What can be cooked from squid: quick and tasty

To your attention is a very popular problem to check the activity of the brain: how to connect nine points with four lines so that the lines do not overlap, and at the same time the pencil or pen does not come off the paper. Many bright minds tried to solve it, but about one out of 30 who tried to give in to the task, which indicates a fairly high level of puzzle complexity. We invite you to try your hand at solving it - this is a useful activity that stimulates brain activity.

9 points 4 lines - the first step to improve ingenuity

Various logic puzzles and puzzles (connect 9 dots with 4 lines, circles on the table, a maze of numbers and others) are a unique tool for the development of human thinking, which can be used at any age. Moreover, they develop not only thinking in general, such tricky tasks are a test for thinking non-standard, non-trivial, ingenuity. Why, you ask, is it so important for a person to develop just this kind of thinking? People with well-trained non-trivial thinking can find a way out of any current life situation, and with the greatest benefit for themselves. Sounds impressive, doesn't it? And immediately an example of the applied use of developed ingenuity.

A certain citizen knocked on one of the reputable American banks (who most likely heard the problem about 9 points) and said that he needed a small short-term loan - 50 thousand dollars for a couple of weeks. When asked about the subject of the pledge, he said that he was the owner of a very expensive Ferrari, worth about $ 300,000, which he was going to leave as a guarantor for the return of the loan funds.

Both parties were satisfied with the terms of crediting, and the citizen left the bank office with fifty thousand dollars in his pocket, but without his car. After the expiration of the term for the loan, the citizen returned to the bank, paid off the body of the loan and the due interest on it, which in 14 days amounted to something about $ 15. I took my supercar and was about to leave when one of the curious bank employees asked why it was necessary to take such an insignificant amount on such an expensive deposit, because you could have asked for much more? To which the satisfied citizen gave a stunning explanation.

He said that he had to leave on business for two weeks, and he would never have been able to attach such an expensive car for such a period for 15 dollars to any parking lot in the city. Therefore, he found the most convenient and low-cost way to take care of his Ferrari: to give it under the protection of the bank and not worry about its safety, and all this for some 15 dollars. A very direct and illustrative example of how important and useful it is to engage in the development of out-of-the-box thinking, and you can start right now by looking for a solution like connecting 9 points with four lines.

Condition of the 9-point problem

There are nine points to be connected with 4 lines. The location of the points is as in the figure, where each number corresponds to a separate point (numbers are put on 9 points for convenience).

Restrictions. It is necessary to connect nine points with straight lines, they should not be repeated, that is, you cannot "return" along the drawn line. When solving the problem of how to connect nine points with four lines, the writing instrument cannot be torn off the sheet with the points depicted on it. Immediately you need to give a hint: the problem cannot be solved by simple attempts to connect 9 points with 4 lines according to the principle of the sides and diagonals of a square. You need to think more broadly).

Solution

Surely many will say that it is impossible to connect nine points with 4 lines while observing the specified restrictions. However, there is more than one solution.

To connect each of the nine points with lines, it is necessary to refer to the concept of a line or a straight line. How does it differ from a segment? The fact that it does not end at the boundary point, but can freely continue for as long as necessary in each of their sides. We have 4 such lines at our disposal, and now it is clear that they can go beyond the limits indicated by the nine points.

So, the sequence, how to connect 9 points with four lines

1. Draw some straight lines - you can mentally, you can write. Connect one point 3 and 5 through point 4, extend it to above point 6, draw a diagonal line through point 6 and 8, extend it to below point 1. These will be the first two lines of four connecting our 9 points.
2. Draw a line connecting points 1 and 3 through point 2, this is the third of the straight lines. The resulting shape is a triangle with one vertex at point 3 and two others outside of points 5 and 1.
3. The handle is at point 3 and now it remains to draw the final line. Points 3.9 and 7 will connect with it.

You can place points in any order: move point 4 to the place where point 2, etc. You can also connect points with lines of nine designated points starting from any corner. There is a similar task where you need to connect 4 points with lines, but the nine-point puzzle is more interesting.

Rice. 4. Connect nine points with four lines

All ingenious is simple! Why doesn't everyone find a solution !? The problem is in the implicit (hidden, disguised) premise that the lines should rest on the vertices of the figure outlined by nine points. As soon as such restrictions are removed, having clearly stated this to the subject, then the latter seems to have an epiphany, and the solution is found instantly ...

A similar implicit premise is the basis for the desire of many managers to cut costs. They proceed from the fact that the amount of income (sales volume) is much more difficult to manage than the amount of expenses, and they strive to reduce the latter as much as possible. Not considering that some expenses are very important, so to speak, generating income, and a reduction in such expenses will inevitably lead to a drop in sales. On the other hand, an increase in profit-generating expenses is likely to lead to outstripping revenue growth.

Eliyahu Goldratt describes this situation very well in his book "Goldratt Rules".

The approach to resolving conflicts should be to try to eliminate the interfering initial premise, which neutralizes the conflict situation itself. Resolving conflict paves the way for the desired change. We can focus on increasing the size of the pie, instead of fighting for a larger share in the process of carving up a small piece. This will be the win-win solution.

It should be taken into account from the outset that changes are possible in any relationship, thanks to which each of the parties comes to meet their needs. It doesn't matter if there is such an opportunity at the moment. It is important, in any tension in a relationship, to be sure that such an opportunity exists. Look for her, not the fault of the other side. If we allow ourselves to judge others, emotions blind us. At the same time, what are the chances of focusing energy and time on the search for changes that will restore harmony? Are insignificant.

Finding a solution in which both parties win involves finding a prerequisite to be eliminated. But finding it is not always easy. A win-win solution increases the size of the pie. The larger the cake, the larger the piece we get. ... when conflicts arise, you need to focus on developing a solution in which both parties benefit. And taking into account that subconsciously we always strive for our own victory, shouldn't we consciously seek a solution that will ensure the benefit of the other side? Will this approach increase the chances of our own success as well?

It is amazing how everything is connected with each other - the statement that harmony exists in any relationship; a win-win approach; advice to start by looking for more (or more) interest from the other party; the ability to identify the biggest gains in solving hidden problems. All this complements each other, forming a single picture.

Let's summarize:

The situation where the gain of one side turns into the loss of the other is not immutable.

If we move from a one-dimensional view to a two-dimensional (or, moreover, to a multidimensional one), we can find options when both sides benefit.

Since we function within the framework of various systems, and these systems have emergent properties, we should strive for a large number of dimensions of the manifestation of these properties.

There is an implicit premise at the heart of the win-lose one-dimensional view; it is necessary to open it and translate the situation into a (two-dimensional) win-win plane.

Similar information:

1. IV. Learning new material. Despite the fact that the definition of a circle is not given to students, it is necessary to acquaint them with the property of points of a circle

Non-standard in its reasoning, the problem of how to connect 9 dots with 4 lines makes you break stereotypes and turn on creativity.

How to position the dots and drawing correctly?

On a piece of paper, it is better if it is in a box, you need to draw 9 points. They should be arranged three in a row. The diagram will look like a square, in the center of which there is a point, and in the middle of each of the sides it is also there. It is better if this drawing is positioned away from the edges of the sheet. This placement of the square will be required in order to correctly solve the problem of how to connect 9 points with 4 lines.

The task

Requirements that must be taken into account:

Observing these rules, you need to connect 9 points with 4 lines. Very often, after a couple of minutes of thinking about this picture, a person begins to assert that this task has no answer.

The solution of the problem

The main thing is to forget everything that was taught in school. They give stereotypical ideas that only get in the way here.

The main reason that the task of how to connect 9 points with 4 lines, cannot be guessed in the following case: they end at drawn points.

This is fundamentally wrong. The points are the ends of the segments, and the problem clearly speaks of lines. This is what you need to take advantage of.

You can start from any vertex of the square. The main thing is that the angle, which one is specific, is not important. Let the points be marked on the left, moving to the right, and from above, moving down. That is, in the first row there are 1, 2 and 3, the second consists of 4, 5 and 6, and the third is formed by 7, 8 and 9.

Let the beginning be at the first point. Then, to connect 9 points with 4 lines, you need to do the following.

1. Guide the beam diagonally to points 5 and 9.
2. You need to stop at the last - this is the end of the first line.
3. Then there are two ways, they are both equal and will lead to the same result. The first will go to the number 8, that is, to the left. The second is towards the six or up. Let it be the last option.
4. The second line starts at point 9 and goes through 6 and 3. But it does not end at the last digit. It needs to be continued upwards for another segment, as if there was another point drawn there. This will be the end of the second line.
5. Now again the diagonal, which will pass through the numbers 2 and 4. It is easy to guess that the second number is not the end of the third line. It needs to be continued as it was with the second. So the third line ended.
6. It remains to draw the fourth through points 7 and 8, which should end in the number 9.

This completes the task and all conditions are met. To some, this figure resembles an umbrella, while others claim that she is an arrow.

If you write down a short plan of how to connect 9 points with 4 lines, you get the following: start at 1, continue at 5, turn at 9, draw at 6 and 3, extend to (0), turn at 2 and 4, continue to ( 0), roll up to 7, 8 and 9. Here (0) denotes the ends of segments that have no numbers.

As a conclusion

Now you can still puzzle over a more difficult problem. There are already 16 points in it, located similarly to the task considered. And you need to connect them already with 6 lines.

If this task turned out to be too tough, then you can try to solve others, with the same requirements, but differing in the set of points and lines, from the following list:

• 25 points in the order of a square, like all subsequent ones, and 8 straight lines;
• 36 points by 10 lines that are not interrupted because the pen cannot be torn off the sheet;
• 49 points connected by 12 lines.

Actual

Miscellaneous
Miscellaneous

If you got to this page, then you probably already tried to solve the "9 point test", namely, connect the nine points with four straight lines without lifting the pen from the sheet of paper. If you have not been able to solve this puzzle, do not despair. On this page, you can find several solutions to this famous difficult nine-point problem that has strained the minds of many thousands, if not millions of people.

The task

Condition:

Condition: you need to connect the drawn nine points with four straight lines without lifting the pen from the sheet of paper.

This task is not as easy as it might seem. To solve it, you need to think outside the box and apply your creative thinking, otherwise nothing will work. If you try to act head-on and start connecting all the points with standard lines, then you can waste a lot of time and still not solve the problem of nine points. Our standard thinking, which we are taught in school, directs us to seek a solution based on only six typical lines: 4 sides of a square and 2 of its diagonals. Most people think that the solution to the 9-point puzzle should lie within this framework. But he's not there. You can't even find it if you connect 2 more lines between the centers of the sides of the square:

In general, only 20 straight lines can be drawn between all nine points: 4 sides of a square; 2 diagonals; 6 lines connecting the centers of the sides of the large square; 8 lines connecting the centers of the sides of a large square with its corners. How to draw all the line segments connecting our 9 points is shown in the picture below:

But, even using this scheme, it is impossible to find 4 lines that could connect all nine points without taking your hands off.

Correct solution of the "test of 9 points"

The solution to this puzzle lies somewhat wider than our standard perception of the problem. In order to find the right approach on your own, remember that:

1. Only one straight line can be drawn through any 2 points.
2. A straight line is not a line, and therefore, we do not have to limit ourselves to our nine blue circles when drawing lines.

Thus, let's try to extend the lines beyond the bounding square until recently. Here you can see that our search area has increased significantly. With a little work, you can come to one of the right decisions.

A sequence of connections of nine points with four lines:

1. First, draw a line connecting point # 1 and point # 7 through point # 4. Do not stop movement and draw further for about as much as from point number 4 to point number 7.
2. Then move diagonally to the right and up, connecting points # 8 and # 6. Do not stop at point number 6 and continue the line until the mental line passing through the top of our square.
3. Draw a line from right to left sequentially through points # 3, # 2 and # 1. Stop at point # 1.
4. Now draw the final segment through points # 1, # 5 and # 9. All 9 points, indeed, are connected by four lines, as required in the problem statement.

Other options. This method is not the only one, you can start from any angle and move in one of two directions. On the 4brain website, there are at least 12 such options for solving the problem "9 points 4 lines":

Just think, a problem that many cannot solve in any way has 12 ways to solve it. Also see a simplified version of this problem: how to connect 4 points with three lines, so that the lines close into a whole figure.

Get creative with this puzzle

Most of the people who solved this problem could not get out of the standard thinking, which in this test is expressed by a square formed by nine points. It is comfortable for us to look at any life task directly, in the most simple way. On the other hand, a person can spend a lot of time and effort in order to, using a standard approach, find the right solution, when it is better to look for this solution, initially approaching the process creatively.

In our life, we are often faced with such problems of "nine points and four lines", and in order to solve them, develop your creative thinking, including with the help of our training. After all, the 9-point problem has other solutions (read on for more on this).

Other solutions

By changing our frame or applying a lateral gap, you can find other options for solving this problem. For example, the method of hyperbolization when creating a lateral discontinuity can lead us to the idea that no one specifies that standard geometry conditions should be applied in the problem (about the infinite smallness of points and infinite thinness of lines). Let our line be so wide that it can cross several points at once in its width. Then we will not be able to connect all 9 points with 4 lines, but even one.

In addition, even in our image of 4 dots, which is given in our condition for a puzzle about 9 dots, the dots themselves are large enough to be connected with 3 lines like this:

Or maybe you shouldn't limit yourself to two-dimensional space at all or use the concept of space curvature. We can also focus on the phrase "without lifting the pen from the sheet of paper," and simply putting the pen on its side, move it and thus draw just 3 parallel lines.

Creativity is not boring, and moreover, you can create with humor.Perhaps this problem is familiar to you. Perhaps you, like many others, think that there is only one solution. So forget it and find a new one.

Here they are - 9 magic points:

Task: without lifting the pencil from the paper, draw 4 intersecting straight lines that will touch all nine points only 1 time.

we too often draw boundaries that don't really exist. And we stay in them. We play by these rules. We use phantom criteria. We forecast the development of the project based on the trends and opportunities that have taken place in the past, without looking for and comparing new ones. We do not abandon the established paradigm without permission.

You could connect the dots with four lines outside the square. Like this:

How do you like the solution? Like? Doesn't it seem elegant to you and the only one possible? In fact, the most serious limitation in solving this problem is precisely the conclusion that there is only ONE answer. In reality, you can find several completely different solutions to this problem.

But how to break the paradigm and find different results?

There is a technique called"Forced exit".We need to forget about posing the problem and work on solving its distant version. This is the path to new paradigms, perspectives and results.

And the first modified task will be ... the same 9 points

Task: this time draw 3 intersecting straight lines that should touch each point only 1 time. If you cannot find a solution, try to determine what frameworks, conclusions and criteria are hindering you and stopping your search.
Let's take a look together.

First, tell me, what do you see when looking at the area of ​​dots? I hope you have already given up the habit of drawing a square and other shapes. Now you may be blocked by the fact that you see these dots on the sheet of paper. In order to find several ways to solve the "3 lines" problem, you need to represent these points in space. This is the only way 3 straight lines can leave a piece of paper.

Secondly, don't you think that these lines should go through the center of each of the 9 points? This nonexistent condition prevents you from thinking.

Third, how do you define the point itself? At school we were taught that point Is an element of geometric space, characterized only by position, belonging, and not by size or shape. But these circles, which in our problem are called points, just have both shape and size. Not entirely fair on our part, right? Well, that's life. In real life, points vary greatly in size and shape. On bigboards, they grow to the size of a human head, and on a clown costume, they shrink to polka dots. So add reality to your dots before you fall prey to another bad habit that gets in the way of creative thinking.

It is about using narrow definitions that constrain the thinking process like a funnel. We are stuck in old paradigms.

Thanks to the missing boundaries, refined assumptions, and extended definitions, we found the following solution to the 3 straight lines problem:

Leave the sheet of paper in your mind. The first line runs tangentially to the first point, crosses the second almost in the center, and slightly touches the third point. Extend this line further, beyond the edge of the paper, until another line can do the same with the middle column of dots. The third straight line should behave in the same way.

Here is a solution based on the postulate of non-Euclidean geometry that parallel lines intersect at infinity. The answer consists of three parallel lines, each touching a different series of points, and then connecting all three lines at infinity. A neat paradigm shift, isn't it? It is possible that finding a solution will require you to leave your comfort zone.

The habit that reduces creativity to zero: often, we pick out a “fair” idea before making a choice from several decisions. Don't let “decency” get in the way of your search.

The next problem is for 9 points.

Task: use 2 intersecting straight lines that touch all 9 points only 1 time.

Impossible, you say? You can use one more revision of your unfounded assumptions, non-existent boundaries, contrived criteria, narrow definitions, thought funnels and patterns.

One block lurks in defining the line that you stick to. From the school curriculum: line- this is an innumerable number of points that are located on one line, which has neither beginning nor end, i.e. have only one property - length. In real life, lines have a width. Think of the traffic flows on the highways or the trolleybus line in front of the intersection. So, this time too, the penchant for ready-made terms has led you to conclude that only thin lines can be used.
This is what happens when the definitions are expanded - a solution consisting of one wide and one narrow lines!

In search of a solution to our last problem, try the “forced exit” technique.

Task: one straight line must touch all nine points.

In general, there are at least a hundred acceptable solutions. Some of them are presented here to spark new paradigms and thought funnels and whet your appetite for more.

• Use one wide line that touches each point.
• Pass the large 3-D line through nine points from top to bottom so that it goes through the paper and touches each point.
• Fold the paper so that you can make one line that touches each point. (Did you assume that you were not allowed to fold the paper?)
• Cut the paper so that each point is on a separate piece. Place the particles in one line that will touch each point. (Did you think you couldn't cut the paper?)
• Roll the piece of paper into a cone and draw a straight line that spirals around the surface of the cone and touches all nine points. (Did it ever occur to you that you can do whatever you want with paper?)
• Place a sheet of paper with nine points on the Earth's equator and carefully draw a straight line around the Earth enough times so that it eventually touches every point. Or put paper on the edge of the universe and draw your straight, circular line around the universe until it touches every point. (Didn't you assume that you could use fantasy? Note that we have expanded our thought funnel from nine points to a window overlooking the edge of the universe.)
• Write “ONE” on top of the first row of dots, “STRAIGHT” above the middle row of dots, and “LINE” above the bottom row of dots.
• Draw a line on the thin edge of the paper. Look at nine points across this lateral line.
• Move the straight line like windshield wipers in a car - and you will touch all points. (Did you feel like you couldn't move the line, or that the line had to touch all points at the same time?)
• Cut a straight line into 1000 pieces and scatter them over nine points (Was it forbidden to cut lines?)
• Cut so that one point is on a separate piece of paper. Line up the dots in a turret, one above the other. Click on all points with a pencil. Not only have you touched all the points on one straight line, but you have eliminated both the points and the problems. In one fell swoop.
• Wait. Here's another food for thought. Imagine you are sitting with your dots at the table, and then the king of beasts comes in and swallows them all at once. Or how about nine people, each named Dot, eaten by one lion?
• I cannot resist making an even stranger decision. Change points into clothespins and hang them on one straight line of clothesline. (Did you assume you can't convert points or lines to something else?)
• Or you can turn the dots into tennis balls and play tennis with them until each one touches the tennis net, which is one straight line.
• Or change the line into the shade of the sundial so that it eventually touches all points as the sun moves across the sky.
• Or convert a straight line to a sunbeam and use a glass prism to break it down into many colored lines that touch all nine points. Enough for now?

Based on the book"R&D CREATIVITY & INNOVATION HANDBOOK" A Practical Guide To Improve Creative Thinking & Innovation By