How to connect three dots with one line. Models of logical decisions. As a conclusion

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The most difficult puzzles that we want to tell you about have recently gained insane popularity on the Internet. As a rule, tasks such as "connect the dots" are considered one of the most difficult. Firstly, you need to think outside the box, and secondly, you need to try to calculate many different combinations.

If you think that this is "kindergarten" for you, then try to cope with these tasks. The percentage of those Internet users for whom this task turned out to be within the power is extremely low.

Do you know what was the most difficult? The number of lines is strictly fixed. Just wait until you know the rest of the requirements.

Many people call these puzzles "Sudoku Dots".

If you cope with only 1 task out of 5, refresh your knowledge of geometry.

2/5 or 3/5 - You are in the top!

At first everything will be very simple, but then the real hell will begin ...

Note: the lines must not cross!

According to the creators of these quizzes, only 20% of people are able to cope with 4 of them. The 5th task is only possible for geniuses!

Many people argue that the rules for these tasks are not well defined.

According to the creators: ‘’There are several ways to solve these problems. You just need to use your creativity.''

Moreover, there is another good reason why the rules are not explained to the very end. After seeing the correct answers of these tasks, you will understand that if all the rules were explained, the task would become meaningless.

Be sure to test your knowledge and creative thinking. Don't be disappointed if something doesn't work out for you. Often we are not able to control or develop our original ideas about solving certain problems.

Today is a great chance to discover your real potential!

1. The first task will not seem too difficult for you.

Connect 9 dots with 4 straight lines

Answer

2.Make sure all lines are connected!

Now: connect all the dots with3 lines

Answer

3.Make sure all lines are connected!

Connect 16 dots with 6 straight lines.

Answer

4.And another masterpiece...

Cut the paper into two pieces so that the dot is in the middle.

On the first image you see a section.On the second - moving!

5.Last bonus!

Write the numbers from 1 to 9 so that each side of the triangle is 17!

Answer

6. Did you succeed?

If you coped with 1 task out of 5 - refresh your knowledge of geometry.

2/5 or 3/5 - You are in the top!

4/5 or 5/5 - You are a real genius.

Rice. 4. Connect nine dots with four lines

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

A similar implicit premise underpins 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 minimize the latter. Not taking into account that some expenses are very important, so to speak, generating income, and the reduction of such expenses will inevitably lead to a drop in sales. On the other hand, an increase in profit-generating spending is likely to lead to outpacing revenue growth.

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

The approach to conflict resolution should be to try to eliminate the interfering initial premise, which neutralizes the conflict situation itself. Removing the conflict opens the way to the desired changes. We can focus on increasing the size of the pie, instead of fighting for a larger share in the process of sharing a small piece. This will be a solution in which everyone wins.

It must be initially borne in mind that in any relationship changes are possible, thanks to which each of the parties comes to meet their needs. It doesn't matter if there is such a possibility at the moment. It is important for any tension in a relationship to be sure that such a possibility exists. Seek her, not the other side's fault. If we allow ourselves to judge others, our emotions blind us. What are the chances of concentrating energy and time on the search for changes that will restore harmony? Insignificant.

Finding a solution that benefits both parties involves finding a premise to be eliminated. But finding it is not always easy. A win-win solution increases the size of the overall pie. The larger the pie, the larger the piece we can get. …when conflicts arise, the focus should be on finding a solution where both parties win. And given that subconsciously we always strive for our own victory, shouldn't we consciously look for a solution that will ensure the victory of the other side? Wouldn't such an approach increase the chances of our own success as well?

It is amazing how everything is interconnected - the statement that harmony exists in any relationship; an approach that benefits both parties; advice to start by looking for a large (or greater) interest of the second party; the ability to identify the biggest gains in solving hidden problems. All this complements each other, forming a single picture.

Let's summarize briefly:

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

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

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

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

Related 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 the points of a circle

9 points 4 lines

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

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 a large square; 8 lines connecting the centers of the sides of a large square with its corners. How to draw all the segments connecting our 9 points is shown in the figure below:

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

The correct solution to the "9-point test"

Spoiler

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

• Through any 2 points, only one straight line can be drawn.
• A straight line is not a line segment, and therefore we don't have to be limited to our nine blue circles when drawing lines.

Thus, let's try to continue the lines beyond the square that limited us until recently. It can be seen that the scope of our search has increased significantly. With a little effort, you can come to one of the right decisions.

The sequence of connecting nine points with four lines:

You can watch a video of this problem:

Get creative with this puzzle

Most people who have solved this problem have not been able to get beyond the standard thinking, which in this test is expressed by a square formed by nine points. We are comfortable looking at any life task directly, most simply. On the other hand, a person can spend a lot of time and effort in order to find the right solution using a standard approach, when it is better to look for this solution, having initially approached the process creatively.

Even in our 4-dot image given in our 9-dot puzzle condition, the circle dots themselves are large enough to be connected with 3 lines like this:

If you have reached this page, then you have probably already tried to solve the “9 dots test”, namely, to connect nine dots with four straight lines without lifting the pen from the sheet of paper. If you didn't manage to solve this puzzle, don't despair. On this page, you can find several solutions to this famous nine-dot puzzle that has puzzled the minds of 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 out. If you try to act head-on and start connecting all the dots with standard lines, then you can spend a lot of time and still not solve the nine dots problem. Our standard thinking, which we are taught in school, directs us to look for 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 dot 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 a large square; 8 lines connecting the centers of the sides of a large square with its corners. How to draw all the segments connecting our 9 points is shown in the figure below:

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

The correct solution to the "9-point test"

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

1. Through any 2 points, only one straight line can be drawn.
2. A straight line is not a line segment, and therefore we don't have to be limited to our nine blue circles when drawing lines.

Thus, let's try to continue the lines beyond the square that limited us until recently. It can be seen that the scope of our search has increased significantly. With a little effort, you can come to one of the right decisions.

The sequence of connecting nine points with four lines:

1. First, draw a line connecting point #1 and point #7 through point #4. Do not stop moving and draw further for about as long as from point #4 to point #7.
2. Then move diagonally to the right and up, connecting points No. 8 and No. 6. Do not stop at point number 6 and continue the line to the mental straight line passing through the upper side of our square.
3. Draw a line from right to left consecutively 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 condition of the problem.

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 “9 points 4 lines” problem:

Just think, a problem that many people can’t solve in any way has 12 ways to solve it. See also 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 people who have solved this problem have not been able to get beyond the standard thinking, which in this test is expressed by a square formed by nine points. We are comfortable looking at any life task directly, most simply. On the other hand, a person can spend a lot of time and effort in order to find the right solution using a standard approach, when it is better to look for this solution, having initially approached the process creatively.

In our life, we often encounter such problems about “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 problem of 9 points has other solutions (read more about 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 conclusion that no one specifies that the standard geometry conditions (about the infinitely smallness of points and the infinitely thinness of lines) should be applied in the problem. Let our line be so wide that it can immediately cross several points along its width. Then we will not only be able to connect all 9 points with 4 lines, but even one.

Also, even in our 4-dot image given in our 9-dot puzzle condition, the circle dots themselves are large enough to be connected with 3 lines like this:

Or maybe you should not 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.

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 correctly position the dots and pattern?

On a sheet of paper, it is better if it is in a box, you need to draw 9 dots. They should be arranged three in a row. The diagram will look like a square, in the center of which there is a dot, and in the middle of each of the sides there is also one. It is better if this pattern is placed 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:

Following these rules, you need to connect 9 points with 4 lines. Very often, after a couple of minutes of thinking about this drawing, a person begins to assert that this task does not have an answer.

The solution of the problem

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

The main reason that the task is about how to connect 9 dots with 4 lines, not unraveled in the following case: they end at the drawn points.

This is fundamentally wrong. The points are the ends of the segments, and the problem explicitly talks about lines. This is what you should definitely use.

You can start at any corner of the square. The main thing is the angle, which one specifically, it doesn’t matter. Let the points be marked on the left, moving to the right, and from above, moving down. That is, the first row contains 1, 2 and 3, the second consists of 4, 5 and 6, and the third is formed by 7, 8 and 9.

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

1. Lead the beam diagonally to points 5 and 9.
2. You need to stop at the last one - this is the end of the first line.
3. Then there are two ways, they are both equivalent and will lead to the same result. The first will go to the number 8, that is, to the left. The second - to the six or up. Let there 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 up for another segment, as if another point was 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 must be continued, as it was with the second. Thus ended the third line.
6. It remains to draw the fourth through points 7 and 8, which should end in the number 9.

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

If we write down a short plan of how to connect 9 points with 4 lines, then we get the following: start at 1, continue at 5, turn at 9, draw at 6 and 3, extend to (0), turn to 2 and 4, continue to ( 0), turn to 7, 8 and 9. Here (0) are the ends of the segments that do not have numbers.

As a conclusion

Now you can still break your head over a more complex problem. There are already 16 points in it, located similarly to the considered task. 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 on 10 lines that are not interrupted because the pen cannot be torn off the sheet;
• 49 dots connected by 12 lines.