How to solve sudoku easier. Problem Solving Example - The Hardest Sudoku
It often happens that you need something to occupy yourself, entertain yourself - while waiting, or on a trip, or simply when there is nothing to do. In such cases, a variety of crosswords and scanwords can come to the rescue, but their minus is that the questions are often repeated there and remembering the correct answers, and then entering them “on the machine” is not difficult for a person with a good memory. Therefore, there is an alternative version of crossword puzzles - this is Sudoku. How to solve them and what is it all about?
What is Sudoku?
Magic square, Latin square - Sudoku has a lot of different names. Whatever you call the game, its essence will not change from this - this is a numerical puzzle, the same crossword puzzle, only not with words, but with numbers, and compiled according to a certain pattern. Recently, it has become a very popular way to brighten up your leisure time.
The history of the puzzle
It is generally accepted that Sudoku is a Japanese pleasure. This, however, is not entirely true. Three centuries ago, the Swiss mathematician Leonhard Euler developed the Latin Square game as a result of his research. It was on its basis that in the seventies of the last century in the United States they came up with numerical puzzle squares. From America, they came to Japan, where they received, firstly, their name, and secondly, unexpected wild popularity. It happened in the mid-eighties of the last century.
Already from Japan, the numerical problem went to travel the world and reached, among other things, Russia. Since 2004, British newspapers began to actively distribute Sudoku, and a year later, electronic versions of this sensational game appeared.
Terminology
Before talking in detail about how to solve Sudoku correctly, you should devote some time to studying the terminology of this game in order to be sure of the correct understanding of what is happening in the future. So, the main element of the puzzle is the cage (there are 81 of them in the game). Each of them is included in one row (consists of 9 cells horizontally), one column (9 cells vertically) and one area (square of 9 cells). A row may otherwise be called a row, a column a column, and an area a block. Another name for a cell is a cell.
A segment is three horizontal or vertical cells located in the same area. Accordingly, there are six of them in one area (three horizontally and three vertically). All those numbers that can be in a particular cell are called candidates (because they claim to be in this cell). There can be several candidates in the cell - from one to five. If there are two of them, they are called a pair, if there are three - a trio, if four - a quartet.
How to solve Sudoku: rules
So, first, you need to decide what Sudoku is. This is a large square of eighty-one cells (as mentioned earlier), which, in turn, are divided into blocks of nine cells. Thus, there are nine small blocks in total in this large Sudoku field. The player's task is to enter numbers from one to nine in all Sudoku cells so that they do not repeat either horizontally or vertically, or in a small area. Initially, some numbers are already in place. These are hints given to make it easier to solve Sudoku. According to experts, a correctly composed puzzle can only be solved in the only correct way.
Depending on how many numbers are already in Sudoku, the degrees of difficulty of this game vary. In the simplest, accessible even to a child, there are a lot of numbers, in the most complex there are practically none, but that makes it more interesting to solve.
Varieties of Sudoku
The classic type of puzzle is a large nine-by-nine square. However, in recent years, various versions of the game have become more and more common:
Basic solution algorithms: rules and secrets
How to solve Sudoku? There are two basic principles that can help solve almost any puzzle.
- Remember that each cell contains a number from one to nine, and these numbers should not be repeated vertically, horizontally and in one small square. Let's try by elimination to find a cell, only in which it is possible to find any number. Consider an example - in the figure above, take the ninth block (lower right). Let's try to find a place for the unit in it. There are four free cells in the block, but one cannot be placed in the third in the top row - it is already in this column. It is forbidden to put a unit in both cells of the middle row - it also already has such a figure, in the area next door. Thus, for this block, it is permissible to find a unit in only one cell - the first in the last row. So, acting by the method of elimination, cutting off extra cells, you can find the only correct cells for certain numbers both in a specific area, and in a row or column. The main rule is that this number should not be in the neighborhood. The name of this method is "hidden loners".
- Another way to solve Sudoku is to eliminate extra numbers. In the same figure, consider the central block, the cell in the middle. It cannot contain the numbers 1, 8, 7 and 9 - they are already in this column. The numbers 3, 6 and 2 are also not allowed for this cell - they are located in the area we need. And the number 4 is in this row. Therefore, the only possible number for this cell is five. It should be entered in the central cell. This method is called "loners".
Very often, the two methods described above are enough to quickly solve a Sudoku.
How to solve Sudoku: secrets and methods
It is recommended to adopt the following rule: write small in the corner of each cell those numbers that could be there. As new information is obtained, the extra numbers must be crossed out, and then in the end the correct solution will be seen. In addition, first of all, you need to pay attention to those columns, rows or areas where there are already numbers, and as many as possible - the fewer options left, the easier it is to handle. This method will help you quickly solve Sudoku. As experts recommend, before entering the answer into the cell, you need to double-check it again so as not to make a mistake, because because of one incorrectly entered number, the whole puzzle can “fly”, it will no longer be possible to solve it.
If there is such a situation that in one area, one row or one column in any three cells, it is permissible to find the numbers 4, 5; 4, 5 and 4, 6 - this means that in the third cell there will definitely be the number six. After all, if there were a four in it, then in the first two cells there could only be five, and this is impossible.
Below are other rules and secrets on how to solve Sudoku.
Locked Candidate Method
When you work with any one particular block, it may happen that a certain number in a given area can only be in one row or in one column. This means that in other rows/columns of this block there will be absolutely no such number. The method is called “locked candidate” because the number is, as it were, “locked” within one row or one column, and later, with the advent of new information, it becomes clear exactly in which cell of this row or this column this number is located.
In the figure above, consider block number six - the center right. The number nine in it can only be in the middle column (in cells five or eight). This means that in other cells of this area there will definitely not be a nine.
Method "open pairs"
The next secret, how to solve Sudoku, says: if in one column / one row / one area in two cells there can be only two any identical numbers (for example, two and three), then they are located in no other cells of this block / row / column will not. This often makes things a lot easier. The same rule applies to the situation with three identical numbers in any three cells of one row/block/column, and with four - respectively, in four.
Hidden Pair Method
It differs from the one described above in the following way: if in two cells of the same row/region/column, among all possible candidates, there are two identical numbers that do not occur in other cells, then they will be in these places. All other numbers from these cells can be excluded. For example, if there are five free cells in one block, but only two of them contain the numbers one and two, then they are exactly there. This method works for three and four numbers/cells as well.
x-wing method
If a specific number (for example, five) can only be located in two cells of a certain row/column/region, then that is where it is located. At the same time, if in the adjacent row/column/area the placement of a five is permissible in the same cells, then this figure is not located in any other cell of the row/column/area.
Difficult Sudoku: Solving Methods
How to solve difficult sudoku? The secrets, in general, are the same, that is, all the methods described above work in these cases. The only thing is that in complex sudoku situations are not uncommon when you have to leave logic and act by the “poke method”. This method even has its own name - "Ariadne's Thread". We take some number and substitute it in the right cell, and then, like Ariadne, we unravel the ball of threads, checking whether the puzzle will fit. There are two options here - either it worked or it didn't. If not, then you need to “wind up the ball”, return to the original one, take another number and try all over again. In order to avoid unnecessary scribbling, it is recommended to do all this on a draft.
Another way to solve complex sudoku is to analyze three blocks horizontally or vertically. You need to choose some number and see if you can substitute it in all three areas at once. In addition, in cases with solving complex Sudokus, it is not only recommended, but it is necessary to double-check all the cells, return to what you missed before - after all, new information appears that needs to be applied to the playing field.
Math Rules
Mathematicians do not remain aloof from this problem. Mathematical methods, how to solve Sudoku, are as follows:
- The sum of all the numbers in one area/column/row is forty-five.
- If three cells are not filled in some area / column / row, while it is known that two of them must contain certain numbers (for example, three and six), then the desired third digit is found using example 45 - (3 + 6 + S), where S is the sum of all filled cells in this area/column/row.
How to increase guessing speed?
The following rule will help you solve Sudoku faster. You need to take a number that is already in place in most blocks / rows / columns, and using the exclusion of extra cells, find cells for this number in the remaining blocks / rows / columns.
Game Versions
More recently, Sudoku remained only a printed game, published in magazines, newspapers and individual books. Recently, however, all sorts of versions of this game have appeared, such as board sudoku. In Russia, they are produced by the well-known company Astrel.
There are also computer variations of Sudoku - and you can either download this game to your computer or solve the puzzle online. Sudoku comes out for completely different platforms, so it doesn't matter what exactly is on your personal computer.
And more recently, mobile applications with the Sudoku game have appeared - both for Android and for iPhones, the puzzle is now available for download. And I must say that this application is very popular among cell phone owners.
- The minimum possible number of clues for a Sudoku puzzle is seventeen.
- There is an important recommendation on how to solve Sudoku: take your time. This game is considered relaxing.
- It is advised to solve the puzzle with a pencil, not a pen, so that you can erase the wrong number.
This puzzle is a truly addictive game. And if you know the methods of how to solve Sudoku, then everything becomes even more interesting. Time will fly by for the benefit of the mind and completely unnoticed!
I would like to say that Sudoku is a really interesting and exciting task, a riddle, a puzzle, a puzzle, a digital crossword, you can call it whatever you like. The solution of which will not only bring real pleasure to thinking people, but will also allow developing and training logical thinking, memory, and perseverance in the process of an exciting game.
For those who are already familiar with the game in all its manifestations, the rules are known and understood. And for those who are just thinking of starting, our information may be useful.
The rules of Sudoku are not complicated, they are found on the pages of newspapers or they can be easily found on the Internet.
The main points fit into two lines: the main task of the player is to fill in all the cells with numbers from 1 to 9. This must be done in such a way that none of the numbers is repeated twice in the column line and the 3x3 mini-square.
Today we bring you several options for electronic games, including more than a million built-in puzzle options in every game player.
For clarity and a better understanding of the process of solving the riddle, consider one of the simple options, the first level of Sudoku-4tune difficulty, 6 ** series.
And so, a playing field is given, consisting of 81 cells, which in turn make up: 9 rows, 9 columns and 9 mini-squares 3x3 cells in size. (Fig.1.)
Don't let the mention of the electronic game bother you in the future. You can meet the game in the pages of newspapers or magazines, the basic principle is preserved.
The electronic version of the game provides great opportunities for choosing the level of difficulty of the puzzle, the options for the puzzle itself and their number, at the request of the player, depending on his preparation.
When you turn on the electronic toy, key numbers will be given in the cells of the playing field. which cannot be transferred or modified. You can choose the option that is more suitable for the solution, in your opinion. Reasoning logically, starting from the figures given, it is necessary to gradually fill the entire playing field with numbers from 1 to 9.
An example of the initial arrangement of numbers is shown in Fig. 2. Key numbers, as a rule, in the electronic version of the game are marked with an underscore or a dot in the cell. In order not to confuse them in the future with the numbers that will be set by you.
Looking at the playing field. You need to decide what to start with. Typically, you want to define a row, column, or mini-square that has the minimum number of empty cells. In our version, we can immediately select two lines, upper and lower. In these lines, only one digit is missing. Thus, a simple decision is made, having determined the missing numbers -7 for the first line and 4 for the last, we enter them in the free cells of Fig.3.
The resulting result: two filled lines with numbers from 1 to 9 without repetition.
Next move. Column number 5 (from left to right) has only two free cells. After not much thought, we determine the missing numbers - 5 and 8.
To achieve a successful result in the game, you need to understand that you need to navigate in three main directions - a column, a row and a mini-square.
In this example, it is difficult to navigate only by rows or columns, but if you pay attention to the mini-squares, it becomes clear. You cannot enter the number 8 in the second (from the top) cell of the column in question, otherwise there will be two eights in the second mine-square. Similarly, with the number 5 for the second cell (bottom) and the second lower mini-square in Fig. 4 (not the correct location).
Although the solution seems to be correct for a column, nine digits in a column, without repetition, it contradicts the main rules. In mini-squares, numbers should also not be repeated.
Accordingly, for the correct solution, it is necessary to enter 5 in the second (top) cell, and 8 in the second (bottom). This decision is in full compliance with the rules. See Figure 5 for the correct option. 
Further solution, seemingly simple task, requires careful consideration of the playing field and the connection of logical thinking. You can again use the principle of the minimum number of free cells and pay attention to the third and seventh columns (from left to right). They left three cells empty. Having counted the missing numbers, we determine their values - these are 2.3 and 9 for the third column and 1.3 and 6 for the seventh. Let's leave the filling of the third column for now, since there is no certain clarity with it, unlike the seventh. In the seventh column, you can immediately determine the location of the number 6 - this is the second free cell from the bottom. What is the conclusion?
When considering the mini-square, which includes the second cell, it becomes clear that it already contains the numbers 1 and 3. From the digital combination we need 1,3 and 6, there is no other alternative. Filling in the remaining two free cells of the seventh column is also not difficult. Since the third row already has a filled 1 in its composition, 3 is entered into the third cell from the top of the seventh column, and 1 into the only remaining free second cell. For an example, see Figure 6.
Let's leave the third column for a clearer understanding of the moment. Although, if you wish, you can make a note for yourself and enter the proposed version of the numbers necessary for installation in these cells, which can be corrected if the situation is clarified. Electronic games Sudoku-4tune, 6** series allow you to enter more than one number in the cells, for a reminder.
We, having analyzed the situation, turn to the ninth (lower right) mini-square, in which, after our decision, there are three free cells left.
After analyzing the situation, you can notice (an example of filling a mini-square) that the following numbers 2.5 and 8 are not enough to completely fill it. Having considered the middle, free cell, you can see that only 5 of the required numbers fits here. Since 2 is present in the upper cell column, and 8 in the row in the composition, which, in addition to the mini-square, includes this cell. Accordingly, in the middle cell of the last mini-square, enter the number 2 (it is not included in either the row or column), and enter 8 in the upper cell of this square. Thus, we have completely filled the lower right (9th) mini- square with numbers from 1 to 9, while the numbers are not repeated in the columns or in the rows, Fig.7.
As the free cells are filled, their number decreases, and we are gradually approaching the solution of our puzzle. But at the same time, the solution of the problem can both be simplified and complicated. And the first way to fill the minimum number of cells in rows, columns or mini-squares ceases to be effective. Because the number of explicitly defined digits in a particular row, column, or mini-square is reduced. (Example: third column left by us). In this case, it is necessary to use the method of searching for individual cells, setting numbers in which there is no doubt.
In electronic games Sudoku-4tune, 6 ** series, the possibility of using hints is provided. Four times per game, you can use this function and the computer itself will set the correct number in the cell you have chosen. The 8** series models do not have this function, and the use of the second method becomes the most relevant.
Consider the second method in our example.
For clarity, let's take the fourth column. The unfilled number of cells in it is quite large, six. Having calculated the missing numbers, we determine them - these are 1,4,6,7,8 and 9. To reduce the number of options, you can take as a basis the average mini-square, which has a fairly large number of certain numbers and only two free cells in this column. Comparing them with the numbers we need, it can be seen that 1,6, and 4 can be excluded. They should not be in this mini-square to avoid repetition. It remains 7,8 and 9. Note that in the line (fourth from the top), which includes the cell we need, there are already numbers 7 and 8 from the three remaining ones that we need. Thus, the only option for this cell remains - this is the number 9, Fig. 8. The fact that all the numbers considered and excluded by us were originally given in the task does not raise doubts about the correctness of this solution. That is, they are not subject to any change or transfer, confirming the uniqueness of the number we have chosen to install in this particular cell.
Using two methods at the same time, depending on the situation, analyzing and thinking logically, you will fill in all the free cells and come to the correct solution to any Sudoku puzzle, and this riddle in particular. Try to complete the solution of our example in Fig. 9 yourself and compare it with the final answer shown in Fig. 10.
Perhaps you will determine for yourself any additional key points in solving puzzles, and develop your own system. Or take our advice, and they will be useful for you, and will allow you to join a large number of fans and fans of this game. Good luck.
In previous articles, we have considered different approaches to problem solving using examples of Sudoku puzzles. The time has come to try, in turn, to illustrate the possibilities of the considered approaches on a rather complicated example of problem solving. So, today we will start the most "incredible" variant of Sudoku. You, if you please, look at the terminology and preliminary information in, otherwise it will be difficult for you to understand the content of this article.
Here is what I found about this super-complex option on the Internet:
University of Helsinki professor Arto Inkala claims (2011) that he has created the world's most difficult Sudoku crossword puzzle. He created this most difficult puzzle for three months.
According to him, the crossword puzzle he created cannot be solved using logic alone. Arto Inkala claims that even the most experienced players will spend at least a few days on the solution. The professor's invention was called AI Escargot (AI - the initials of the scientist, Escargot - from the English "snail").
To solve this difficult task, 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, "brute force sequences" - it still smacks of a machine version of solving problems, and those who solved the Arto Incal problem with their own brains talk about it in different ways. Someone solved it for a couple of months, someone announced that it took only 15 minutes. Well, a world chess champion could probably do it in such a time, and a psychic, if there are any on our plane, probably even faster. And the one who accidentally picked up a few good numbers the first time to fill in the empty cells could also quickly solve the problem. Let's say one of the thousand solvers of the problem could be lucky in this way.
So, about enumeration: if you successfully choose two or three correct numbers, then it may not be necessary to sort through eight sequences (and these are dozens of options). This was my thought when I decided to start solving this problem. To begin with, being already prepared in the framework of the methods of previous articles, I 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 it was necessary to take a fresh look at it. I have arranged (in Excel) the original table (on the right) and the working table, the meaning of which I already had the opportunity to talk about in my first Sudoku article:
The worksheet, let me remind you, contains previously valid 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 have already forgotten how exactly I solved this problem a few days earlier, I begin to comprehend it in a new way. First of all, I paid attention to two numbers 67 in the cells of the fourth block and combined them with the mechanism of cell rotation (movement), 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 the numbers 6 and 7 cannot be in the same column and cannot rotate asynchronously, they can only follow one after the other during rotation. Also, if you look closely, the seven and four seem to move synchronously in all three columns. Therefore, I make a plausible assumption that the lower left cell of block 4 should contain the number 7, and the upper right cell, respectively, 6.
But for the time being, I accept this result only as a possible guideline in testing other options. And I pay the main attention to the number 59 in the cell of the 4th block. It can be either the number 5 or 9. Nine promises to destroy a lot of extra numbers, i.e. to simplify the further course of solving the problem, and I start with this option. But rather quickly I come to a "dead end", i.e. then you have to make some choice again and how to know how long my choice will be checked. My guess is that if the nine had ever really been the right choice, then Inkala would hardly have left such an obvious option in plain sight, although the mechanism of his program could have allowed such a lapse. 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 long to check. When I had the number 6 in the upper right cell of block 4, as it was supposed to be according to the previously selected landmark, 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 up with an inconsistent result, i.e. the choice of nine is not correct. Then I choose the number 1 and again check what comes of it.
At some point, I come to the situation:
where again you 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 would not take a lot of time. But I still, as I already said, stopped at 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 was actually assumed when choosing the number 7 for the lower left cell of the 4th block. At the same time, two or nine, whether any of them is the required digit in the middle left cell of this block, should move asynchronously to the pair 4 and 7, respectively. In this case, I gave preference to the number 2, since it "promised" to eliminate many extra digits from the numbers of cells and, accordingly, a quick check of the admissibility of this option. And the 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, not my opinion, the more preferable of the numbers - 2. The result came out as follows:
Then I had to once again make a semi-arbitrary choice, so to speak: I chose a deuce 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 rows 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 remember - I chose 2 instead of the number 26, perhaps because this option, in my opinion, was quickly tested. However, there were already few options left, and it was possible to quickly check any of them. It was also possible, instead of the variant with a deuce, to assume that the numbers 7 and 8 rotate synchronously in the last three columns (columns), and from this it followed that only the number 8 could be in the upper left cell of the 9th block, which also leads to a quick decoupling of the problem .
It must be said that the Arto Incal problem does not allow a purely logical solution within the capabilities of an ordinary person - this is how it is conceived - but still allows you to notice some promising options for enumeration of possible substitutions of numbers and significantly reduce this enumeration. Try to start the enumeration from positions other than 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 choice of suitable substitutions of numbers. About two months ago, I already tried to solve this problem without having the preparation that I described in previous articles. I checked ten options for her solution and left 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 considering the choice, from my point of view, of the most indicative options for readers and also with preliminary consideration of the text of the future article. And the final result is the following:
Actually, this article has no independent value, it is written only to illustrate how the acquired skills and theoretical considerations described in previous articles allow solving rather complex problems. And the articles were, let me remind you, not about Sudoku, but about the mechanisms for solving problems using Sudoku as an example. Items are completely different to me. However, since many people are interested in sudoku, I thus decided to draw attention to a more significant issue, not related to sudoku itself, but to problem solving.
As for the rest, I wish you success in solving all problems.
The first thing that should be determined in the methodology of problem solving is the question of actually understanding what we achieve and can achieve in terms of problem solving. Understanding is usually thought of as something that goes without saying, and we lose sight of the fact that understanding has a definite starting point of understanding, only in relation to which we can say that understanding really takes place from a specific moment we have determined. Sudoku here, in our consideration, is convenient in that it allows, using its example, to some extent to model the issues of understanding and solving problems. However, we will start with several other and no less important examples than Sudoku.
A physicist studying special relativity might talk about Einstein's "crystal clear" propositions. I came across this phrase on one of the sites on the Internet. But where does this understanding of "crystal clarity" begin? It begins with the assimilation of the mathematical notation of postulates, from which all multi-level mathematical constructions of SRT can be built according to known and understandable rules. But what the physicist, like me, does not understand is why the postulates of SRT work in this way and not otherwise.
First of all, the vast majority of those discussing this doctrine do not understand what exactly lies in the postulate of the constancy of the speed of light in the translation from its mathematical application to reality. And this postulate implies the constancy of the speed of light in all conceivable and inconceivable senses. The speed of light is constant relative to any resting and moving objects at the same time. The speed of the light beam, according to the postulate, is constant even with respect to the oncoming, transverse and receding light beam. And, at the same time, in reality we only have measurements that are indirectly related to the speed of light, interpreted as its constancy.
Newton's laws for a physicist and even for those who simply study physics are so familiar that they seem so understandable as something taken for granted and it cannot be otherwise. But, say, the application of the law of universal gravitation begins with its mathematical notation, according to which even the trajectories of space objects and the characteristics of orbits can be calculated. But why these laws work in this way and not otherwise - we do not have such an understanding.
Likewise with Sudoku. On the Internet, you can find repeatedly repeated descriptions of "basic" ways to solve Sudoku problems. If you remember these rules, then you can understand how this or that Sudoku problem is solved by applying the "basic" rules. But I have a question: do we understand why these "basic" methods work in this way and not otherwise.
So we move on to the next key point in problem solving methodology. Understanding is possible only on the basis of some model that provides a basis for this understanding and the ability to perform some natural or thought experiment. Without this, we can only have rules for applying the learned starting points: the postulates of SRT, Newton's laws, or "basic" ways in Sudoku.
We do not and in principle cannot have models that satisfy the postulate of the unrestricted constancy of the speed of light. We do not, but unprovable models consistent with Newton's laws can be invented. And there are such "Newtonian" models, but they somehow do not impress with productive possibilities for conducting a full-scale or thought experiment. But Sudoku provides us with opportunities that we can use both to understand the actual problems of Sudoku, and to illustrate modeling as a general approach to solving problems.
One possible model for Sudoku problems is the worksheet. It is created by simply filling in all the empty cells (cells) of the table specified in the task with the numbers 123456789. Then the task is reduced to the sequential removal of all extra digits from the cells until all the cells of the table are filled with single (exclusive) digits that satisfy the condition of the problem.
I'm creating such a worksheet in Excel. First, I select all the empty cells (cells) of the table. I press F5-"Select"-"Empty cells"-"OK". A more general way to select the desired cells: hold Ctrl and click the mouse to select these cells. Then for the selected cells I set the color to blue, size 10 (original - 12) and font Arial Narrow. This is all so that subsequent changes in the table are clearly visible. Next, I enter the numbers 123456789 into empty cells. I do it as follows: I write down and save this number in a separate cell. Then I press F2, select and copy this number with the Ctrl + C operation. Next, I go to the table cells and, sequentially bypassing all the empty cells, enter the number 123456789 into them using the Ctrl + V operation, and the worksheet is ready.
Extra numbers, which will be discussed later, I delete as follows. With the operation Ctrl + mouse click - I select cells with an extra number. Then I press Ctrl + H and enter the number to be deleted in the upper field of the window that opens, and the lower field should be completely empty. Then it remains to click on the "Replace All" option and the extra number is removed.
Judging by the fact that I usually manage to do more advanced table processing in the usual "basic" ways than in the examples given on the Internet, the worksheet is the most simple tool in solving Sudoku problems. Moreover, many situations concerning the application of the most complex of the so-called "basic" rules simply did not arise in my worksheet.
At the same time, the worksheet is also a model on which experiments can be carried out with the subsequent identification of all the "basic" rules and various nuances of their application arising from the experiments.
So, before you is a fragment of a worksheet with nine blocks, numbered from left to right and top to bottom. In this case, we have the fourth block filled with numbers 123456789. This is our model. Outside the block, we highlighted in red the "activated" (finally defined) numbers, in this case, fours, which we intend to substitute in the table being drawn up. The blue fives are figures that have not yet been determined regarding their future role, which we will talk about later. The activated numbers assigned by us, as it were, cross out, push out, delete - in general, they displace the same numbers in the block, so they are represented there in a pale color, symbolizing the fact that these pale numbers have been deleted. I wanted to make this color even paler, but then they could become completely invisible when viewed on the Internet.
As a result, in the fourth block, in cell E5, there was one, also activated, but hidden four. "Activated" because she, in turn, can also remove extra digits if they are on her way, and "hidden" because she is among other digits. If the cell E5 is attacked by the rest, except for 4, activated numbers 12356789, then a "naked" loner will appear in E5 - 4.
Now let's remove one activated four, for example from F7. Then the four in the filled block can be already and only in cell E5 or F5, while remaining activated in row 5. If activated fives are involved in this situation, without F7=4 and F8=5, then in cells E5 and F5 there will be a naked or hidden activated pair 45.
After you have sufficiently worked out and comprehended different options with naked and hidden singles, twos, threes, etc. not only in blocks, but also in rows and columns, we can move on to another experiment. Let's create a bare pair 45, as we did before, and then connect the activated F7=4 and F8=5. As a result, the situation E5=45 will occur. Similar situations very often arise in the process of processing a worksheet. This situation means that one of these digits, in this case 4 or 5, must necessarily be in the block, row and column that includes cell E5, because in all these cases there must be two digits, not one of them.
And most importantly, we now already know how frequently occurring situations like E5=45 arise. In a similar way, we will define situations when a triple of digits appears in one cell, etc. And when we bring the degree of comprehension and perception of these situations to a state of self-evidence and simplicity, then the next step is, so to speak, a scientific understanding of situations: then we will be able to do a statistical analysis of Sudoku tables, identify patterns and use the accumulated material to solve the most complex problems .
Thus, by experimenting on the model, we get a visual and even "scientific" representation of hidden or open singles, pairs, triples, etc. If you limit yourself to operations with the described simple model, then some of your ideas will turn out to be inaccurate or even erroneous. However, as soon as you move on to solving specific problems, the inaccuracies of the initial ideas will quickly come to light, but the models on which the experiments were carried out will have to be rethought and refined. This is the inevitable path of hypotheses and refinements in solving any problems.
I must say that hidden and open singles, as well as open pairs, triples and even fours, are common situations that arise when solving Sudoku problems with a worksheet. Hidden couples were rare. And here are the hidden triples, fours, etc. I somehow didn’t come across when processing worksheets, just like the methods for bypassing the “x-wing” and “swordfish” contours that have been repeatedly described on the Internet, in which there are “candidates” for deletion with any of the two alternative ways of bypassing contours. The meaning of these methods: if we destroy the "candidate" x1, then the exclusive candidate x2 remains and at the same time the candidate x3 is deleted, and if we destroy x2, then the exclusive x1 remains, but in this case the candidate x3 is also deleted, so in any case, x3 should be deleted , without affecting the candidates x1 and x2 for the time being. More generally, this is a special case of the situation: if two alternative ways lead to the same result, then this result can be used to solve a Sudoku problem. In this, more general, situation, I met situations, but not in the "x-wing" and "swordfish" variants, and not when solving Sudoku problems, for which knowledge of only "basic" approaches is sufficient.
The features of using a worksheet can be shown in the following non-trivial example. On one of the sudoku solver forums http://zforum.net/index.php?topic=3955.25;wap2 I came across a problem presented as one of the most difficult sudoku problems, not solvable in the usual ways, without using enumeration with assumptions about the numbers substituted in the cells . Let's show that with a working table it is possible to solve this problem without such enumeration:
On the right is the original task, on the left is the working table after the "deletion", i.e. routine operation of removing extra digits.
First, let's agree on notation. ABC4=689 means that cells A4, B4 and C4 contain the numbers 6, 8 and 9 - one or more digits per cell. It's the same with strings. Thus, B56=24 means that cells B5 and B6 contain the numbers 2 and 4. The ">" sign is a conditional action sign. Thus, D4=5>I4-37 means that due to the message D4=5, the number 37 should be placed in cell I4. The message can be explicit - "naked" - and hidden, which should be revealed. The impact of the message can be sequential (transmitted indirectly) along the chain and parallel (act directly on other cells). For example:
D3=2; D8=1>A9-1>A2-2>A3-4,G9-3; (D8=1)+(G9=3)>G8-7>G7-1>G5-5
This entry means that D3=2, but this fact needs to be revealed. D8=1 passes its action on the chain to A3 and 4 should be written to A3; at the same time, D3=2 acts directly on G9, resulting in G9-3. (D8=1)+(G9=3)>G8-7 – combined influence of factors (D8=1) and (G9=3) leads to the result G8-7. Etc.
The records may also contain a combination of type H56/68. It means that the numbers 6 and 8 are prohibited in cells H5 and H6, i.e. they should be removed from these cells.
So, we start working with the table and for a start we apply the well-manifested, noticeable condition ABC4=689. This means that in all other (except A4, B4 and C4) cells of block 4 (middle, left) and the 4th row, the numbers 6, 8 and 9 should be deleted:
Apply B56=24 in the same way. Together we have D4=5 and (after D4=5>I4-37) HI4=37, and also (after B56=24>C6-1) C6=1. Let's apply this to a worksheet:
In I89=68hidden>I56/68>H56-68: i.e. cells I8 and I9 contain a hidden pair of digits 5 and 6, which forbids these digits from being in I56, resulting in the result H56-68. We can consider this fragment in a different way, just as we did in experiments on the worksheet model: (G23=68)+(AD7=68)>I89-68; (I89=68)+(ABC4=689)>H56-68. That is, a two-way "attack" (G23=68) and (AD7=68) leads to the fact that only the numbers 6 and 8 can be in I8 and I9. Further (I89=68) is connected to the "attack" on H56 together with previous conditions, which leads to H56-68. In addition to this "attack" is connected (ABC4=689), which in this example looks redundant, however, if we worked without a working table, then the impact factor (ABC4=689) would be hidden, and it would be quite appropriate to pay attention to it specially.
Next action: I5=2>G1-2,G6-9,B6-4,B5-2.
I hope it is already clear without comments: substitute the numbers that come after the dash, you can't go wrong:
H7=9>I7-4; D6=8>D1-4,H6-6>H5-8:
Next series of actions:
D3=2; D8=1>A9-1>A2-2>A3-4,G9-3;
(D8=1)+(G9=3)>G8-7>G7-1>G5-5;
D5=9>E5-6>F5-4:
I=4>C9-4>C7-2>E9-2>EF7-35>B7-7,F89-89,
that is, as a result of "crossing out" - deleting extra digits - an open, "naked" pair 89 appears in cells F8 and F9, which, together with other results indicated in the record, we apply to the table:
H2=4>H3-1>F2-1>F1-6>A1-3>B8-3,C8-5,H1-7>I2-5>I3-3>I4-7>H4-3
Their result:
This is followed by fairly routine, obvious actions:
H1=7>C1-8>E1-5>F3-7>E2-9>E3-8,C3-9>B3-5>B2-6>C2-7>C4-6>A4-9>B4- 8;
B2=6>B9-9>A8-6>I8-8>F8-9>F9-8>I9-6;
E7=3>F7-5,E6-7>F6-3
Their result: the final solution of the problem:
One way or another, we will assume that we figured out the "basic" methods in Sudoku or in other areas of intellectual application on the basis of a model suitable for this and even learned how to apply them. But this is only part of our progress in problem solving methodology. Further, I repeat, follows not always taken into account, but an indispensable stage of bringing the previously learned methods to a state of ease of their application. Solving examples, comprehending the results and methods of this solution, rethinking this material on the basis of the accepted model, again thinking through all the options, bringing the degree of their understanding to automaticity, when the solution using the "basic" provisions becomes routine and disappears as a problem. What it gives: everyone should feel it on their own experience. And the bottom line is that when the problem situation becomes routine, the search mechanism of the intellect is directed to the development of more and more complex provisions in the field of the problems being solved.
And what is "more complex provisions"? These are just new "basic" provisions in solving the problem, the understanding of which, in turn, can also be brought to a state of simplicity if a suitable model is found for this purpose.
In the article Vasilenko S.L. "Numeric Harmony Sudoku" I find an example of a problem with 18 symmetric keys:
Regarding this task, it is stated that it can be solved using "basic" methods only up to a certain state, after reaching which it remains only to apply a simple enumeration with a trial substitution into the cells of some supposed exclusive (single, single) digits. This state (advanced a little further than in Vasilenko's example) looks like:
There is such a model. This is a kind of rotation mechanism for identified and unidentified exclusive (single) digits. In the simplest case, some triple of exclusive digits rotates in the right or left direction, passing by this group from row to row or from column to column. In general, at the same time, three groups of triples of numbers rotate in one direction. In more complex cases, three pairs of exclusive digits rotate in one direction, and a triple of singles rotate in the opposite direction. So, for example, the exclusive digits in the first three lines of the problem under consideration are rotated. And, most importantly, this kind of rotation can be seen by considering the location of the numbers in the processed worksheet. This information is enough for now, and we will understand other nuances of the rotation model in the process of solving the problem.
So, in the first (upper) three lines (1, 2 and 3) we can notice the rotation of the pairs (3+8) and (7+9), as well as (2+x1) with unknown x1 and the triple of singles (x2+4+ 1) with unknown x2. In doing so, we may find that each of x1 and x2 can be either 5 or 6.
Lines 4, 5 and 6 look at the pairs (2+4) and (1+3). There should also be a 3rd unknown pair and a triple of singles of which only one digit 5 is known.
Similarly, we look at rows 789, then the triplets of columns ABC, DEF and GHI. We will write down the collected information in a symbolic and, I hope, quite understandable form:
So far, we need this information only to understand the general situation. Think it over carefully and then we can move forward further to the following table specially prepared for this:
I highlighted the alternatives with colors. Blue means "allowed" and yellow means "prohibited". If, say, allowed in A2=79 allowed A2=7, then C2=7 is forbidden. Or vice versa – allowed A2=9, forbidden C2=9. And then permissions and prohibitions are transmitted along a logical chain. This coloring is done in order to make it easier to view different alternatives. In general, this is some analogy to the "x-wing" and "swordfish" methods mentioned earlier when processing tables.
Looking at the B6=7 and, respectively, B7=9 options, we can immediately find two points that are incompatible with this option. If B7=9, then in lines 789 a synchronously rotating triple occurs, which is unacceptable, since either only three pairs (and three singles asynchronously to them) or three triples (without singles) can rotate synchronously (in one direction). In addition, if B7=9, then after several steps of processing the worksheet in the 7th line we will find incompatibility: B7=D7=9. So we substitute the only acceptable of the two alternatives B6=9, and then the problem is solved by simple means of conventional processing without any blind enumeration:
Next, I have a ready-made example using a rotation model to solve a problem from the World Sudoku Championship, but I omit this example so as not to stretch this article too much. In addition, as it turned out, this problem has three solutions, which is poorly suited for the initial development of the digit rotation model. I also puffed a lot on Gary McGuire's 17-key problem pulled from the Internet to solve his puzzle, until, with even more annoyance, I found out that this "puzzle" has more than 9 thousand solutions.
So, willy-nilly, we have to move on to the "most difficult in the world" Sudoku problem developed by Arto Inkala, which, as you know, has a unique solution.
After entering two quite obvious exclusive numbers and processing the worksheet, the task looks like this:
The keys assigned to the original problem are highlighted in black and larger font. In order to move forward in solving this problem, we must again rely on an adequate model suitable for this purpose. This model is a kind of mechanism for rotating numbers. It has already been discussed more than once in this and previous articles, but in order to understand the further material of the article, this mechanism should be thought out and worked out in detail. Approximately as if you had worked with such a mechanism for ten years. But you will still be able to understand this material, if not from the first reading, then from the second or third, etc. Moreover, if you persist, then you will bring this "difficult to understand" material to the state of its routine and simplicity. There is nothing new in this regard: what is very difficult at first, gradually becomes not so difficult, and with further incessant elaboration, everything becomes the most obvious and does not require mental effort in its proper place, after which you can free your mental potential for further progress on the problem being solved or on other problems.
A careful analysis of the structure of Arto Incal's problem shows that the whole problem is built on the principle of three synchronously rotating pairs and a triple of asynchronously rotating pairs of singles: (x1+x2)+(x3+x4)+(x5+x6)+(x7+x8+ x9). The rotation order can be, for example, as follows: in the first three lines 123, the first pair (x1+x2) goes from the first line of the first block to the second line of the second block, then to the third line of the third block. The second pair jumps from the second row of the first block to the third row of the second block, then, in this rotation, jumps to the first row of the third block. The third pair from the third row of the first block jumps to the first row of the second block and then, in the same direction of rotation, jumps to the second row of the third block. A trio of singles moves in a similar rotation pattern, but in the opposite direction to that of pairs. The situation with columns looks similar: if the table is mentally (or actually) rotated by 90 degrees, then the rows will become columns, with the same character of movement of singles and pairs as before for rows.
Turning these rotations in our minds in relation to the Arto Incal problem, we gradually come to understand the obvious restrictions on the choice of variants of this rotation for the selected triple of rows or columns:
There should not be synchronously (in one direction) rotating triples and pairs - such triples, in contrast to the triple of singles, will be called triplets in the future;
There should not be pairs asynchronous with each other or singles asynchronous with each other;
There should not be both pairs and singles rotating in the same (for example, right) direction - this is a repetition of the previous restrictions, but it may seem more understandable.
In addition, there are other restrictions:
There must not be a single pair in the 9 rows that matches a pair in any of the columns and the same for columns and rows. This should be obvious: because the very fact that two numbers are on the same line indicates that they are in different columns.
You can also say that very rarely there are matches of pairs in different triples of rows or a similar match in triples of columns, and also there are rarely matches of triples of singles in rows and / or columns, but these are, so to speak, probabilistic patterns.
Research blocks 4,5,6.
In blocks 4-6, pairs (3+7) and (3+9) are possible. If we accept (3+9), then we get an invalid synchronous rotation of the triplet (3+7+9), so we have a pair (7+3). After substituting this pair and subsequent processing of the table by conventional means, we get:
At the same time, we can say that 5 in B6=5 can only be a loner, asynchronous (7+3), and 6 in I5=6 is a paragenerator, since it is in the same line H5=5 in the sixth block and, therefore, it cannot be alone and can only move in sync with (7+3.
and arranged the candidates for singles by the number of their appearance in this role in this table:
If we accept that the most frequent 2, 4 and 5 are singles, then according to the rules of rotation, only pairs can be combined with them: (7 + 3), (9 + 6) and (1 + 8) - a pair (1 + 9) discarded since it negates the pair (9+6). Further, after substituting these pairs and singles and further processing the table using conventional methods, we get:
Such a recalcitrant table turned out to be - it does not want to be processed to the end.
You will have to strain yourself and notice that there is a pair (7 + 4) in columns ABC and that 6 moves synchronously with 7 in these columns, therefore 6 is a pairing, so only combinations (6 + 3) are possible in column "C" of the 4th block +8 or (6+8)+3. The first of these combinations does not work, because then in the 7th block in column "B" an invalid synchronous triple will appear - a triplet (6 + 3 + 8). Well, then, after substituting the option (6 + 8) + 3 and processing the table in the usual way, we come to the successful completion of the task.
The second option: let's return to the table obtained after identifying the combination (7 + 3) + 5 in rows 456 and proceed to the study of columns ABC.
Here we can notice that the pair (2+9) cannot take place in ABC. Other combinations (2+4), (2+7), (9+4) and (9+7) give a synchronous triple - a triplet in A4+A5+A6 and B1+B2+B3, which is unacceptable. There remains one acceptable pair (7+4). Moreover, 6 and 5 move synchronously 7, which means they are steam-forming, i.e. form some pairs, but not 5 + 6.
Let's make a list of possible pairs and their combinations with singles:
The combination (6+3)+8 does not work, because otherwise, an invalid triple-triplet is formed in one column (6 + 3 + 8), which has already been discussed and which we can verify once again by checking all the options. Of the candidates for singles, the number 3 scores the most points, and the most likely of all the above combinations: (6 + 8) + 3, i.e. (C4=6 + C5=8) + C6=3, which gives:
Further, the most likely candidate for singles is either 2 or 9 (6 points each), but in any of these cases, candidate 1 (4 points) remains valid. Let's start with (5+29)+1, where 1 is asynchronous to 5, i.e. put 1 from B5=1 as an asynchronous singleton in all columns of ABC:
In block 7, column A, only options (5+9)+3 and (5+2)+3 are possible. But we better pay attention to the fact that in lines 1-3 the pairs (4 + 5) and (8 + 9) have now appeared. Their substitution leads to a quick result, i.e. to the completion of the task after the table has been processed by normal means.
Well, now, having practiced on the previous options, we can try to solve the Arto Incal problem without involving statistical estimates.
We return to the starting position again:
In blocks 4-6, pairs (3+7) and (3+9) are possible. If we accept (3 + 9), then we get an invalid synchronous rotation of the triplet (3 + 7 + 9), so for substitution in the table we have only the option (7 + 3):
5 here, as we see, is a loner, 6 is a paraformer. Valid options in ABC5: (2+1)+8, (2+1)+9, (8+1)+9, (8+1)+2, (9+1)+8, (9+1) +2. But (2+1) is asynchronous to (7+3), so there are (8+1)+9, (8+1)+2, (9+1)+8, (9+1)+2. In any case, 1 is synchronous (7 + 3) and, therefore, paragenerating. Let's substitute 1 in this capacity in the table:
The number 6 here is a paragenerator in bl. 4-6, but the conspicuous pair (6+4) is not on the list of valid pairs. Hence the quad in A4=4 is asynchronous 6:
Since D4+E4=(8+1) and according to the rotation analysis forms this pair, we get:
If cells C456=(6+3)+8, then B789=683, i.e. we get a synchronous triple-triplet, so we are left with the option (6+8)+3 and the result of its substitution:
B2=3 is single here, C1=5 (asynchronous 3) is a pairing, A2=8 is also a pairing. B3=7 can be both synchronous and asynchronous. Now we can prove ourselves in more complex tricks. With a trained eye (or at least when checking on a computer), we see that for any status B3=7 - synchronous or asynchronous - we get the same result A1=1. Therefore, we can substitute this value into A1 and then complete our, or rather Arto Incala, task by more usual simple means:
One way or another, we were able to consider and even illustrate three general approaches to solving problems: determine the point of understanding the problem (not a hypothetical or blindly declared, but a real moment, starting from which we can talk about understanding the problem), choose a model that allows us to realize understanding through a natural or mental experiment and - thirdly - to bring the degree of understanding and perception of the results achieved in this case to a state of self-evidence and simplicity. There is also a fourth approach, which I personally use.
Each person has states when the intellectual tasks and problems facing him are solved more easily than is usually the case. These states are quite reproducible. To do this, you need to master the technique of turning off thoughts. At first, at least for a fraction of a second, then, more and more stretching this disconnecting moment. I can’t tell further, or rather recommend, something in this regard, because the duration of the application of this method is a purely personal matter. But I resort to this method sometimes for a long time, when a problem arises in front of me, to which I do not see options for how it can be approached and solved. As a result, sooner or later, a suitable prototype of the model emerges from the storerooms of memory, which clarifies the essence of what needs to be resolved.
I solved the Incal problem in several ways, including those described in previous articles. And always in one way or another I used this fourth approach with switching off and subsequent concentration of mental efforts. I got the fastest solution to the problem by simple enumeration - what is called the "poke method" - however, using only "long" options: those that could quickly lead to a positive or negative result. Other options took more time from me, because most of the time was spent on at least a rough development of the technology for applying these options.
A good option is also in the spirit of the fourth approach: tune in to solving Sudoku problems, substituting only a single digit per cell in the process of solving the problem. That is, most of the task and its data are "scrolled" in the mind. This is the main part of the process of intellectual problem solving, and this skill should be trained in order to increase your ability to solve problems. For example, I am not a professional Sudoku solver. I have other tasks. But, nevertheless, I want to set myself the following goal: to acquire the ability to solve Sudoku problems of increased complexity, without a worksheet and without resorting to substituting more than one number into one empty cell. In this case, any way to solve Sudoku is allowed, including a simple enumeration of options.
It is no coincidence that I recall the enumeration of options here. Any approach to solving Sudoku problems involves a set of certain methods in its arsenal, including one or another type of enumeration. Moreover, any of the methods used in Sudoku in particular or in solving any other problems has its own area of its effective application. So, when solving relatively simple Sudoku problems, the most effective are the simple "basic" methods described in numerous articles on this topic on the Internet, and the more complex "rotation method" is often useless here, because it only complicates the course of a simple solution and at the same time what -does not provide new information that appears in the course of solving the problem. But in the most difficult cases, like Arto Incal's problem, the "rotation method" can play a key role.
Sudoku in my articles is just an illustrative example of approaches to problem solving. Among the problems I have solved, there are also an order of magnitude more difficult than Sudoku. For example, computer models of boilers and turbines located on our website. I wouldn't mind talking about them either. But for the time being, I have chosen Sudoku in order to show my young fellow citizens in a rather visual way the possible ways and stages of moving towards the ultimate goal of the problems being solved.
That's all for today.
Hello! In this article, we will analyze in detail the solution of complex Sudoku using a specific example. Before starting the analysis, we agree to call the small squares numbers, numbering them from left to right and from top to bottom. All the basic principles of solving Sudoku are described in this article.
As usual, we will first look at open singles. And there were only two such b5-5, e6-3. Next, we place possible candidates on all empty fields.
Candidates will be placed in small green print to distinguish them from the numbers already standing. We do this mechanically, simply sorting through all the empty cells and entering in them the numbers that can be in them.
The fruit of our labors can be seen in Figure 2. Let's turn our attention to the cell f2. She has two candidates 5 and 9. We will have to go with the guessing method, and in case of an error, return to this choice. Let's put number five. Let's remove the five from the candidates of row f, column 2 and square four.
We will constantly remove possible candidates after setting the number, and in this article we will no longer focus on that!
We look further at the fourth square, we have a tee - these are cells e1, d2, e3, which have candidates 2, 8 and 9. Let's remove them from the rest of the unfilled cells of the fourth square. Move on. In square six, the number five can only be on e8.
More at the moment there are no pairs, no tees, let alone fours. Therefore, let's go the other way. Let's go through all the verticals and horizontals in order to remove unnecessary candidates.
And so on the second vertical, the number 8 can only be on the cells -h2 and i2, let's remove the figure eight from the other unfilled cells of the seventh square. On the third file, the number eight can only be on e3. What we got is shown in Figure 3.
There is nothing more to grab on to. We got a pretty tough nut, but we'll crack it anyway! And so, consider again our pair e1 and d2, arrange it in this way d2-9, e1 -2. And in case of our mistake, we will return again to this pair.
Now we can safely write a deuce into the cell d9! And there are seven in the square, nine can only be on h1. After that, on the vertical 1, a five can only be on i1, which in turn gives the right to place a five on the h9 cell.
Figure 4 shows what we have done. Now consider the next pair, these are d3 and f1. They have candidates 7 and 6. Looking ahead, I will say that the arrangement variant d3-7, f1-6 is erroneous and we will not consider it in the article, so as not to waste time.
Figure 5 illustrates our work. What is left for us to do next? Of course, again go through the options for setting numbers! We put a triple in the cell g1. Save as always so you can come back. One is set on i3. now in the seventh square we get a pair of h2 and i2, with the numbers 2 and 8. This gives us the right to exclude these numbers from the candidates for the entire unfilled vertical.
Based on the last thesis, we arrange. a2 is a four, b2 is a three. And after that we can put down the entire first square. c1 - six, a1 - one, b3 - nine, c3 - two.
Figure 6 shows what happened. On i5 we have a hidden loner - the number three! And i2 can only have the number 2! Accordingly, on h2 - 8.
Now let's turn to the cells e4 and e7, this is a pair with candidates 4 and 9. Let's arrange them like this: e4 four, e7 nine. Now a six is placed on f6, and a nine is placed on f5! Further on c4 we get a hidden loner - the number nine! And we can immediately put four from 8, and then close the horizontal with: c6 eight.