Let's play Sudoku - Game 3
Last time we solved our second Sudoku game board each little step by little step while explaining what the game is all about and how to play it in every small details.
After going through the last session, we hope that you have successfully solved the Sudoku game we have given you at the end as an exercise. But remember, playing only one or two Sudoku games is not enough. If you are serious about enjoying the game continued over a long time you must collect whatever beginner level games you can get from various sources and go on trying to solve those.
In this third session also we would have a recap but we will keep it brief.
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 a bit faster than before as you have already experienced Sudoku game playing to some extent.
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 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 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.
As you go on filling up empty cells, the number of empty cells gets reduced and it gets more and more easy to find a valid square with only one digit placement possibility. Near the end suddenly you can fill up all the remaining cells one after the other quickly, as possibility of digit placement in each cell is highly reduced at this stage.
Generally the more a Sudoku game board has cells filled up already, the easier it is to solve it. Conversely if you attempt a Sudoku game with lots of empty cells, finding a valid cell at each step may be very difficult.
But at this beginning stage we won't worry about that, as we know each game that we will solve, should not pose great difficulties.
Once you get fully comfortable with the basic techniques and strategies of finding a valid cell and valid digit, you would be ready for the next stage of difficulty. We will mention when we try a game of higher level of difficulty.
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 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, the valid cell and its valid digit have the characteristics that, the digit you are considering does not appear at all in any of the row, column and the 9-cell square that contain the valid cell. The valid cell is the only place where you can put the digit. Additionally, you can't put any other digit in the valid cell. This situation is unique for the particular digit placement.
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.
In Sudoku game playing, the start may always give you some trouble.
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 (or columns) of a 9-cell square zone (say R1, R2, R3 or C4, C5, C6 but not C3, C4 and C5 as the columns in this case cross over two sets of 9-cell squares) and if in any such three rows (or columns) we find a digit to have appeared twice then these two rows (or columns) are technically invalid zones for the particular digit. You can place the digit then only in the third row (or column) unccupied.
We have thus narrowed the possibility of the particular digit placement to one row.
But again, as the other two appearances of the digit are in two 9-cell squares, the digit can only appear in the third 9-cell square, thus leaving only three possible cells for it. We have narrowed the possibility of placement to three cells.
Now we change direction of scanning by 90 degrees and scan those three columns or rows for occurrence of the digit.
This is why we call this cross-scanning or row column sweep. If you scanned three rows first, finally you have to scan three columns.
If you are lucky you will eliminate by this process two out of three cells for placing the digit 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 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
A quick scan resulted in the most favorable situation at the start by blocking the cell R1C9 for the digit 4 from all sides. The rows R2, R3 have each a 4 as well as the columns C7, C8. These invalid zones for digit 4 are highlighted by grey shading. In the 9-cell square R1R2R3-C7C8C9 the digit 4 can be placed in only the particular cell R1C9 and nowhere else. This is the result we have got by row-column sweep or cross-scanning, our most basic technique of finding a valid cell.
Continuing our search for digit 4, next we box-in or lock the cell R5C6 for 4 by surrounding rows R4, R6 and the columns C4, C5. In this 9-cell square at the center of the board, the digit 4 has no other place to go except the green colored cell R5C6.
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.
The third valid cell of 4 in R9C1 could be identified again by cross scanning of rows R7, R8 each containing a 4, and the columns C2, C3 also each containing a 4.
Freezing the board at this stage, we copy it to a new place in our spreadsheet, remove the cell shadings and start analyzing fresh for finding the next valid cells.
At this stage we get the valid cell R2C6 for digit 1 by easy scanning of columns C4 and C5 each containing a 1. 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 1.
In the same way by two column scan we get the valid cell R9C5 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 2 and 6. We examine the first empty cell R1C5 for placing either 2 or 6 and find a 2 in this row already existing. Thus, cell R1C5 becomes the valid cell for 6.
The only remaining digit 2 goes into the only remaining cell R5C5. This technique we call as the exclusion technique.
Again we apply the digit subset analysis technique on remaining three empty cells of column C6 for digits 9, 2 and 8. As in the containing row of cell R4C6 we have 2 and 8, out of three possibilities 9, 2 and 8, only 9 can put in this cell. Thus R4C6 gets 9. We have colored it blue to indicate application of a technique different from cross-scanning.
Turning our attention to row R1 we find only two empty cells and two remaining digits 3 and 8. As column C7 has a 8, R1C7 must have 3 and R1C8 must have 8.
We will freeze the board at this stage and go over to the next 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 R8C4 for 9 by scanning a single column C6. Next we get R5C9 for 3 by cross-scanning of rows R4, R6 and columns C7, C8. This placement of 3 enabled us to get cell R6C9 for 8 by cross-scanning of columns C7, C8 and row R4. Again we get valid cell R5C7 for 5 by scanning a single row R4 and a single column C8.
We freeze the board here and go over to the next stage.
First cell we get is R8C1 for digit 3 by row scanning of R7, R9. Next cell we get is R9C2 for 9 by cross-scanning of column C3 and row R7.
Now we don't find any easy cross-scanning possibility and take recourse to the powerful technique of digit subset analysis on row R8. This row has only three digits 1, 2, 8 left. We examine the possibility of putting one of these in each of the three empty cells in row R8 and look vertically to see which of these three digits are already there in the intersecting column.
This digit subset analysis technique analyzes the remaining digits and empty cells for suitability in a row or column of interest. But while inspecting suitability we must look across the row or column we are considering, to check which of these digits in the subset are already there. The digit that does not appear in the cross row or column gets its place in the empty cell being examined.
This is a very powerful technique and usually we get out of a spot of difficulty using this technique.
Applying this technique now on row R8, we find in a moment examining R8C8 for suitability of digits 1, 2 amd 8, that the column C8 has 2 and 8. This fixes the cell R8C8 for digit 1. As we have used a special technique we have colored this cell blue.
Immediately we look right to fix the cell R8C9 for digit 2 and column C9 has a 8 in it. The last digit 8 goes into the remaining cell R8C6 by Exclusion rule which 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.
Turning our attention to now favorable zone column C7, we get cell R3C7 for 1, as row R3 has the 6 out of leftover digits 1 and 6 in column C7. By exclusion rule, R7C7 gets 6, and then R7C9 digit 9.
Appearance of this 9 fixes cell R9C2 for 9 by cross-scanning. You may be able to see it how. Digit 8 goes into R8C6 by exclusion in row R8.
We will freeze at this stage and go over to the next stage. This technique of stopping at a point after getting a good number of successes has the advantage of starting the new stage with a fresh mind and ideas. Copying the whole board beside the previous board in the spreadsheet and removing all the shading are the cleaning process after the ceremony during the previous night.
At the start we get 9 in R3C8 easily. Now as the board is quite well filled up things should be getting easier. Then 6 and 5 in succession in cells R2C9 and R3C9.
Next in 9-cell square R1R2R3-C4C5C6, cell R2C4 for 5 and R3C4 for 8 are easy.
We have forgotten the potential 2 crying for its only available cell R7C6. This immediately fixes the cell R9C3 for 2.
In 9-cell square R4R5R6-C7C8C9, we apply the digit subset analysis. Two digits 1 and 6 are left and two cells R4C9 and R5C8 are left. Column C9 has a 6, and so R4C9 gets digit 1, thus fixing the cell R5C8 for 6 and across the board cell R4C1 for 6 again, by exclusion.
So we can see, we can apply the powerful digit subset analysis technique not only in a heavily filled up row or column, we can very well apply this golden technique in a suitable 9-cell square also.
It is time for freezing this stage and start a new stage from where we left.
In the 9-cell square R1R2R3-C1C2C3 we get the cell R2C2 for 3 by scan of columns C1 and C3, then R2C1 for 8 by singlw row scan of R3 and R3C1 for 2 by one column scan of C3 and R3C3 for 7 by exclusion.
We go over to the next stage.
First we fix the cell R6C1 for 7 by digit subset analysis applied on column C1 examining for digits 1, 7, 9 in each empty cells R5C1, R6C1 and R7C1. The row R6 has 1 and 9 in it and so the cell R6C1 must have 7 in it. This fixes cell R5C2 for 8 by two surronding column scan, and then 9 in R5C1 by single scan of column C3 and finally 1 in R5C3 by exclusion.
All the rest four digits we get by exclusion.
We would repeat now our recommendation regarding 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.
Other Sudoku game plays at absolute beginner level
Sudoku beginner level game play 3
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.