comments below the game board of how you solved a special barrier in finding a valid cell. This may help you to build a complete structure of Sudoku game playing system,
- in coloring rows, columns or a group of cells to highlight them. This coloring regions is a great facility in a spreadsheet program,
- in one spreadsheet, you can place the stages of one game side by side, stages of second game below it, and this way, in one spreadsheet only you can keep record of a large number of Sudoku game playing sessions for same difficulty of game,
- you can play games of different difficulty level in another spreadsheet.
The virtues of spreadsheet for playing Sudoku are practically endless.
Virtues of Sudoku
We consider Sudoku as one of the best brain games for everyone with every ability because,
- Sudoku game boards are available in every level of difficulty, but at each level it poses enough challenge to the player at that ability level to keep itself interesting,
- it allows enhancement of mind power, especially in discovering new useful patterns that lies at the core of problem solving ability,
- it allows gradual progress through the level of difficulties,
- for even a child a new version of Sudoku game can be a great resource to teach the young the beauty of discovering new patterns and thus nurture creativity, We will present this version later,
- varieties of Sudoku game boards are inexhaustible, you can never finish it
- for the aged who still want to flex the muscles of their mind at their leisure, there is no friend like the Sudoku.
There are much more to Sudoku than this short list of pros of Sudoku playing that we could think of now.
The only con that we can think of is, the game is addictive, but what of it? Is there any great thing in this world that is not addictive?
Now 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.
Sudoku for beginners learning and practice sessions
Sudoku for beginners 2 - Learn and practice Sudoku level 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.