Japanese puzzle drawings. How to solve Japanese crossword puzzles? Faculty of Computer Science and Technology
Faculty of Computer Science and Technology
Department of Software Engineering
Speciality Software engineering
Japanese crosswords. Solution algorithm
Now there are a lot of different puzzles that allow you to have fun and spend time with interest. Among them, Japanese puzzles stand out especially: kakuro, sudoku and, of course, Japanese crosswords. Even as a child, I liked to solve Sudoku and I always looked at Japanese crosswords located in the same magazines with bewilderment. They were very complex and incomprehensible for me, although I tried to understand them. Since I couldn't figure it out, I left them. And he returned to them 10 years later, when he was already at the university. There was a lot of free time during the summer holidays and I decided to try to deal with them again and this time I succeeded. Since that time, they are probably the most beloved puzzles.
Among the masters of our university there are several people who covered this topic in their individual section [ , ]. Moreover, Nina Avdzhi did it very well, talking about general description crossword puzzles, the history of their occurrence, the general methodology and principles of solution. She also highlighted the features of black and white and color crossword puzzles. However, the solution algorithm itself and the methods were not described in detail, so in this section I want to describe them in a formalized way, in a form in which they can be used to write a program for solving Japanese crosswords as a person.
Software model design
In order to describe in detail the algorithm for solving Japanese crossword puzzles, you first need to briefly describe the software model and how it works. The Japanese crossword puzzle consists of a main field, on which cells are located, which can have 3 states: filled, empty and indeterminate. This field is divided into rows and columns, next to which there are numbers indicating the number of cells that should be filled. Based on this, a class diagram was developed, shown in Figure 1.
Figure 1 - Class Diagram
The algorithm for solving a Japanese crossword puzzle will not be applied to the entire crossword puzzle at once, but in turn for rows and columns, because in fact there are no significant differences between them. Therefore, the solution algorithm is reduced to the analysis of one line of cells and the numbers corresponding to it, which determine the shading.
The Nonogram class has a queue of line numbers to analyze. After each analysis of the line, the method returns the numbers of the modified cells so that they can be added to the analysis queue, since when modifying the cells, some changes could appear that would advance the solution.
It should be noted that the developed program does not implement all the solution methods as a human. Only methods that work with extreme groups and empty spaces are implemented. Other methods are detailed in .
The source code of the program can be found in .
Edge crossing method
Each numerical group in the line is analyzed and the extreme right and extreme left boundaries of the numerical group are found. If the difference between the right and left borders is greater than or equal to zero, then you can fill in the cells between them (inclusive). It is worth noting that, as can be seen from the figure, groups that intersect only with themselves are painted over in the final line, so cell No. 6 is not painted over, since in different positions (left and right) it belongs to different numerical groups.
Figure 2 - Example of line analysis using edge crossings
Wall push method
Extreme non-empty intervals are analyzed. If there are filled cells at a distance to the numerical group from the edge, then you can paint the cells, starting with the shaded one and up to the cell equal to the size of the numerical group.
If the number of filled cells is equal to the size of the numerical group, then the numerical group can be crossed out, and after the group, cross out the cell (the minimum space between the groups of cells).
Figure 3 - Example of line analysis using wall rebounds
Out of reach method
Extreme numerical groups are analyzed. If the distance from the beginning of an indefinite line fragment to the first filled cell is less than or equal to the extreme numerical group, then it is necessary cross out the cells to which the numerical group does not reach.
It is also necessary to consider the case when there is one uncrossed out group in the line. In this case, it is necessary to cross out all the cells that are further from the filled cells by the length of the numerical group.
This article is for fans of various puzzles. It will discuss how to correctly solve a Japanese crossword puzzle, and where you can find a huge selection of interesting tasks for free.
History of appearance
The birthplace of the puzzle, as the name implies, is Land of the Rising Sun. Authorship is still disputed by two representatives of this country. But whoever came "inventor" this crossword, puzzle fans all over the world are having fun solving these interesting puzzles.
Later, another name for the puzzle appeared - NONOGRAM, on behalf of one of the inventors, a Japanese artist and designer Non Isis. From the beginning of the 90s, the puzzle began to conquer the European continent, and later on the Americas, Australia and Africa.
In less than a decade non-gammas conquer the whole world, does not stand aside and Russia. Puzzles are published in various newspapers and magazines, published as separate brochures and, of course, published on gaming sites on the Internet.
How to unravel
The puzzle is a grid of squares. Abroad playing field, horizontally and vertically, there are rows of numbers indicating how many cells in a given line should be painted over. Puzzles come in two types- black and white and color. The algorithm is almost identical for all variations of the crossword, with slight differences. Consider the basic principles of working with nonograms.
Basic principles of the solution
For example, take a crossword puzzle with a small picture (size 13x12 cells), which we will solve later.
So, the solution algorithm:
Rule 1
There must be at least one empty cell between filled cells of the same color. Explanation for color crosswords - if the cells are of different colors, there may not be a gap.
Rule 2
For convenience, it is advisable to put a “cross”, “dot” or other small sign in the cells that will remain empty (not colored).
Rule 3
Numbers that have already been used to create a drawing are recommended to be crossed out. Before proceeding to the solution, let's carefully study the numbers located on the sides of the field.
Important rules for solving crossword puzzles
Rule 4
If there are values that match the width or height of the field, we start painting from them.
In our example, this is the first vertical column. (value 12 is the same as the number of cells in height) and the last horizontal line (value 13 is equal to the number of cells in width). Thus, it is necessary to start filling out the drawing with these lines.
Rule 5
If there is no number equal to the number of cells along the length or width, it is necessary to find a sequence of numbers, the sum of which is equal to the length/width of the playing field.
In our example, the first horizontal line falls under this norm: 8 + space + 1 + space + 2 = 13.
If the previous 2 options did not work, then move on to the next possibility. Let's call it "overlap". The bottom line is this.
Rule 6
We are looking for a sequence whose sum is as close as possible to the number of uncolored cells. We try to virtually draw it first from left to right (or top to bottom), and then vice versa. Cells that fall on the intersection will be unambiguously painted over. Let's give an example on the penultimate vertical row with the sequence "2; 7". This is not the largest sequence, but, as an option, it will do.
Lines 6 to 9 are in the overlap area - they will be painted over.
Pay attention to the pattern: 2 + space + 7 = 10. The total length of the row is 13 cells. Total 13 - 10 = 3. This suggests that the block of cells is more than 3 pcs. will have an "overlap". In example 7 - 3 = 4. We have 4 filled cells turned out.
Rule 7
If there are filled cells along the perimeter of the field, shade the boundary values.
For our example, let's take a vertical column and fill in all the extreme positions as shown on the slide.
Five more important rules
Rule 8
If there are more empty cells than the length of the last block to be filled, then in the cells that will obviously not be filled in, we put the sign of an empty cell (remember about the crosses and dots?).
For clarity, look at the following figure. The shaded sequence must contain 5 elements of which 4 are already shaded. Therefore, on one side you need to color 1 cell. There are 2 empty fields on the left, 1 on the right. Based on this requirement, the leftmost cell is marked as empty.
Rule 9
If it is impossible to fit a block of cells into an unshaded gap due to its length, such a gap will remain empty.
In our example, there are two unshaded areas. The length of the first is 4, the second is 2. Only the number 4 remains on the left panel. Therefore, a block of 4 squares will not fit in the second gap. We mark it as the one that will remain empty.
Rule 10
If there is a gap between two nearby cells, filling which we get a contradiction with the task condition, then such a gap should remain unfilled.
In our case, there are two figures for 1 and 2 squares. Between them, a section to fill which or not is unknown. If we color this cell, we get a block of 4 cells. But according to the condition, only blocks 1-1-3-1 are possible in this line. Therefore, the existing the interval is marked as "empty".
Rule 11
For multi-colored crossword puzzles, in addition to the above, color matching must be observed at the intersection of horizontal and vertical rows.
The example is simple. The extreme color conditions of the first 3 (green) and last 4 (blue) columns do not match the color sequence of the block of the last horizontal row. In this way, these cells will be marked as "empty".
Last Rule
Rule 12
The most important rule. The process of solving a puzzle should not be torment. It must provide moral satisfaction.
By following this not tricky prescription, you can fully enjoy the wonderful world of hand-drawn crossword puzzles.
This concludes the theoretical part of the article. Let's move on to practical tasks.
Knowing the basic principles of solving a Japanese crossword puzzle, combining them, you can solve nonograms of almost any complexity. Gaining experience, you will develop your own style and methods of solving. Each next puzzle will be solved faster and easier than the previous one. But it's good to start from simple drawings.
Solving black and white crossword puzzles
To consider the main canons of the crossword puzzle solutions were chosen 2 easy tasks: One is black and white, the other is color. Let's solve them using 12 golden rules of decision.
We start with a mono color crossword puzzle. The first step consists of applying Rules #4(the length of the block is equal to the width or length of the field). At the same time, do not forget to cross out the numbers corresponding to the drawn blocks (Rule No. 3). Look at the slide below.
The next step is to draw blocks around the perimeter of the field (Rule #7). We draw blocks on the left horizontally for 8, 2, 1, 1, 1, 2, 2, 1, 1, 1 and 2 cells. Vertically, fill in the cells from below into 2, 1, 1, 3, 4, 4, 4, 2, 1, 1, 7, 8 squares. Do not forget to mark the end of the blocks.
Pay attention to an important detail. In vertical rows No. 3 and 9 (counting from the left edge) all the necessary cells are drawn. Therefore, the remaining ones are marked with a cross, they will without filling.
Drawing these sequences, we see that more 2 sides have the option to fill in boundary blocks. This is the top side and side right. Let's draw the necessary.
It remains to make a few strokes until the task is completed. Please note that 4 cells remain unfilled on the upper horizontal line. According to the task, there should be blocks of 1 and 2 cells 1 + 2 = 3. But we remember that there must be at least one empty cell between blocks of the same color. Total 3 +1 = 4!!!
We finish filling in the field and get the desired picture.
Color nonograms
The distinguishing feature of these puzzles is multicolor. When solving, it is necessary not only to correctly arrange the sequence of cells, but also to color them in the colors required, according to the conditions. The wrong color will negate all efforts. You should also remember the first condition - Between shaded cells one color must be at least one empty, if the cells are of different colors - there may not be a gap.
All of the above affects appearance crossword- not just numbers are written along the edge of the field, these cells also contain a color that should be used when drawing.
As in the case of the black and white nonogram, let's take a step-by-step look at completing the color puzzle. The initial size of the field is 14x14, contains 8 colors.
The algorithm for solving such a puzzle is identical to that used in black and white. Conducting description of Rule No. 11, one of the options for starting the task was given. Using the same norm and also the property "overlap", Let's start the solution in a different way.
In the 12th line horizontally, the numbers are 4 + 2 + 1 + 4 = 11. The field length is 14. Thus, a sequence of more than 3 (14 - 11) can be reflected on the field. Draw a blue cube. Since this is the only figure in the vertical row, we mark the remaining cells of the 11th row vertically with “x”.
As you already understood, you can start drawing in several ways. The result does not change, only the duration of the procedure, its laboriousness, changes. Agree, it is easier to determine the boundaries of color sequences than to calculate the areas of overlap. But, let's repeat, all comes with experience.
Continuation of the solution of the crossword puzzle
Draw on the bottom horizontal row block of 6 squares. Next, draw the boundary blocks. We mark with the symbol "x" those positions, where there is no drawing.
In the next step, pay attention to the 7th vertical row. Taking into account already colored positions 12 cells remain. We check the initial condition 1 + 5 + 2 + 2 + 2 = 12. We boldly paint over the whole row in the colors determined by the condition.
Consistently fill in the boundary values, while not forgetting to cross out the used numerical values and putting down "x" in the identified places. We apply the learned grafts and combine them use nonograms to solve.
As a result, we get a wonderful parrot and a lot of positive emotions. It took just under 3 minutes.
Now you can safely proceed to the independent solution of Japanese puzzles. Below is an overview of the most popular resources containing free crossword puzzles.
Top services with crossword puzzles
For fans of nonograms, as well as those who decided to try their hand at solving Japanese puzzles, our rating of sites on a given topic that provide big choice puzzles.
"Japanese crosswords"
First place in the TOP-5 is occupied by a resource with the same name "Japanese crosswords". Site contains order 20,000 crosswords varying complexity and subject matter. The user can choose both mono color and color options of various sizes and complexity.
A distinctive feature of the site is the name of the puzzles. The user sees only the serial number of the task, not knowing what will be shown in the picture. This creates a certain intrigue in the decision.
A user-friendly interface, a timer and advanced settings for displaying the progress of the solution, along with a large base of nonograms, certainly determine the superiority of the resource.
GrandGames
Honorary second place we give a resource dedicated to puzzles - GrandGames. Unlike the rating leader, the resource is not dedicated to exclusively Japanese crossword puzzles. There are other puzzles here as well.
A large database (up to 10,000 different tasks) of Japanese puzzles, a convenient search menu, a pleasant interface and advanced customization options make the resource silver medalist of our TOP-parade.
Overlay extreme positions
If the number next to the line is only one and is more than half the length, then you can paint over several cells in the middle. To do this, it is necessary to superimpose the extreme left position of the cell groups on the extreme right. Where groups of cells intersect, there will be shaded cells.
If there are several numbers next to the line, we can also superimpose the extreme left position of the cell groups on the extreme right, but we can paint over the cells only in those places where the group of numbers overlapped itself (see example). It is also necessary to take into account the presence of a minimum indent between groups of cells (for black and white crosswords- always one empty cell between the numbers; in color - there is one empty cell between groups of the same color, between groups of different colors - there are no empty cells)
Pushing off the walls
If there is a filled cell in the line, the distance from which to the left border of the crossword puzzle is less than the value of the first digit, then you can paint over several cells on the right. To do this, we count the value of the first digit from the left border of the crossword puzzle - we paint over all the cells that are to the right of the solved one. A similar method works for the last digit and the right border of the crossword - you can paint over the cells to the left of the solved one.
out of reach
If there are filled cells in the line, for which it is unambiguous to say which numbers they belong to, then it becomes possible to put crosses in cells that are "inaccessible" to any numbers. Most often, this method is used when a cell (or several cells) is found that can only refer to the first or last digit.
Does not fit
There are situations when areas limited by crosses appear in a line, in which none of the declared digits can fit. Accordingly, such areas are filled with crosses. We act similarly when this area is formed at the beginning / end of the line, and the first / last digit does not fit in it.
Separation
In situations where there are some filled cells separated by one empty cell, it is necessary to check the possibility of the existence of a filled cell in it - if this leads to a contradiction with the numbers indicated in the line, then this cell must have a cross.
An association
If there are some filled cells in the line that clearly refer to the same digit, then the space between these cells is painted over.
dual position
Sometimes there are situations when a cell shaded in a line can correspond to only two options for the arrangement of cell groups. Cells that are empty in both arrangements are marked with crosses.
And we paint over those cells that are shaded in both arrangements.
Colors at the intersection
In color crosswords, it is additionally necessary to take into account the colors at the intersection of cells. This makes it possible to exclude a large number of possible arrangements of cell groups.
An important feature is that the cells in the first row can be either empty or filled with the color of the first digit in the column. A similar method works for the last row - the cells in it are either empty or painted over with the color of the last digit in the column.
Each of us chooses an occupation to his liking. Some people like to cross stitch. Someone - to make crafts from a variety of materials. Another hobby is crossword puzzles.
Now there are many periodicals devoted entirely to crossword puzzles. Almost every newspaper has a page for those who like to test their erudition. One of the popular types of crosswords in recent times is a Japanese crossword puzzle.
The technique of how to solve Japanese crossword puzzles is quite complicated. But if you figure it out once, you can always keep yourself occupied and train your brains.
What is the difference between a Japanese crossword puzzle and a regular one?
In ordinary crossword puzzles, we guess the words, and in Japanese we need to decipher the hidden picture. The Japanese crossword diagram looks like this:
The numbers indicate how many cells in a row should be crossed out. For example, in the first line there should be nine of them. There are eight in the first column.
What you need to know
- The entire field of the Japanese crossword is usually divided into squares of five cells. That is, you do not need to count one cell at a time, you can count it in fives. Thus, we can calculate that our drawing is 14 by 15 cells in size.
- The order of the numbers does not change. In what order they stand, in this order they will be crossed out in a row or column.
- There must be at least one space between the filled numbers. Maybe more, but there should be a gap of one cell. For convenience, they can be crossed out with crosses or marked with dots.
- It is better to draw crosses with a pencil, because then it will be possible to erase them and see a beautiful picture.
Instructions for solving a Japanese crossword puzzle
We proceed, in fact, to the very technique of how to solve Japanese crossword puzzles. Find the largest numbers first. In our case it is 9 in the first line. Now you need to determine where to cross out these 9 cells in the first line? We need to find out which cells will be 100% crossed out. To do this, we count 9 cells from the left like this:
And now nine cells on the right:
Those cells that are at the intersection will be crossed out:
Now we look at the columns in which the crossed out cells fell. These are the sixth, seventh, eighth and ninth columns. Each of them has the number one - that is, one cell. We have already crossed out one cell, which means that there must be a gap under it. We mark them with crosses, and cross out the number, so as not to be confused later:
We do the same with the next descending numbers. We have 9 in the last column, 8 in the first, and 7 in the last row:
Please note that our last line consists of 14 cells, and therefore seven on the left and seven on the right give exactly half, which means there will be no intersection.
Now it’s already clear how to solve Japanese crossword puzzles? We move on. Now we look horizontally, what the shaded cells give us. In the seventh line, one cell on the right is painted over. So, we cross out the rightmost unit and put a cross in front of the cell - we mark the gap:
Eighth line. Two units - two filled cells. Mark the gaps and cross them out. Congratulations, the eighth line is solved! And this means that we can cross out all the space between them.
Let's see what it gives us. In the seventh and ninth columns we see the number five. Five cells must be filled in after the one, which we have already crossed out. We look at the distance between the crosses in these columns ... Exactly five cells! The question arises why they cannot be at the bottom of the field, after the crosses. Let's go back to the rules again: the numbers are listed in order. That is, if we painted over one cell from the very top, then there must be five, and only then 4 cells one at a time. So, boldly paint over these cells:
Check lines horizontally. Alas, this does not give us anything in the third and fourth lines - it is impossible to determine whether this is one cell painted over or, possibly, two. But we can definitely put an end between them, since there cannot be three in a row in a line:
But in the fifth line, we can put as many as three crosses and cross out two units. In this case, it doesn’t matter which ones, since the entire line consists of ones, and the drawing will not go astray:
Checking the sixth line gives us only a cross between the black bars, the seventh line does not give us anything yet. We skip the eighth, since it has already been solved, and in the ninth we put a cross in the penultimate cell and cross out the unit.
Further, alas, while we can not cross out anything horizontally. Let's go back vertically. Checking the first six columns gives us nothing. At first glance, the seventh one too, but if you look closely ... We have 4 units left. And there are six empty cells in the column. That is, there is exactly enough space to accommodate four filled cells and the gaps between them. The same situation is with the ninth column:
The art of how to solve Japanese crossword puzzles is to constantly double-check yourself. Now let's go back to the horizontal and see what the crossed out cells in the lower field give us. In the ninth line we get a cross. Nothing on the tenth. In the eleventh - also no reliable information, as in the twelfth. But in the thirteenth, we can draw a cell between two already drawn ones, because we have the number 5. It cannot be somewhere on the side, because there are units on the sides. And even if we put units on the sides, we will retreat the gap - and five cells will not fit.
Look at the last two lines. In the very last one, where 7 cells should be drawn, we can cross something out. Since the cells in the seventh and ninth columns are crossed out in the middle, the cell between them will also be painted over. Three out of seven. We retreat hypothetically possible four left and right, and mark everything else with crosses:
And we continue to act in the same spirit. Again and again, checking horizontally and vertically, calculating all the options, we cross out new cells. When you have almost only units left, you need to look at the drawing itself, as a rule, the picture emerges and you can understand what the author wanted to say and where you need to draw the cell. Here is what you should end up with:
This is such a fun emoticon you will get when you understand how to solve Japanese crosswords!
Good luck and have fun!
Have you noticed that lately many around you have begun to solve not ordinary, but Japanese crossword puzzles? And there is an explanation for this. Ordinary crossword puzzles and their light version - crossword puzzles have not forced you to strain your intellect for a long time. From newspaper to newspaper, the same wordings like “parrot of 3 letters” or “clothes for walls” wander. Boring…
What's so good about the "Japanese"? Oh, this is a completely different level, each task is unique, and as a result you get moral satisfaction not from the fact that you remembered all the words you know, but from the fact that you saw the picture you drew yourself, and the more difficult the crossword puzzle, the more detailed it will be drawn all her details.
The rules for solving such crossword puzzles are not complicated. Let's study? So…
A Japanese crossword is a picture encrypted with numbers. The numbers opposite each row (column) indicate the number of filled cells in this row (column). If more than one number is written in a row, this means that in this row (column) there are several groups of filled cells, between which there is at least one unshaded cell. The order of the digits is the same as the order of the shaded groups. Your goal is to determine the place of all groups of numbers on the field and get a drawing as a result. A crossword puzzle can have only one solution, so if something doesn’t fit, we go back a step and carefully check all our steps. That's all the rules.
Everything seems to be simple. But in practice, many questions arise. In magazines and newspapers that publish Japanese crossword puzzles, very primitive pictures are given as an example. And it often happens that it does not come out to solve any of the proposed options on its own. Therefore, I suggest that you start learning from a more complex picture, for example, 15x15 cells in size.
1. We start by looking for the largest digit, or group of digits. This is the line with the number 14.
We count from left to right 14 cells and put a point. We repeat the countdown from right to left and also put a point. We connect them and paint over the whole group. We got 13 shaded cells. Where the 14th cell will be located - on the right or on the left - we do not yet know.
2. We repeat the countdown for the line with the number 9, also from left to right and vice versa. We paint over 3 cells:
3. Now let's look at the bottom row with the numbers 8 and 4. This entry means that this line contains a group of 8 cells, then a gap of at least one cell, and a group of 4 cells. Let's try to calculate them.
From left to right, we count 8 cells, put a point, skip one cell and continue to count 4 cells. We put a point. Now from right to left: count 4 cells (dot), skip one and count 8 cells (dot). We connect in pairs the points related to the eight and the four, and we get groups of 6 and 2 cells. We paint them. In what direction each of the groups will continue is still unknown.
Please note that when we calculate several groups in a row or column, we always skip 1 intermediate cell, although upon completion of solving, you will see that there are sometimes more of them. But we will always use such a counting mechanism if we want everything to work out. Let's go further.
4. We apply the same counting algorithm to the line "4 - 7". You should get groups of one and four cells - these are pieces from 4 and 7, respectively.
5. Now let's see big picture:
Notice the columns. Many of them end with the number 1. This means that the lowest group of cells in these columns is equal to one. Therefore, in the line "8 - 4" we can safely note those "ones" that have automatically emerged from us, and "twos" that can be safely completed. At the same time, we remember that between groups of numbers there must be at least 1 unpainted cell and we agree that we will mark such cells with crosses. Under no circumstances will such cells be painted over.
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6. Next, let's do it yourself:
- column "2-1-6-2" - after the bottom "two" comes the "six". We count 6 cells and paint over it completely. Here everything happened by itself. At the end of the group, do not forget to put a cross;
- column "1-3-5-2" - we do the same with the "five";
- line "9" - we have two filled cells closer to the right edge. From there, we count 9 cells, put a point and connect it with a group of 2 cells. Let's paint over and see that we have 7 out of 9 filled cells. Since we have only one group in this line, we leave 2 cells free from its supposed left edge, and mark the rest with crosses. There will be nothing there anyway;
- check the vertical and notice the “triples” that have appeared (columns “1-1-3-1”, “1-3-1-3-1” and “2-1-2-3-1”), paint over them and do not forget to separate them with crosses;
- in the line "1-6" we count the "six": from right to left we count six cells (point) and from the cross from left to right also 6 cells and put a point. We connect, 5 out of 6 cells are painted over. We do not pay attention to the “one” in this line yet;
- we also recalculate the line "7-1", as a result, we paint over 6 out of 7 cells;
- do the same with lines "1-5" and "7";
- then check the verticals and finish the groups that start immediately after the crosses. After each move, check how the picture changes, draw the positions that appear. You should get the following intermediate picture:
Think logically while solving. If in the line "1-6" for the unit there is only one position, then it is also part of the "two" from the first column. Therefore, leave room for the completion of the “two”, and mark the rest of the column with crosses. Now you can finish the line "14" and count the rows and columns again, marking with crosses those positions where, by no means, can there be filled cells. Draw the line "4-1-1", recalculate the columns "1-3-5-2" and "1-3-1-3-1", and then reason logically and be careful, all the cells will appear with each next step . As a result, we have a drawing of a mouse in a shoe.
I congratulate you on your first success!
I hope you enjoyed it and join our ranks of Japanese crossword lovers!
