To your attention is a very popular task for checking brain activity: how to connect nine dots with four lines so that the lines do not overlap each other, and at the same time the pencil or pen does not come off the paper. Quite a few bright minds tried to solve it, but about one out of 30 who tried to solve the problem, which indicates a fairly high level of complexity of the puzzle. We invite you to try your hand at solving it - this is a useful activity that helps stimulate brain activity.
9 dots 4 lines - the first step to improving ingenuity
Various logical puzzles and puzzles (connect 9 dots with 4 lines, circles on the table, a labyrinth of numbers and others) is 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. And why, you ask, is it so important for a person to develop 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 (who most likely heard the puzzle about 9 points) knocked on one of the reputable American banks and said that he needed a small short-term loan - 50 thousand dollars for a couple of weeks. When asked about the subject of collateral, he said that he owns a very expensive Ferrari, worth about $300,000, which he is going to leave as a guarantor for the return of loan funds.
Both parties agreed on the terms of the loan, and the citizen left the bank office with fifty thousand dollars in his pocket, but without his car. After the expiration of the loan period, the citizen returned to the bank, repaid the body of the loan and the prescribed interest on it, which in 14 days amounted to something about $ 15. I picked up 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 ask 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 to any parking lot in the city. So he found the most convenient and cost-effective way to take care of his Ferrari: give it to 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 develop non-standard thinking, and you can start right now by looking for a solution to how to connect 9 points with four lines.
Condition of the problem about 9 points
There are nine dots that need to be connected with 4 lines. The location of the dots is as in the figure, where each number corresponds to a separate dot (the numbers are put on 9 dots for convenience).
| 3 | 4 | 5 |
| 2 | 9 | 6 |
| 1 | 8 | 7 |
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 must not be torn off the sheet with the points depicted on it. You need to give a hint right away: the problem cannot be solved by simple attempts to connect 9 points with 4 lines according to the principle of sides and diagonals of a square. Think big.)
Solution
Surely, many will say that it is impossible to connect nine points with 4 lines in compliance with the indicated restrictions. However, there is a solution, and not even one.
To connect each of the nine points with lines, you need to refer to the concept of a line or a straight line. How is it different from a cut? The one that does not end at the boundary point, but can freely continue for as long as you like in each direction. We have 4 such lines at our disposal and now it is clear that they can go beyond the limits indicated in nine points.
So, the sequence, how to connect 9 points with four lines
- Lay a few straight lines - you can mentally, you can write. Connect one point 3 and 5 through point 4, extend it to a place above point 6, draw a diagonal line through 6 and 8, extend it to a place below point 1. These will be the first two lines of four connecting our 9 points.
- Draw a line connecting point 1 and 3 through point 2, this is the third of the straight lines. The resulting figure is a triangle with one vertex at point 3 and two others that go beyond points 5 and 1.
- The pen is at point 3 and now it remains to draw the final line. Points 3,9 and 7 will be connected with it.
You can arrange points in any sequence: move point 4 to the place where point 2 is, etc. You can also connect the points with lines of nine designated points starting from any corner. There is a similar task where you need to connect 4 dots with lines, but the nine-dot puzzle is more interesting.
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:
- 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:
- 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.
- 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.
- Draw a line from right to left consecutively through points #3, #2 and #1. Stop at point #1.
- 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.
FROM Let's face it, this is a trick question. Rather, the solution itself, like most similar tasks, is not based on logic, but rather on creativity.
Creativity is what distinguishes talented people from the countless gray mass.
And every person who stands in the way of systematic development would like to have such a talent!
Someone will say that you need to be born with this quality, and all the great creative personalities have shown their extraordinaryness since childhood.
But we are sure: desire and work, self-education and the experience gained will help you reach a new, non-trivial level of thinking. The main thing- practice!
The clue without further ado: how to connect nine dots using four lines?
This mystery has intrigued hundreds of thousands of people, and maybe more. The following conditions must be met: cross all nine points that form a square using straight lines (no more than four).
A hand, or rather, a pencil, in this case, cannot be torn off the sheet. The next line should start where the previous one ended.
At first glance, it is not so difficult, but in fact, each subsequent attempt often moves the inquisitive mind away from a positive result.
The thing is that from childhood we were taught to think, starting from certain patterns and rules.
First of all, logical thinking developed, on the principles of which our world is built. Yes, not so.
Here you need to go beyond logic and stop thinking in terms of the four sides of the square and its diagonals.
We analyze the problem based on knowledge about the object (square), but we should just remember that a straight line is not necessarily limited by the shape, that is, it is possible and necessary to go beyond the boundaries.
We conditionally number each point from 1 to 9.
- We draw the first line, starting from point 1 through 4, 7 and go beyond the boundaries of the figure.
- Without taking your hand off the sheet, we make a corner and strive for the point numbered 8 and 6 and go beyond the framework in the same way.
- Then we turn and go through 3, 2, 1.
- We turn through the corner of the square, passing the rest of the way through the points numbered 1.5 and 9. It turns out a kind of arrow-cursor that can be directed to any of the four corners as you wish.
There is also a "hardcore" method for those who know spatial thinking.
Draw nine circles on a square sheet (sticker) (as in the problem). Under the 7th and 8th point, apply glue.
Take the base of a cylindrical shape. A tube of decorative cosmetics (lipstick or foundation) is ideal. Connect the place under 7 and 8 with a dot to the place under 2 and 3.
Draw one solid line starting at point #1 and spiraling down.
When you return the leaf to its original form, you will see that three lines have been drawn on it, covering all the points, which fits into the conditions of the puzzle.
"Advanced" individuals can solve it even without the help of glue, the main thing is to imagine the end result.
To solve this and similar puzzles, it is worth developing and discovering unusual approaches to the problem.
Try the fun exercises below.
Tip: number the dots right on the paper, so it will be easier to figure out the solution.
Games for home entertainment
At one time, Steve Jobs, the man synonymous with the word "creativity", emphasized that people who have the skill of creative thinking do not invent, but rather notice the connection between several things.
This is what makes it possible to synthesize something new.
Therefore, first of all, it is worth “pumping” such observation, over the surrounding phenomena and things more often.
We suggest the following exercise: look around and name as many things as possible that are in the same room with you and begin with the same letter, including mental concepts.
For example, "m":
- Furniture, lightning (on clothing), chalk (pet food)
- Opinion, peacefulness, mannerisms
- Milk, materials (upholstery), T-shirt
- Ointment, makeup, gauze, etc.
A simple version of the game: the letters "v", "s", "p", "k". If you are confident in your abilities, choose - "t", "a", "d".
Do not limit yourself and your innate imagination.
If desired, in one room you can find about 40+ words. Experts find about 100 words in each room.
The next game was very popular back in the 17th century. If you are offered to have fun with "nonsense" - do not rush to refuse, its other name is "burime".
To immerse yourself in the process, you will need a few pieces of paper, a pen and a good company that does not mind practicing collective poetry writing.
The topic and limitations are specified in advance.
H Most likely, obvious combinations of cognates, pronouns, verb forms and hackneyed platitudes (hi-lunch, love-carrots) are excluded. Sometimes a specific topic is discussed.
It happens like this: someone writes a line, and the other completes the next verse until a full-fledged work is obtained.
This is for all ages, even the smallest.
It develops spatial, which will certainly come in handy in the adult future.
Have your child sit at a table and give him a black pencil and a piece of paper. Put on some nice music and ask him to close his eyes.
Let the child draw, intertwine random lines with each other, without thinking about accuracy.
Sometimes it is better to create several drawings in this way that overlap one another.
Later, sit down with him and use colored pencils to highlight animal-like shapes, objects, all kinds of images. Let the child be the source of ideas.
Tip: puzzles with matches (sticks) will be an excellent workout for the mind. Such little puzzles will be interesting for both children and adults. They are available to everyone!
Creative Thinking Exercises
Get up. Take any book from the shelf. On two different pages, blindly select a few words.
Now try to find everything they have in common. For example, the word “carpet” and “tree”: they both lie on the ground, their images are found in fairy tales (flying carpet, the tree on which the scientist cat walks), etc.
If you are playing with a child, choose simpler words: cat-dog, tomato-pear, table-chair.
Write on a piece of paper a dozen of any nouns: “strawberry”, “fish”, “water”, etc. Now imagine that this sheet is the requirements of the customer, and you yourself are a builder-architect.
Build a house using these as basic requirements.
For example, the wallpaper will be red "strawberry" color, and the walls of the house will shine in the sun like fish scales. Let the house itself stand on the top of the mountain, where the sky is just infinitely blue, like water, etc.
Sitting in a room, find within your sight an object that is familiar and interesting to you. For example, "apple".
Books will help you Think of five adjectives that fit perfectly with the subject:
- Green
- Sour
- delicious
- Soft
- juicy
And now we complicate the task and come up with five more adjectives, but those that absolutely do not fit in meaning: prickly, rough, plush, pewter, slender.
Some words are not so easy to work with, but the task is more interesting: a well, a train, a wind, a wall.
Take a pencil in your hands and draw a column of crosses in a notebook in a box.
Width and height are arbitrary, just make sure they are far enough apart.
Then we turn these crosses into small pictures, adding the necessary details (fish, crossed axes, sword, dragonfly, etc.).
In the same way, draw the letter "o", "t", "v" and come up with new, interesting images. At an advanced level, you can turn sketches into small plots with action.
Come up with a whole story! This is not as difficult as it might seem at first glance.
Use your computer wisely Tip: read the words backwards: fairy tale-akzaks, bottle-aklytub, spoon-akzhol. This, of course, is a useful activity, it will help to pass the time of waiting in line or in public transport.
Online games that improve creative thinking
IQ-ball
You are a small, round, living ball with a sucker foot flying out of your body.
The goal is to get candy at each level, overcoming all sorts of obstacles. Fixed and moving elements, time limit, inertia will interfere with you.
You can not push off or cling to the “foot” from all surfaces. You will have to think quickly, the reach of the goal depends on it.
Get ready for 25 exciting levels.
"Black cat"
Before you is a field created from circles. A black cat sits in the middle of it. With a mouse click, you can fill in a mini-area that the cat can no longer pass through.
You make one move, the next is a cunning animal.
Your task is not to let him run out of the edge of the playing field, because this means losing. Here you will have to connect all your intelligence and creative thinking, and most importantly - choose the right strategy for fighting.
In this case, you can advise not to rush, but to think through your move in advance, mark the circles through one.
In this case, you will always have time to close the path to the furry animal.
They are various images that you can easily find on the Internet.
These are not just sketches, but pictures with inherent potential. One and the same "doodle" can carry several meanings at the same time. For example:
- Facade
- Wedge
- Cloak
- Diamond etc.
A positive result of the game is the acceleration of thought processes, the versatile development of imagination and creativity. Such unpretentious fun can capture you for a long time.
"Memory Matrix"
Many adults and children are familiar with this game. Before you is a field that is filled with squares for a few seconds.
With each subsequent level, the field grows and the task becomes more complicated. The game develops memory, creativity and the ability to quickly concentrate well.
Tip: try playing Lines 98. It develops logical thinking in parallel.
Tricky, developing tasks
Draw a rectangular island on a piece of paper, in the middle of which countless treasures are hidden. It is surrounded by a moat of the same shape.
You are a treasure hunter who is outside of this land. The arsenal is only two boards, each one is slightly less than the width of the moat.
It is impossible to jump over or fly over it, there are no ropes for fastening the boards together, as well as nails, and each individually can easily fall into the abyss.
The goal is to get to the treasure. The answer to this riddle is based on the principles of geometry: “lay” the first board on the corner of the moat so that it does not fall through.
By doing this, you reduce the width of the moat, and the second board will freely reach the island with the treasure.
Put a thick dot in the middle of the sheet. The goal is to drawThere is a regular circle around it, but so that the beginning of the line starts from the very point.
The solution: fold the corner of the paper, put a dot in front of the corner itself, without lifting your hand, draw a line from the dot to the rest of the sheet, align the corner and continue moving until you draw a circle.
And finally, a simple question: why all over the world they make exclusively round pizza, but deliver it in square boxes?
The contradiction here is only at first glance. And the answer is this: the pizza is round so that the corners do not burn, which inevitably happens when baking rectangular dishes.
In the case of a box, several factors are important:
- This makes it easier to get food out of it.
- It is much cheaper and easier to produce square boxes than round ones.
- The pizza looks amazing
Tip: train your brain with small tasks at least a few times a week and very soon you will feel that it has become much easier for you to find non-standard solutions at work and in life, to think outside the strict framework of logical thinking.
You probably love logical and creative tasks if you are reading this article. And, admittedly, this task is indeed quite difficult, although there are at least twelve (as smart people say) ways to solve it.
Condition
What do we see? We have nine dots arranged in the shape of a three by three square - three rows horizontally, three columns vertically. I don't know what you thought when you read the title of the problem, but I'm willing to argue - when you saw the picture, you were a little puzzled. Most people, after thinking about the task for a few minutes, will be forced to admit that they have no idea how to do it.
Why not resolved
And for that it is a creative task, so as not to be solved easily. From childhood, we are driven into the framework - the average person is the easiest way to teach the minimum necessary for life and survival. We are accustomed to minimal logic, but it is really minimal, and contains, as a rule, only one solution to each question. But this option is allegedly 100%, and creativity - well, this ... maybe there will be no use from it. Let me guess - the shape of the dots immediately made you think of a square. But no, guys, this is the simplest solution - but there are five lines, and you try to make four.
Solution
Discard completely what is commonly called logic. Who said that you at least once drew the figure we needed at school on geometry? What makes you think that the lines must end at one of the points?
From the top left point, draw a line down through the entire left column, but let the line continue on, not end at the bottom row. Draw a similar line from the same point horizontally to the right. Now, from the same point, draw a line through the center point to the opposite one. You already guessed, right? The last line should pass through the middle point of the bottom row and the middle point of the left column, connecting to the first two lines outside the square of dots.
It doesn't matter which corner of the square of dots you start drawing the lines from, but the result should be an umbrella. But admit it - you did not even think about it.
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: