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What does sudoku mean. How to solve complex sudoku using the example of diagonal sudoku

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. This the hardest puzzle he created 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 an inconsistent (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 to assume instead of the variant with a deuce 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 resolution 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 other positions 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.

• tutorial

1. Basics

Most of us hackers know what sudoku is. I will not talk about the rules, but immediately move on to the methods.
To solve a puzzle, no matter how complex or simple, cells that are obvious to fill are initially searched for.

1.1 " The last Hero»

Consider the seventh square. Only four free cells, so something can be quickly filled.
"8 " on the D3 blocks padding H3 and J3; similar " 8 " on the G5 closes G1 and G2
With a clear conscience we put " 8 " on the H1

1.2 "Last Hero" in a row

After viewing the squares for obvious solutions, move on to the columns and rows.
Consider " 4 " on the field. It is clear that it will be somewhere in the line A .
We have " 4 " on the G3 that covers A3, there is " 4 " on the F7, cleaning A7. And another one " 4 " in the second square prohibits its repetition on A4 and A6.
"The Last Hero" for our " 4 " this is A2

1.3 "No Choice"

Sometimes there are multiple reasons for a particular location. " 4 " in J8 would be a great example.
Blue the arrows indicate that this is the last possible number squared. Red and blue the arrows give us the last number in the column 8 . Greens the 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 "And who, if not me?"

Filling in numbers is easier to do using the methods described above. However, checking the number as the last possible value also yields results. The method should be used when it seems that all the numbers are there, but something is missing.
"5 " in B1 is set based on the fact that all numbers from " 1 " before " 9 ", Besides " 5 " is in the row, column and square (marked in green).

In jargon it is " naked loner". If you fill in the field with possible values ​​​​(candidates), then in the cell such a number will be the only possible one. Developing this technique, you can search for " hidden loners" - numbers unique for a particular row, column or square.

2. "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 the correct solutions of the puzzle will be only in these cells and only with these values, while all other candidates from the general block can be removed.


In this example, there are several "naked pairs".
red in line BUT cells are highlighted A2 and A3, both containing " 1 " and " 6 ". I don't know exactly how they are located here yet, but I can safely remove all the others " 1 " and " 6 " from string A(marked in yellow). Also A2 and A3 belong to a common 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 trio". When such a group is found, these three candidates can be removed from other cells of the block.

Candidate combinations for "naked trio" may 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 the solution " 3 " for cell E7.

2.3 "Fab Four"

"Naked Four" a very rare occurrence, especially in its full form, and yet produces results when detected. The solution logic is the same as "naked triplets".

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 occupy only those cells and no others. We remove the candidates highlighted in yellow.

3. "Everything hidden becomes clear"

3.1 Hidden pairs

A great way to open the field is to search hidden pairs. This method allows you to remove unnecessary candidates from the cell and give rise to more interesting strategies.

In this puzzle we see that 6 and 7 is in the first and second squares. Besides 6 and 7 is in the column 7 . Combining these conditions, we can assert that in the cells A8 and A9 there will be only these values ​​and we remove all other candidates.


More interesting and complex example hidden pairs. The pair [ 2,4 ] in 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 a row E. Candidates highlighted in yellow are removed.

3.1 Hidden triplets

We can develop hidden couples before hidden triplets or even hidden fours. The Hidden Three consists of three pairs of numbers located in one block. Such as, and. However, as in the case with "naked triplets", each of the three cells does not have to contain three numbers. will work Total three numbers in three cells. For example , , . Hidden triplets will be masked by other candidates in the cells, so first you need to make sure that troika applicable to a specific block.


In this complex example, there are two hidden triplets. The first, marked in red, in the column BUT. Cell A4 contains [ 2,5,6 ], A7 - [2,6 ] and cell A9 -[2,5 ]. These three cells are the only ones where there can be 2 , 5 or 6, so they will be the only ones there. Therefore, we remove unnecessary candidates.

Second, in a column 9 . [4,7,8 ] are unique to cells B9, C9 and F9. Using the same logic, we remove candidates.

3.1 Hidden fours

Perfect 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 appear 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:

1. Pair or Three in a square - if they are located in one line, then you can remove all other similar values ​​​​from the corresponding line.
2. 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.
3. Pair or Three in a row - if they are located in the same square, then you can remove all other similar values ​​​​from the corresponding square.
4. Pair or Three in a column - if they are located in the same square, then you can remove all other similar values ​​\u200b\u200bfrom 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.


Special puzzle. Very difficult to solve, but if you look closely, you can see a few 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 parsing and comparing rows and columns with the contents of the squares (rules №3 , №4 ).
Consider the line BUT. "2 "are possible only in A4 and A5. following the rule №3 , remove " 2 " them B5, C4, C5.


Let's continue to solve 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, we obtain the solution " 2 " for C7.

Game history

The numerical structure was invented in Switzerland in the 18th century; on its basis, a numerical crossword puzzle was developed in the 20th century. However, in the United States, where the game was directly invented, it did not become widespread, unlike Japan, where the puzzle not only took root, but also gained great popularity. It was in Japan that it acquired the familiar name "Sudoku", and then spread throughout the world.

Rules of the game

The crossword puzzle has a simple structure: a matrix of 9 squares, called sectors, is given. These squares are arranged three in a row and have a size of 3x3 cells. The Sudoku matrix looks like a square, consisting of 3 rows and 3 columns, which divide it into 9 sectors containing 9 cells each. Some of the cells are filled with numbers - the more numbers you know, the easier the puzzle.

Purpose of the game

You need to fill in all the empty cells, while there is only 1 rule: the numbers should not be repeated. Each sector, row and column must contain numbers from 1 to 9 without repetition. It is better to fill in empty cells with a pencil: it will be easier to make changes in case of a mistake or start over.

Solution Methods

Consider a simple version of Sudoku. For example, there is only 1 left in a sector or line empty cage, - it is logical that it is necessary to enter in it the number that is not in the number series.

Next, it is worth examining the rows and columns that have the same numbers in 2 sectors. Since the numbers should not be repeated, it is possible to check in which cells the same number can be located in the 3rd sector. Often there is only 1 cell in which you just need to enter the number.

Thus, part of the crossword field will be filled. Then you can start learning strings. Let's say there are 3 free cells in a line, you understand what numbers should be entered there, but you don't know where exactly. You need to try the substitution. Often there are options when a number cannot be located in 2 other cells, because either it is in the corresponding column or in the sector.

Difficult Sudoku

In complex sudoku, these methods only work halfway, there comes a point when it is completely impossible to determine in which cell to enter the number. Then you need to make an assumption and check it. If there are 2 cells in a row, column or sector in which it is equally possible to enter a number, then you need to enter it with a pencil and follow the filling logic further. If your assumption is wrong, then at some point the crossword puzzle will show an error, and there will be a repetition of numbers. Then it becomes obvious that the number should be in the second cell, you need to go back and correct the mistake. In this case, it is better to use a colored pencil to make it easier to find the moment from which you need to solve the crossword puzzle again.

Little secret

It’s easier and faster to solve Sudoku if you first outline with a pencil what numbers can be in each cell. Then you do not have to check all the sectors every time, and in the process of filling, those cells in which only 1 variant of the valid number remains will be immediately obvious.

Sudoku is not only exciting game, which allows you to pass the time, is a puzzle that develops logical thinking, the ability to retain a large amount of information and attention to detail.

• tutorial

1. Basics

Most of us hackers know what sudoku is. I will not talk about the rules, but immediately move on to the methods.
To solve a puzzle, no matter how complex or simple, cells that are obvious to fill are initially searched for.

1.1 "The Last Hero"

Consider the seventh square. Only four free cells, so something can be quickly filled.
"8 " on the D3 blocks padding H3 and J3; similar " 8 " on the G5 closes G1 and G2
With a clear conscience we put " 8 " on the H1

1.2 "Last Hero" in a row

After viewing the squares for obvious solutions, move on to the columns and rows.
Consider " 4 " on the field. It is clear that it will be somewhere in the line A .
We have " 4 " on the G3 that covers A3, there is " 4 " on the F7, cleaning A7. And another one " 4 " in the second square prohibits its repetition on A4 and A6.
"The Last Hero" for our " 4 " this is A2

1.3 "No Choice"

Sometimes there are multiple reasons for a particular location. " 4 " in J8 would be a great example.
Blue the arrows indicate that this is the last possible number squared. Red and blue the arrows give us the last number in the column 8 . Greens the 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 "And who, if not me?"

Filling in numbers is easier to do using the methods described above. However, checking the number as the last possible value also yields results. The method should be used when it seems that all the numbers are there, but something is missing.
"5 " in B1 is set based on the fact that all numbers from " 1 " before " 9 ", Besides " 5 " is in the row, column and square (marked in green).

In jargon it is " naked loner". If you fill in the field with possible values ​​​​(candidates), then in the cell such a number will be the only possible one. Developing this technique, you can search for " hidden loners" - numbers unique for a particular row, column or square.

2. "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 the correct solutions of the puzzle will be only in these cells and only with these values, while all other candidates from the general block can be removed.


In this example, there are several "naked pairs".
red in line BUT cells are highlighted A2 and A3, both containing " 1 " and " 6 ". I don't know exactly how they are located here yet, but I can safely remove all the others " 1 " and " 6 " from string A(marked in yellow). Also A2 and A3 belong to a common 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 trio". When such a group is found, these three candidates can be removed from other cells of the block.

Candidate combinations for "naked trio" may 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 the solution " 3 " for cell E7.

2.3 "Fab Four"

"Naked Four" a very rare occurrence, especially in its full form, and yet produces results when detected. The solution logic is the same as "naked triplets".

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 occupy only those cells and no others. We remove the candidates highlighted in yellow.

3. "Everything hidden becomes clear"

3.1 Hidden pairs

A great way to open the field is to search hidden pairs. This method allows you to remove unnecessary candidates from the cell and give rise to more interesting strategies.

In this puzzle we see that 6 and 7 is in the first and second squares. Besides 6 and 7 is in the column 7 . Combining these conditions, we can assert that in the cells A8 and A9 there will be only these values ​​and we remove all other candidates.


More interesting and complex example hidden pairs. The pair [ 2,4 ] in 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 a row E. Candidates highlighted in yellow are removed.

3.1 Hidden triplets

We can develop hidden couples before hidden triplets or even hidden fours. The Hidden Three consists of three pairs of numbers located in one block. Such as, and. However, as in the case with "naked triplets", each of the three cells does not have to contain three numbers. will work Total three numbers in three cells. For example , , . Hidden triplets will be masked by other candidates in the cells, so first you need to make sure that troika applicable to a specific block.


In this complex example, there are two hidden triplets. The first, marked in red, in the column BUT. Cell A4 contains [ 2,5,6 ], A7 - [2,6 ] and cell A9 -[2,5 ]. These three cells are the only ones where there can be 2 , 5 or 6, so they will be the only ones there. Therefore, we remove unnecessary candidates.

Second, in a column 9 . [4,7,8 ] are unique to cells B9, C9 and F9. Using the same logic, we remove candidates.

3.1 Hidden fours

Perfect 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 appear 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:

1. Pair or Three in a square - if they are located in one line, then you can remove all other similar values ​​​​from the corresponding line.
2. 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.
3. Pair or Three in a row - if they are located in the same square, then you can remove all other similar values ​​​​from the corresponding square.
4. Pair or Three in a column - if they are located in the same square, then you can remove all other similar values ​​\u200b\u200bfrom 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.


Special puzzle. Very difficult to solve, but if you look closely, you can see a few 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 parsing and comparing rows and columns with the contents of the squares (rules №3 , №4 ).
Consider the line BUT. "2 "are possible only in A4 and A5. following the rule №3 , remove " 2 " them B5, C4, C5.


Let's continue to solve 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, we obtain the solution " 2 " for C7.

Sudoku is very interesting puzzle. It is necessary to arrange the numbers from 1 to 9 in the field in such a way that each row, column and block of 3 x 3 cells contains all the numbers, and at the same time they should not be repeated. Consider step by step instructions how to play sudoku, basic methods and solution strategy.

Solution algorithm: from simple to complex

The algorithm for solving the Sudoku mind game is quite simple: you need to repeat the following steps until the problem is completely solved. Gradually move from the most simple steps to more complex ones, when the first ones no longer allow opening a cell or excluding a candidate.

Single Candidates

First of all, for a more visual explanation of how to play Sudoku, let's introduce a numbering system for blocks and cells of the field. Both cells and blocks are numbered from top to bottom and from left to right.

Let's start looking at our field. First you need to find single candidates for a place in the cell. They can be hidden or explicit. Consider possible candidates for the sixth block: we see that only one of the five free cells there is a unique number, therefore, the four can be safely entered in the fourth cell. Considering this block further, we can conclude: the second cell should contain the number 8, since after the exclusion of the four, the eight in the block does not occur anywhere else. With the same justification, we put the number 5.

Carefully review all possible options. Looking at the central cell of the fifth block, we find that there can be no other options besides the number 9 - this is a clear single candidate for this cell. The nine can be crossed out from the rest of the cells of this block, after which the remaining numbers can be easily put down. Using the same method, we pass through the cells of other blocks.

How to discover hidden and explicit "naked couples"

Having entered the necessary numbers in the fourth block, let's return to the empty cells of the sixth block: it is obvious that the number 6 should be in the third cell, and 9 in the ninth.

The concept of "naked pair" is present only in the game of Sudoku. The rules for their detection are as follows: if two cells of the same block, row or column contain an identical pair of candidates (and only this pair!), then the other cells of the group cannot have them. Let's explain this on the example of the eighth block. Putting possible candidates in each cell, we find an obvious "naked pair". The numbers 1 and 3 are present in the second and fifth cells of this block, and there and there there are only 2 candidates each, therefore, they can be safely excluded from the remaining cells.

Completion of the puzzle

If you learned the lesson on how to play Sudoku and followed the above instructions step by step, then you should end up with something like this picture:

Here you can find single candidates: a one in the seventh cell of the ninth block and a two in the fourth cell of the third block. Try to solve the puzzle to the end. Now compare your result with the correct solution.

Happened? Congratulations, this means that you have successfully mastered the lessons on how to play Sudoku and learned how to solve the simplest puzzles. There are many variations of this game: Sudoku different sizes, Sudoku with additional areas and additional conditions. The playing field can vary from 4 x 4 to 25 x 25 cells. You may come across a puzzle in which the numbers cannot be repeated in an additional area, for example, diagonally.

Start with simple options and gradually move on to more difficult ones, because with training comes experience.