Sudoku is a mathematical puzzle whose birthplace is considered to be rising sun- Japan. Time flies with this incredibly exciting and educational mystery. The article will provide ways, methods and strategies on how to solve Sudoku.
History of the game name
Oddly enough, Japan is not the birthplace of the game. In fact, the puzzle was invented by the famous mathematician Leonhard Euler in the 18th century. From the course of higher mathematics, many should remember the famous “Euler circles”. The scientist was fascinated by the fields of combinatorics and propositional logic; he called his squares of various orders “Latin” and “Greco-Latin”, since he mainly used letters to compose them. But the puzzle gained real popularity after regular publication in the Japanese magazine Nikoli, where it received the name Sudoku in 1986.
What does a riddle look like?
The puzzle is a square field with dimensions of 9 by 9 cells. Depending on the complexity and type of the puzzle, the computer leaves a given number of square cells filled. Sometimes beginners are interested in the question: “How many variations of a puzzle can you make?”
According to the rules of combinatorics, the number of permutations can be found by calculating the factorial of the number of elements. So, Sudoku uses numbers from 1 to 9, which means it is necessary to calculate the factorial of 9. With some simple calculations we get 9! = 1*2*3*4*5*6*7*7*9 = 362,880 - options various combinations lines. Next, you need to use the matrix permutation formula and calculate the number of possible positions of rows and columns. The calculation formula is quite complicated; you just need to point out that by replacing only one column/row triple, you can increase the total number of options by 6 times. Multiplying the values we get 46,656 - ways of permutations in the riddle matrix for only 1 combination. It is not difficult to guess that the final number will be 362,880 * 46,656 = 16,930,529,280 game options - decide not to overdecide.
However, according to Bertham Felgenhauer's calculations, the puzzle has many more solutions. Bertham's formulas are very complex, but they give a total number of permutations of 6,670,903,752,021,072,936,960 options.
Rules of the game
The rules of Sudoku vary depending on the type of puzzle. But all options have in common the requirement of classical Sudoku: numbers from 1 to 9 should not be repeated vertically and horizontally of the field, as well as in each selected three-by-three section.
There are other types of games, such as odd-even, diagonal, windoku, girandole, area and Latin sudoku. In Latin, letters of the Latin alphabet are used instead of numbers. The even-odd variant should be solved like a regular Sudoku, only taking into account the multi-colored areas. Cells of one color should contain even numbers, and cells of the second color should contain odd numbers. In the diagonal puzzle to classical rules“vertical, horizontal, three by three” two more diagonals of the field are added, in which there should also be no repetitions. A variation of the area is a type of colored Sudoku that lacks the three-by-three divisions of the classic type of game. Instead, using color or bold borders, arbitrary areas of 9 cells are selected in which numbers must be placed.
How to solve Sudoku correctly?
The main rule of the riddle is: there is only one correct number for each cell of the field. If you select the wrong number at some stage, further decision will become impossible. The numbers will begin to repeat vertically and horizontally.
The simplest example of a statement is a situation with 8 known numbers horizontally, vertically, or in a three-by-three area. The ways to solve Sudoku in this case are obvious - enter the missing number of the sequence from 1 to 9 into the required square. In the example in the image above, this will be the number 4.
Sometimes two cells of a three-by-three area remain unfilled. In this case, each cell has two possible options filling, but only one is correct. You can make the right choice by considering empty areas not only as part of the area, but also as part of the vertical and horizontal. For example, in a three-by-three square, 2 and 3 are missing. You need to select one cell and consider the vertical and horizontal intersection of which it is. Let's say there is already one 3 vertically, but both sequences are missing 2. Then the choice is obvious.
Entry-level riddles are difficult, as a rule, they provide the opportunity to fill several cells with the only correct values at once. You just need to carefully examine the playing field. But the choice of methods/methods for solving Sudoku is not always so simple.
What does "predetermined choice" mean in Sudoku?
Sometimes the choice is not the only one, but nevertheless predetermined. Let's call this number “unique candidate”. Finding such an arrangement of numbers on the puzzle field is not difficult, but it will require some experience in solving the puzzle. An example of how to correctly solve Sudoku with a unique candidate is described in detail for the playing field option in the image below.
At first glance, the highlighted red square could contain any number except 5. However, in fact, the unique candidate for the location is the number 4. It is necessary to consider all the verticals and horizontals of the three-by-three area in question. So, in verticals 2 and 3 there are fours, which means 4 of the small field can be in one of the three squares of the first column. The top square is already occupied by the number 5, the number of locations for the symbol 4 is reduced. It is also not difficult to find a four in the lower horizontal line of the area, therefore, out of 3 options for the location of the number, only one remains.
Search for a unique candidate on the playing field
The example considered was obvious, since there were simply no other numbers on the field. Finding a unique candidate in a particular puzzle is not easy. The playing field in the image below will serve as a clear example to explain the method of how to solve Sudoku by searching for a unique candidate.
Although the description of the solution option does not seem simple, its application in practice does not cause difficulties. A unique candidate is always sought in a specific three-by-three area. In this regard, the player is only interested in three verticals and three horizontals of the playing field. All others are considered unimportant and are simply discarded. In the example, you need to find the location of the unique candidate number 7 for the central region. The corner squares of the field in question are occupied by numbers, and the number 7 is already present in the central vertical. This means that the only possible squares for placing the unique candidate 7 are cells 1 and 3 of the middle row of the “three by three” area.
How to solve difficult Sudoku?
Each type of game has 4 difficulty levels. They differ in the number of digits in the initial version of the field. The more there are, the easier it is to solve Sudoku. As in other games, fans organize competitions and entire Sudoku championships.
The most complex versions of the game involve a large number of options for filling each cell. Sometimes there may be the maximum possible number - 8 or 9. In such situations, it is recommended to write down all the options in pencil along the edges and corners of the cell. Listing all combinations, with detailed study, can already help eliminate overlapping numbers and reduce the number of variations for a single cell.
Color Puzzle Solving Strategies
A more complex version of the game is color Sudoku riddles. Such puzzles are considered difficult due to the introduction of additional conditions. In fact, color is not only an element of complication, but also a kind of hint that should not be neglected when deciding. This also applies to the odd-even game.
But color can also be used when solving ordinary Sudoku, marking more likely cases of substitution. In the above picture of the puzzle, the number 4 can only be placed in the blue and orange squares, all other options are obviously wrong. Highlighting these areas will allow you to distract yourself from the number 4 and switch to searching for other values, but you won’t be able to completely forget about the cells.
Sudoku for kids
It may sound strange, but children love solving Sudoku. The game develops logic very well and imaginative thinking. Scientists have already proven that playing prevents the death of brain cells. People who regularly solve puzzles have more high level IQ.
For very young children who do not yet know numbers, variants of Sudoku with symbols have been developed. The riddle is absolutely semantically independent. Parents should definitely teach their kids to play Sudoku if they want to develop their children’s logic, concentration and thinking. The game is useful for maintaining mental abilities at any age. Researchers compare the effect of a puzzle on the human brain with the effect physical exercise for muscle development. Psychologists say that Sudoku relieves depression and helps treat dementia.
VKontakte Facebook Odnoklassniki
For those who like to solve Sudoku puzzles on their own and slowly, a formula that allows you to quickly calculate the answers may seem like an admission of weakness or cheating.
But for those who find solving Sudoku too much effort, this could literally be the perfect solution.
Two researchers developed mathematical algorithm, which allows you to solve Sudoku very quickly, without guessing and backtracking.
Complex network researchers Zoltan Torozkay and Maria Erksi-Ravaz of the University of Notre Dame were also able to explain why some Sudoku puzzles are more difficult than others. The only downside is that you need a PhD in mathematics to understand what they offer.
Can you solve this puzzle? It was created by mathematician Arto Incala and is claimed to be the hardest Sudoku in the world. Photo from nature.com
Torozkay and Erksi-Ravaz began analyzing Sudoku as part of their research into optimization theory and computational complexity. They say that most Sudoku enthusiasts use a "brute force" approach based on guessing techniques to solve these problems. Thus, Sudoku fans arm themselves with a pencil and try all possible combinations of numbers until the correct answer is found. This method will inevitably lead to success, but it is labor-intensive and time-consuming.
Instead, Torozkay and Erksi-Ravaz proposed a universal analog algorithm that is completely deterministic (does not use guesswork or brute force) and always finds the correct solution to the problem, and quite quickly.
The researchers used a "deterministic analog solver" to complete this sudoku puzzle. Photo from nature.com
The researchers also found that the time it took to solve a puzzle using their analog algorithm correlated with the difficulty level of the task as judged by humans. This inspired them to develop a ranking scale for the difficulty of a puzzle or problem.
They created a scale from 1 to 4, where 1 is “easy,” 2 is “moderately difficult,” 3 is “difficult,” and 4 is “very difficult.” A puzzle rated 2 takes on average 10 times longer to solve than a puzzle rated 1. According to this system, the most complex riddle of the known ones still has a rating of 3.6; More complex Sudoku problems are not yet known.
The theory begins by mapping the probabilities for each individual square. Photo from nature.com
"I wasn't interested in Sudoku until we started working on more general class feasibility of Boolean problems, says Torozkay. - Since Sudoku is part of this class, the 9th order Latin square turned out to be a good test field for us, which is how I got to know them. I, and many researchers who study such problems, are fascinated by the question of how far we humans can go in solving Sudoku, deterministically, without brute force, which is a choice at random, and if the guess is wrong, we need to go back a step or several steps back and start over. Our analogue decision model is deterministic: there is no random selection or return."
Chaos Theory: The degree of difficulty of the puzzles is shown here as chaotic dynamics. Photo from nature.com
Torozkay and Erksi-Ravaz believe that their analog algorithm has the potential to be applied to the solution large quantity various tasks and problems in industry, computer science and computational biology.
The research experience also made Torozkai a big fan of Sudoku.
“My wife and I have several Sudoku apps on our iPhones, and we must have played them thousands of times by now, competing for the fastest time on each level,” he says. “She often intuitively sees combinations of patterns that I don’t notice.” I have to get them out. It becomes impossible for me to solve many of the puzzles that our scale categorizes as difficult or very difficult without writing down the probabilities in pencil.”
Torozkai and Erksi-Ravaz's methodology was first published in Nature Physics and later in Nature Scientific Reports.
- Tutorial
1. Basics
Most of us hackers know what Sudoku is. I won’t talk about the rules, but will go straight to the methods.To solve a puzzle, no matter how complex or simple, the cells that are obvious to fill are initially looked for.
1.1 " The Last Hero»
Let's look at the seventh square. There are only four free cells, which means something can be filled quickly.
"8
"on D3 blocks filling H3 And J3; exactly the same" 8
"on G5 closes G1 And G2
With a clear conscience we put " 8
"on H1
1.2 "The Last Hero" in line
After looking at the squares for obvious solutions, we move on to the columns and rows.
Let's consider " 4
" on the field. It is clear that it will be somewhere in the line A
.
We have" 4
"on G3 what's yawning A3, There is " 4
"on F7, cleaning A7. And one more" 4
" in the second square prohibits its repetition for A4 And A6.
"The Last Hero" for our " 4
" This A2
1.3 "No choice"
Sometimes there are multiple reasons for a particular location. " 4 " V J8 would be a great example.
Blue the arrows indicate that this is the last possible number in the square. Reds And blue the arrows give us last number in column 8 . Greens arrows give the last possible number in the line J.
As you can see, we have no choice but to put this " 4 "in place.
1.4 “Who else if not me?”
It is easier to fill in the numbers using the methods described above. However, checking the number as the last possible value also gives results. The method should be used when it seems that all the numbers are there, but something is missing.
"5 " V B1 is placed based on the fact that all numbers are from " 1 "before" 9 ", except " 5 " is in row, column and square (marked in green).
In the jargon it's " Naked loner". If you fill the field with possible values (candidates), then in the cell such a number will be the only possible one. By developing this technique, you can search for " Hidden singles" - numbers unique to a specific row, column or square.
2. "The Naked Mile"
2.1 "Naked" couples
""Naked" couple" - a set of two candidates located in two cells belonging to one common block: row, column, square.It is clear that right decisions puzzles will only be in these cells and only with these values, while all other candidates from the general block can be removed.
There are several "naked couples" in this example.
Red in line A cells highlighted A2 And A3, both containing " 1 " And " 6 "I don't know yet exactly how they are located here, but I can easily remove all the others." 1 " And " 6 " from line A(marked in yellow). Also A2 And A3 belong to the same square, so we remove " 1 " from C1.
2.2 "Threesome"
"Naked Threes"- a complicated version of “naked couples”.Any group of three cells in one block containing All in all three candidates is "naked threesome". When such a group is found, these three candidates can be removed from other cells in the block.
Combinations of candidates for "naked three" could be like this:
// three numbers in three cells.
// any combinations.
// any combinations. 
In this example everything is pretty obvious. In the fifth square of the cell E4, E5, E6 contain [ 5,8,9
], [5,8
], [5,9
] respectively. It turns out that in general these three cells have [ 5,8,9
], and only these numbers can be there. This allows us to remove them from other block candidates. This trick gives us a solution" 3
" for cell E7.
2.3 "The Fab Four"
"The Naked Four" very rare occurrence, especially in full form, and still produces results when detected. The logic of the solution is the same as in "naked threes".
In the above example, in the first square of the cell A1, B1, B2 And C1 generally contain [ 1,5,6,8
], so these numbers will only occupy these cells and no others. We remove candidates highlighted in yellow.
3. “Everything secret becomes clear”
3.1 Hidden pairs
A great way to expand the field is to search hidden pairs. This method allows you to remove unnecessary candidates from the cell and allow the development of more interesting strategies.
In this puzzle we see that 6 And 7 is in the first and second squares. Besides this 6 And 7 is in the column 7 . Combining these conditions, we can state that in cells A8 And A9 There will be only these values and we will remove all other candidates.
A more interesting and complex example hidden pairs. The pair [ 2,4 ] V D3 And E3, cleaning 3 , 5 , 6 , 7 from these cells. Highlighted in red are two hidden pairs consisting of [ 3,7 ]. On the one hand, they are unique for two cells in 7 column, on the other hand - for the row E. Candidates highlighted in yellow are removed.
3.1 Hidden triplets
We can develop hidden couples to hidden triplets or even hidden fours. Hidden threesome consists of three pairs of numbers located in one block. Such as, and. However, as is the case with "naked threesomes", each of the three cells does not have to contain three numbers. Will work total three numbers in three cells. For example , , . Hidden Threes will be masked by other candidates in the cells, so you first need to make sure that troika applicable to a specific block.
In this complex example there are two hidden threesomes. The first one, marked in red, in the column A. Cell A4 contains [ 2,5,6 ], A7 - [2,6 ] and cell A9 -[2,5 ]. These three cells are the only ones that can contain 2, 5 or 6, so those are the only ones that will be there. Therefore, we remove unnecessary candidates.
Second, in the column 9
. [4,7,8
] are unique to cells B9, C9 And F9. Using the same logic, we remove candidates.
3.1 Hidden fours
Great example hidden fours. [1,4,6,9 ] in the fifth square can only be in four cells D4, D6, F4, F6. Following our logic, we remove all other candidates (marked in yellow).
4. “Non-rubber”
If any of the numbers appears twice or thrice in the same block (row, column, square), then we can remove that number from the conjugate block. There are four types of pairing:
- Pair or Three squared - if they are located on one line, then you can remove all other similar values from the corresponding line.
- Pair or Three in a square - if they are located in one column, then you can remove all other similar values from the corresponding column.
- Pair or Three in a row - if they are located in one square, then you can remove all other similar values from the corresponding square.
- Pair or Three in a column - if they are located in one square, then you can remove all other similar values from the corresponding square.
4.1 Pointing pairs, triplets
Let me show you this puzzle as an example. In the third square" 3
"is only in B7 And B9. Following the statement №1
, we remove candidates from B1, B2, B3. Likewise, " 2
" from the eighth square removes a possible value from G2.
A special puzzle. Very difficult to solve, but if you look closely, you can notice several pointing pairs. It is clear that it is not always necessary to find them all in order to advance in the solution, but each such find makes our task easier.
4.2 Reducing the irreducible
This strategy involves carefully analyzing and comparing rows and columns with the contents of the squares (rules №3 , №4 ).
Consider the line A. "2 "are possible only in A4 And A5. Following the rule №3 , remove " 2 " their B5, C4, C5.
Let's continue solving the puzzle. We have a single location " 4 " within one square in 8 column. According to the rule №4 , we remove unnecessary candidates and, in addition, get a solution" 2 " For C7.
Sudoku is an interesting puzzle for training logic, unlike scanword puzzles, which require erudition and memory. Sudoku has many countries of origin, one way or another, it was played in Ancient China, in Japan, North America... In order for you and me to learn the game, we have made a selection How to solve Sudoku from easy to difficult.
To begin with, let's tell you that Sudoku is a square measuring 9x9, which in turn consists of 9 squares measuring 3x3. Each square must be filled with numbers from one to nine so that each number is used only once along a vertical and horizontal line, and only in a 3x3 square.
When you fill in all the cells, you should have all the numbers from 1 to 9 in each of the 9 squares. So, along the horizontal line all the numbers are from 1 to 9. And along the vertical line the same thing, see the picture:
It would seem simple rules, but in order to answer the question of how to solve Sudoku, and even more so, if you want to know how to solve complex Sudoku (especially for those who are just starting their journey), you need to solve at least a couple of easy problems. Then it will be clear what we are talking about. Below are the games. Try printing them out and filling them out so that everything fits together:
How to solve difficult Sudoku
I hope you have read the text above and solved the task that you need in order to understand what will be discussed next. If yes, then let's continue.
This part of the article will answer the questions:
How to solve difficult Sudoku?
How to solve Sudoku: methods?
How to solve Sudoku: methods and methods of cells and fields?
So, you were given two games, by solving which you acquired skills and received general idea. In order to save your time, I will tell you a couple of life hacks for quickly solving Sudoku.
1. Always start with number 1 and go first along the lines and then along the squares. This way you definitely won’t get confused and will prevent yourself from making many mistakes.
2. Always check which number is missing where there are fewer empty cells left. This will save time. And be sure to pay attention to how many and what numbers are missing in the 3 by 3 square (both horizontal and vertical lines).
3. If there are a lot of empty cells in a square and you reach a dead end, try dividing the square along lines in your mind. Think about what numbers may be there, and from this you can understand what numbers will be on the same lines in other squares (and perhaps even understand what numbers will be in other squares on another line).
4. Don’t be afraid of anything, it’s better to make a mistake and understand why than to do nothing!
5. More practice and you will become a master.
And if people who solve Sudoku also have abstract intelligence, which gives powerful potential to its owner, then one can move far forward. Read more about such people.
Below you will find a selection of “How to solve difficult Sudokus”, after which you will be able to do a lot!
In previous articles, we looked at different approaches to solving problems using Sudoku puzzles as examples. The time has come to try, in turn, to illustrate the capabilities of the considered approaches using a fairly complex example of problem solving. So, today we will start with the most “incredible” version of Sudoku. Please look at the terminology and preliminary information, otherwise it will be difficult for you to understand the content of this article.
Here is the information I found about this super complex option on the Internet:
University of Helsinki professor Arto Inkala claims (2011) that he created the world's most difficult Sudoku crossword puzzle. He spent three months creating this complex puzzle.
According to him, the crossword puzzle he created cannot be solved using logic alone. Arto Incala argues that even the most experienced players It will take at least a few days to decide. The professor’s invention was called AI Escargot (AI – the initials of the scientist, Escargot – from the English “snail”).
To solve this difficult problem, according to Arto Incala, you need to keep eight sequences in your head at the same time, unlike ordinary puzzles, where you need to remember one or two sequences.
Well, “sequences of searches” – this still smacks of a machine version of problem solving, and those who solved Arto Incal’s problem with their own brains talk about it differently. Someone solved it for a couple of months, someone announced that it only took 15 minutes. Well, the world chess champion could probably cope with the task in such a time, and a psychic, if such a thing lives on our plane, perhaps even faster. And someone who accidentally picked up several on the first try could quickly solve the problem. lucky numbers to fill empty cells. Let's say, one out of a thousand problem solvers might be similarly lucky.
So, about brute force: if you successfully choose two or three correct digits, then you may not need to brute force eight sequences (which means dozens of options). This was my thought when I decided to begin solving this problem. To begin with, I, having already been prepared within the framework of the methods of previous articles, decided to forget about what I knew so far. There is such a technique that the search for a solution should proceed freely, without schemes and ideas imposed on it. And the situation was new for me, so I needed to look at it in a new way. I have placed (in Excel) the original table (on the right) and the work table, the meaning of which I already had the opportunity to talk about in my first article about Sudoku:
Let me remind you that the worksheet contains pre-allowed combinations of numbers in initially empty cells.
After the usual almost routine processing of tables, the situation became a little simpler:
I began to study this situation. Well, since I’ve already forgotten how exactly I solved this problem a few days earlier, I’m starting to think about it anew. First of all, I paid attention to the two numbers 67 in the cells of the fourth block and combined them with the mechanism of rotation (movement) of cells, which I talked about in the previous article. After going through all the options for rotating the first three columns of the table, I came to the conclusion that numbers 6 and 7 cannot be in the same column and cannot rotate asynchronously; during the rotation process, they can only follow one another. Also, if you look closely, the seven and four seem to move synchronously along all three columns. Therefore, I make a plausible assumption that the number 7 should be placed in the lower left cell of block 4, and the number 6 in the upper right cell, respectively.
But for now I accept this result only as a possible guideline for testing other options. And I pay main attention to the number 59 in the cell of the 4th block. There can be either the number 5 or 9. Nine promises to destroy a lot of extra numbers, i.e. simplify the further course of solving the problem, and I start with this option. But quite quickly I reach a “dead end”, i.e. Then I have to make some choice again and who knows how long my choice will be checked. I guess if there really was a nine at one time the right choice, then Incala would hardly have left such an obvious option in plain sight, although the mechanism of his program could have allowed such a blunder. In general, one way or another, I decided to first thoroughly check the option with the number 5 in the cell with the number 59.
But later, when I solved the problem, I, so to speak, to clear my conscience, nevertheless returned to the option with the number 9 in order to determine how long it would take to check it. It didn't take very long to check. When I had the number 6 in the upper right cell of block 4, as expected according to the pre-selected reference point, then the number 19 appeared in the right middle cell (6 out of 169 was removed). I chose the number 9 in this cell for further testing and quickly came to a contradictory result, i.e. the choice of nine is incorrect. Then I choose number 1 and again check what comes out of it.
At some point I come to the situation:
where again I have to make a choice - the number 2 or 8 in the upper middle cell of block 4. I check both options (2 and 8) and in both cases I end up with a contradictory (not meeting the Sudoku condition) result. So I could check the option with the number 9 in the middle bottom cell of block 4 from the very beginning and it wouldn’t take much time. But I still, as I already said, settled on the number 5 in the mentioned cell. This led me to the following result:
The location of the numbers 4 and 7 in the first three columns (columns) indicates that they rotate synchronously, which is what was actually expected when choosing the number 7 for the lower left cell of the 4th block. In this case, a two or a nine, whether any of them is the required number in the middle left cell of this block, must accordingly move asynchronously with the pair 4 and 7. Preference in in this case I gave the number 2, since it “promised” to eliminate many extra digits from the cell numbers and, accordingly, quick check the admissibility of this option. And nine quickly led to a dead end - it required the selection of new numbers. Thus, in the left middle cell of the block with the number 29, I put down, in my opinion, the more preferable number - 2. The result came out as follows:
Next, I had to once again make a semi-arbitrary choice: I chose two in the cell with the number 26 in the ninth block. To do this, it was enough to notice that 5 and 2 in the three lower lines rotate synchronously, since 5 did not rotate synchronously with either 1 or 6. True, 2 and 1 could also rotate synchronously, but for some reason - definitely not I remember - I chose 2 instead of the number 26, perhaps because this option, in my opinion, was quickly checked. However, there were already few options left, and it was possible to quickly check any of them. It was also possible, instead of the option with two, to assume that the numbers 7 and 8 rotate synchronously in the last three columns (columns), and from this it followed that in the upper left cell of the 9th block there could only be the number 8, which also leads to a quick solution to the problem .
It must be said that Arto Incal's task does not allow purely logical solution within the limits of possibilities ordinary person- this is how it is intended, but still allows you to notice some promising options for searching through possible substitutions of numbers and significantly reduce this search. Try to start the search from positions other than those in this article, and you will see that almost all options very quickly lead to a dead end and you need to make more and more new assumptions regarding the further selection of suitable substitutions of numbers. About two months ago I already tried to solve this problem, without the preparation that I described in previous articles. I checked ten options for her solution and abandoned further attempts. The last time, already being more prepared, I solved this problem for half a day or a little more, but at the same time thinking about the choice from my point of view of the most indicative options for readers and also with preliminary thinking about the text of the future article. And the final result of the solution was as follows:
Actually, this article does not have independent meaning, it is written only to illustrate how the acquired skills and theoretical considerations described in previous articles can solve quite complex problems. And the articles, let me remind you, were not about Sudoku, but about mechanisms for solving problems using Sudoku as an example. The subjects, as for me, are completely different. However, since Sudoku is of interest to many, I thus decided to draw attention to a more significant issue that concerns not Sudoku itself, but problem solving.
For the rest, I wish you success in solving all your problems.