13th Sudoku puzzle at 3rd level of hardness and its solution explained
This is the 13th puzzle session at third level of hardness. This is a difficult puzzle with multiple bottlenecks that needed to be overcome and broken through.
Digit subset analysis, 2, 3 and 4 cell Cycles and digit lockdown techniques are the primary techniques and resources used for overcoming the bottlenecks.
Solution of each step is explained fully.
You'll get links to all our Sudoku game plays at the end.
The 13th Sudoku puzzle
The following is the Sudoku puzzle that should intensively engage your mind for some time.
You may go through the next five sections for learning strategies and techniques for solving Sudoku puzzles in brief. Or, you may skip.
Following these five 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 filling each empty cell with valid digit subsets was needed to be taken early. Only then the possible breakthrough points could be discovered. Strategically for faster solution, it is better to delay this time consuming task as much as possible.
In this strategy a few of the cells of interest are DS filled and analyzed.
Use of digit lockdown technique indicated the hardness of the puzzle.
The main strategy should always be to adopt the easier and faster technique and path to the solution by looking for key patterns all the time. Digit lockdown, Cycles, Valid cell by DSA are some of the key patterns.
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.
On filling up of every empty cell DS or full DS evaluation
We have not discussed the filling up of every empty cell with their valid digit subsets or DSs.
Let us see this in a little detail.
For example, to evaluate the DS for cell R4C1, look at the row R6 with only three digits missing in it—3, 6 and 8. Now cross-scan column C1 to identify any digit among [3,6,8] appearing in the column.
So the valid digit subset or DS for the cell would be just two digits [3,6]—8 cancelled. None of these two digits appear in the home square.
Basically for evaluating the valid DS for a cell,
You have cross-scan the row and column as well as check against the home square digits to identify the missing digits that may fill the cell.
This is a tedious and error-prone process.
In solving a very hard Sudoku puzzle, there would be no option than to go through the full empty cell DS evaluation. But it should be done when it has to be done and as late as possible.
Two strategic approaches are adopted to minimize the overall work load in this process—
- First try to find valid digit find and fill as much as as possible so that the number of possible valid digits in empty cells as well as number of empty cells are reduced, and
- Identify promising zones to evaluate the DS of a few cells trying for a breakthrough and so reduce the full DS evaluation load.
The second is a dynamic approach that depends on your experience and skill in identifying promising zones.
Single digit lockdown and its use
Occasionally, after evaluating valid DSs for a large number of empty cells, you may find if you look closely, that,
A single digit appears only in the DSs of two or three cells in a 9 cell square, in a column or a row, and in no other DSs in the 9 cell square.
This is what we call as single digit lockdown.
If it happens in a row inside a 9 cell square, in no cell in the row outside the square the digit can appear.
And so you can eliminate all occurrences of the locked digit from the DSs in the row outside the 9 cell square. If you can do that, usually it would give you the much needed breakthrough. It is a very powerful structure.
As an example, if DSs in R9C7 and R9C9 in row R9 and in the bottom right 9 cell square, are [1,4,8] and [1,4] and digit 4 appears only in these two DSs in the 9 cell square, you know that the digit 4 is locked in R9 inside the bottom right 9 cell square.
Then if the DS in R9C1 is [3,4,7], happily delete the locked out 4 from this DS to reduce it to just [3,7].
You may think, what is the point of it, what would it achieve after all!
Well, in similar situation in the process of solving this very puzzle game, the reduced DS in R9C1 formed a cycle (3,7) in column C1 and helped to pinpoint a valid digit 4 in cell R2C1 and that started a deluge of valid cell finds. This, you would find in solving this puzzle, to be the major breakthrough point.
Let us solve the puzzle now.
The Sudoku 13th 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 R2C4 is easily identified for filling by digit 2 by scan of rows R1 and R3. 2 in these two leaves the only cell R2C2 in top middle small square eligible for getting digit 2. This first fill in a stage is colored turquoise blue as a convention.
Next five valid cells obtained by direct scans are—
R9C5 9 by cross-scan R7, C4, C6 -- R8C2 9 by cross-scan R7, R9, C3 -- R1C1 9 by cross-scan R2, C2, C3 -- R3C8 9 by cross-scan R1, R2, C7 -- R5C9 9 by cross-scan.
If you check and fill on your own Sudoku game board (either drawn on paper or better, in a spreadsheet), you would be able to check the progress and even may detect any mistakes we have made unknowingly.
Now we'll resort to valid cell identification by DSA (Digit Subset Analysis) scan, Cancellation, Exclusion and Cycle formation—
R6C9 2 by cancellation of [3,7] from DS [2,3,7] in R6 by scan C9 -- Cycle (3,7) in R6C1, R6C3 -- R4C1 6 by cancellation of [1,5,8] from square DS [1,5,6,8] scan R4, C1 -- R5C1 1 by cancellation of [5,8] from DS [1,5,8] against C1 -- R5C3 8 by cancellation of 5 against C3 -- R5C2 5 by exclusion -- R4C9 8 by cancellation of 3 of DS [3,8] in R4 scan C9.
And a few more to end this stage—
Cycle (2,3,7) in R5C4, R5C5, R5C6 -- Cycle (4,6) in R5C7, R5C8 row R5 -- R4C7 3 by exclusion -- Cycle (4,6) in R2C8 and R5C8 in column C8 -- R8C8 5 by cancellation of [3,5] in R8 from DS [2,3,5].
After the empty cell Digit subsets have nearly all been evaluated, discover two powerful puzzle solving structures. If you are with us, you may try on your own to identify these from the resulting board picture shown below.
First is the digit 4 locked in two cells R9C7 and R9C9 of R9. The digits are colored red.
And the second is the 4 cell Cycle (3,4,6,7) in R2 cells R2C1, R2C3, R2C7, R2C8.
We prefer to use the digit locked resource and will show the effect of it in next stage. The reason of this choice lies in its potentially large number of valid cell fill capacity that is assessed by looking a few steps ahead.
Stage 3 of Sudoku puzzle solution: 3rd level game play 13
As the digit 4 is locked away in bottom right square, the cell R9C1 cannot have 4 in its valid digit subset. Its DS becomes [3,7] and immediately creates a cycle (3,7) with R6C1.
This is the main breakthrough and turning point in the puzzle. The result is—
R2C1 4 as the digits [3,7] are locked in the cycle below the cell in column C1.
The next series of valid cell identifications are—
R2C8 6 by cancellation -- R5C8 4 by cancellation -- R5C7 6 by cancellation -- R2C7 7 by cancellation -- R2C3 3 by cancellation of 8 in DS [3,8] -- R2C2 8 by exclusion -- R6C3 7 by cancellation -- R6C1 3 by exclusion -- R9C1 7 by exclusion -- R8C3 1 by cancellation -- R7C2 3 by cancellation -- R7C3 4 by exclusion in left bottom square.
And the last few valid digit fills in this stage—
R1C3 6 by exclusion in C3 -- R8C7 8 by cancellation -- R8C6 7 by cancellation -- R8C9 6 by exclusion.
The results are shown below.
Final stage of Sudoku puzzle solution: 3rd level puzzle 13
Discover in the above figure that in R7 digit 1 appears only in DS of R7C4. There is no other valid place in the row for digit 1. So digit 1 must be placed in R7C4. This is the second breakthrough. It is not so difficult compared to the first, but not so easy to spot either.
As a result, all the rest of the empty cells could be filled easily and quickly—
R9C4 8 by cancellation -- R9C6 2 by cancellation -- R7C6 5 by cancellation -- R7C5 6 by exclusion -- R7C8 2 by exclusion -- R9C8 3 by exclusion -- R3C5 7 by cancellation -- R1C5 5 by cancellation -- R5C5 2 by exclusion -- R5C6 3 by cancellation -- R5C4 7 by exclusion -- R1C4 3 by cancellation of cross-scan R1, C4 -- R3C4 6 by exclusion -- R3C2 1 by cancellation of cross-scan R3, C2 -- R1C2 7 by exclusion -- R3C7 4 by cancellation -- R3C6 8 by exclusion -- R1C6 4 by exclusion in C6 -- R1C9 1 by exclusion -- R9C9 4 by exclusion in C9 -- R9C7 1 by exclusion.
The final solved game is shown below.
Check for the validity of the solution if you need.
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 and complex, we have to stop at step 3 to collect detailed DS information of all empty cells.
The primary breakthrough was using digit 1 lockdown, and later the secondary breakthrough, by identifying the lone 1 in a cell in R7.
Usually, this second situation is equivalent to identify by cross-scans. But in this case, no cross-scan would have revealed its presence and that's why we name this special situation and use as an independent technique—lone digit identification.
At the point of the primary breakthrough, two options were open, the digit lockdown use or the 4 cell Cycle use. The first option has been chosen by looking ahead a few steps for each of the two options. This is analysis of consequence of making a decision at the present point of time (or Consequence analysis). In chess game play, consequence analysis forms an integral part of the ability to succeed.
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 13
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.