12th Sudoku puzzle at 3rd level of hardness and its solution explained
This is the 12th puzzle session at third level of hardness.The hardness of this puzzle arose primarily from the need to use Digit subset analysis (or DSA) and Cycle technique frequently for valid digit identification for a cell.
Though the problem didn't pose a real hard bottleneck that needed special efforts to break though, for a large portion of solving process, careful analysis was needed. Solution of each step is explained fully.
The game below is followed by three sections on Strategies and techniques that you may skip if you so desire.
At the end are the links of all our Sudoku game plays.
The 12th Sudoku puzzle at third level hardness
The following is the Sudoku puzzle that should intensively engage your mind for some time.
You may go through the next three sections for learning strategies and techniques for solving Sudoku puzzles in brief. Or, you may skip.
Following these three concept sections, the solution of the puzzle is explained step by step in details. Please spend your time fruitfully on the game before going through the solutions.
Overall strategy adopted and techniques used
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.
Hardness level being higher now, easy breaks by row-column scan are not many in the beginning.
So the 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 the next 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.
The last resort of populating each empty cell with valid digit subsets was not needed to be taken.
A faster approach of DS analysis for favorable cells in favorable zones that have fairly large number of digits could produce results consistently.
Full DS pouplation of each empty cell is a time-consuming process, and so DS population wherever needed was adopted and it worked.
Digit lockdown or other advanced techniques were not needed.
The main strategy for solving a hard Sudoku puzzle is to use the technique that would produce best results fastest. Easy to say, not so easy to do—comes with practice.
Structure and use of a cycle
Form of a cycle:
In a Cycle the digits involved are locked within the few cells forming the cycles, they 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) generally simplifying the situation and occasionally providing a breakthrough.
Use of a cycle:
In the example of cycle 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.
This is how a new valid cell is broken through which otherwise we were not able to find out in any other way.
In any hard Sudoku game solution, creating, analyzing and using the structure 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 3,5 and 9 eliminated 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 only the left out digit 1 of the DS [1,3,5,9] can occupy the cell R4C1.
This is how we identify a valid cell by Digit Subset Analysis.
You may also refer to our first and second game play sessions at level 2 where we first explained use of a cycle and DSA.
Let us solve the puzzle now.
The Sudoku 12th puzzle at third level of hardness
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 still better—created in a spreadsheet.
The first valid cell R7C5 is easily identified for filling by digit 9 by cross-scan of column C6 and row R9. 9 in these two leaves the only cell R7C5 in bottom middle small square eligible for getting digit 9. This first fill in a stage is colored turquoise blue as a convention.
Next 2 direct fills by cross-scan are—
R9C4 3 scan R7, R8, C5, C6 -- R5C7 3 scan R6, C8, C9.
The next 2 fills are by DSA or digit subset analysis followed by formation of 1st cycle—
In central square, R6C4 9 by cancellation of digits 5, 6 and 8 in DS [5,6,8,9] in the cell by cross-scan of R6 and C4 -- R5C4 5 by cancellation of 6, 8 from DS [5,6,8] in the cell by scan of C4 again -- Cycle (6,8) in R4C6, R5C6 by exclusion.
The 2nd cycle formation and next fill—
Cycle (1,7) on column C4 in R2C4, R3C4 by scan R2, R3 cancelling 2 from DS [1,2,7] in these two cells of C4 -- R1C4 2 by exclusion in C4.
3rd joined cycle of (1,7) is formed giving a series of more fills—
Cycle (1,7) at R2C7 by scan R2, C7 -- R2C9 6 by scan of top right square and R2 -- R2C3 3 by cancellation of 5 DS [3,5] by scan C3 -- R2C1 5 by exclusion -- R9C2 5 scan C1, C3 -- R9C5 4 with cancellation of [1,2,5] in bottom middle square DS [1,2,4,5] by cross-scan of R9, C5 -- R3C5 6 by cancellation of 5 from DS [4,5] in C5 by scan R3 -- R1C5 5 by exclusion in C5 -- R1C6 4 by exclusion in top middle square.
We'll close this stage at this point and go over to the next stage for ease of explanation.
The results of these actions in the puzzle game board is shown below. Follow the green-colored fills done at this stage.
Third stage of solution: 3rd level Sudoku game puzzle 12
This stage starts with formation of three Cycles in quick succession started first by Digit subset analysis—
Cycle (7,8) formed at R1C7, R1C8 by DSA [6,7,8,9] in R1 and top right square DS scan --Cycle (6,9) formed at R1C2, R1C3 by exclusion in R1 -- Cycle (7,8) formed at R3C1, R3C2 by exclusion.
Now it is time for a series of valid fills—
R2C7 1 by cancellation of 7 in square because of cycle (7,8) -- R2C4 7 in R2 -- R3C4 1 -- R3C9 4 exclusion in square -- R4C1 3 by cross-scan C2, C3, R5, R6 -- R6C1 2 scan C2, C3, R5 -- R4C9 2 scan R5, R6, C8 -- R9C7 2 scan C8, C9, R8 -- R9C6 1 cancellation -- R6C6 5 cancellation -- R5C6 2 exclusion.
Lastly in this stage,
Cycle (4,8) formed at R6C7, R8C7 by scan R6, C7 and R8, C7.
The results are shown below. Follow the light blue colored cells filled at this stage.
We will get continuous number of fills till completion in next stage.
Final stage of solution: 3rd level Sudoku game puzzle 12
First two cells filled are,
As Cycle (4,8) locks 8 in the two cells, R1C7 7 by cancellation of 8 -- R1C8 8 exclusion.
The Cycle (4,8) helped to break through.
Next two cell filling needed a bit care,
R7C1 7 cancellation of [1,5] from DS [1,5,7] in R7 scan R7 -- R9C3 8 cancellation from DS [7,8] in R9 scan home square.
But now the fills are by cancellation or exclusion,
By cancellation R8C1 9 -- R8C3 1 -- R8C9 8 -- R8C7 4 -- R6C7 8 -- R9C8 7 by exclusion -- R7C8 1 by cancellation of 5 in row DS [1,5] -- R7C9 5 exclusion -- R6C9 by cancellation of 7 in DS [1,7] -- R5C9 7 exclusion -- R6C2 6 by cancellation of 6 in DS [4,6] in R6 scan C2 -- R6C8 4 exclusion -- R3C1 8 by cancellation -- R3C2 7 cancellation -- R5C1 4 exclusion -- R1C2 9 cancellation -- R1C3 6 exclusion -- R4C2 8 by DS [1,8] cancellation in R4 -- R5C2 1 exclusion -- R5C3 9 cancellation of 7 from C3 DS [7,9] scan R5 -- R4C3 7 exclusion -- R5C8 6 cancellation of 9 from C8 DS [6,9] by scan R5 -- R5C6 8 by cancellation -- R4C6 6 exclusion -- R4C8 9 by exclusion.
The final solved board is shown below. Follow the dark blue colored cells filled in this final stage.
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 four steps (steps 4, 5, 6 and 7) for finally solving the problem.
As this Sudoku problem is large but not so complex, you didn't have to stop at step 4 to collect detailed DS information of all empty cells. Breakthroughs could be achieved in a continuous stream based on optimal DS analysis.
Nevertheless level of hardness being high in this puzzle, careful attention to details was needed near till the last stage.
Checkout and tally these seven steps with the process of solving this puzzle.
This is what we call the life cycle of Problem Solving, an independent and enormously powerful overlay subject standing on its own.
Watch out for the next 3rd level Sudoku puzzle solution.
Other Sudoku game plays at third level hardness
Sudoku third level game play 12
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.
First and second level Sudoku games
Fourth level game plays
List of fourth level very hard Sudoku game plays are available at Fourth level Sudoku.