Drawing without taking your hand off. Solving the problem of how to draw an envelope without lifting your hand

Instructions

It is assumed that the given figure consists of points connected by straight or curved segments. Consequently, at each such point a certain segment converges. Such figures are usually called graphs.

If an even number of segments converge at a point, then such a point itself is called an even vertex. If the number of segments is odd, then the vertex is called odd. For example, a square in which both are drawn has four odd vertices and one even vertex at the point of intersection of the diagonals.

By definition, a line segment has two ends, and therefore always connects two vertices. Therefore, by summing all the incoming segments for all vertices of the graph, you can only get an even number. Consequently, whatever the graph, there will always be an even number of odd vertices (including zero).

A graph in which there are no odd vertices at all can always be drawn without lifting your hand from the paper. It doesn’t matter which peak you start from.

If there are only two odd vertices, then such a graph is also unicursal. The path must begin at one of the odd vertices and end at another of them.

A figure in which there are four or more odd vertices is not unicursal, and it will not be possible to draw it without repeating lines. For example, the same square with drawn diagonals is not unicursal, since it has four odd vertices. But a square with one diagonal or an “envelope” - a square with diagonals and a “lid” - can be drawn with one line.

To solve the problem, you need to imagine that each drawn line disappears from the figure - it is impossible to go through it a second time. Therefore, when depicting a unicursal figure, you need to ensure that the rest of the work does not fall apart into unrelated parts. If this happens, it will no longer be possible to complete the matter.

Sources:

• How to draw a closed envelope without lifting your hand?

A square is an equilateral and rectangular quadrilateral. It's very easy to draw. Start your workout first on a squared notebook. Using a simple pencil and an invisible square of dots, learn to draw a square without lifting your hand from the paper.

You will need

• - a simple pencil;
• - checkered leaf;
• - sheet A4;
• - ruler.

Instructions

First, let's take it into a cage; it's convenient to draw a square in it. Stepping back approximately 3 cm from the left edge and above, place a dot. From it, to the right, count 5, place another point.
Then from these points down the line we count another 5 cells and put 2 more points. The result is an invisible square. And using a pencil, carefully connect 1,2,3 and. A square measuring 2.5 by 2.5 cm is ready.

You can use such a square on a regular A4 size, with a side of 3 cm. Place the sheet vertically. Step back 10 cm from the top edge of the paper. Use a ruler to place the dots in a straight line. Attach the ruler to the left edge so that the edges of the ruler and paper coincide, this is necessary for the correct drawing of the square. Measure approximately 5 cm from the edge (for the margin) and place the first point. Further to the left, after 3 cm there is another point - the second. Then rotate the ruler 90 degrees. The beginning of the ruler will coincide with the top edge of the paper, and from the first point down, measure 3 cm, place the third point. Move the ruler to the second point and down from it, at a distance of 3 cm we place the fourth point. Now carefully connect all the dots using even lines, without lifting the pencil from the drawing.

We were inspired by Japanese animator and illustrator Kazuhiko Okushita.

The artist creates drawings without lifting the pencil from the paper. A very useful activity! Develops imagination, thinking, sharpens graphics and trains the hand.

Lera couldn't stop))

Children's imagination never sleeps! This is not the whole result of her vigorous activity) But the sharks struck me! Everything was drawn by my daughter without lifting her hand.

And then we came up with a way for Yegor to draw without lifting his hand.

For this drawing you will need: PVA glue - a lot, threads - any thick ones, A3 sheet, paints and brushes.

First, pour the glue into a convenient container, dip the thread into the glue - it should be thoroughly saturated with PVA.

Then we take it out like this.

Or like this))

By the way, it’s very interesting to play with glued hands)

And put the thread on a sheet of paper. Forming a pattern. If your thread breaks, you need to place a new one at the end of the old one. But in principle it is possible in any order.

And Lera liked to peel the dry glue off her hands afterwards)) The activity is very multifaceted)))

And now we’re adding paint!

Egor was so carried away that he even painted with his fingers.

I think everyone should like this drawing! Show us what you got!

Let's say right away that this is a tricky question. Or rather, the solution itself, like most similar problems, is not based on logic, but rather on creativity. M We are sure: desire and work, self-education and experience will help you reach a new, non-trivial level of thinking.

The solution without further ado: how to connect nine dots using four lines?

This mystery has intrigued hundreds of thousands of people. The following conditions must be met: cross all nine points that form a square using straight lines (no more than four).

In this case, you cannot take your hand, or rather a pencil, off the sheet. The next line should begin where the previous one ended.At first glance, it is not so difficult, but in reality, each subsequent attempt often distances the inquisitive mind from a positive result.

The thing is that from childhood we were taught to think based on certain patterns and rules.First of all, logical thinking developed, on the principles of which our world is built. Yes, but not so.

Here you need to go beyond logic and stop thinking within the boundaries of the four sides of the square and its diagonals.

We analyze the problem based on knowledge about the object, but we should simply remember that a straight line is not necessarily limited by the boundaries of the form, that is, it is possible and necessary to go beyond the boundaries.

Let’s conditionally number each point from 1 to 9:

1. We draw the first line, starting from point 1 through 4, 7 and going beyond the boundaries of the figure.
2. Without lifting your hand from the sheet, we make a corner and strive for the point numbered 8 and 6 and go beyond the limits in the same way.
3. Next we turn and go through 3, 2, 1.
4. We turn through the corner of the square, going the rest of the way through points numbered 1, 5 and 9. It turns out to be a kind of arrow-cursor, which can be directed to any of the four corners at your request.

There is also a “hardcore” method for those who have spatial thinking.On a square piece of paper (sticky note), draw nine circles (as in the problem). Apply glue under the 7th and 8th points.

Take a cylindrical base. A tube of decorative cosmetics (lipstick or foundation) is ideal. Connect the place under 7 and 8 with the place under 2 and 3.

Draw one continuous line, starting from point No. 1 and going down in a spiral.When you return the leaf to its original form, you will see that three lines are 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 points directly on paper, this will make it easier to figure out the solution.

Games for home leisure

At one time, Steve Jobs, a man synonymous with the word “creativity,” emphasized that people who master 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 up” such observation, over surrounding phenomena and things more often.

Game No. 1

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, not excluding mental concepts.

For example, "m":

1. Furniture, zipper (on clothes), chalk (pet food)
2. Opinion, peacefulness, mannerism
3. Milk, materials (upholstery), T-shirt
4. 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”.Don't limit yourself and your innate imagination.

If desired, you can find about 40+ words in one room. Experts find approximately 100 words in each room.

Game No. 2

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; another name for it is “burime”.

To dive into the process, you will need a few sheets of paper, a pen, and a good company that doesn’t mind practicing collectively composing poetry. ZThe topic and limitations are specified in advance.

H Most often, obvious combinations of cognates, pronouns, verb forms and hackneyed platitudes (hello-dinner, love-carrot) are excluded. Sometimes a specific topic is discussed.

It happens like this: someone writes a line, and another complements the verse with the next one until a full-fledged work is obtained.

Game No. 3

This is for all ages, even the little ones.It develops spatial skills, which will certainly be useful in the adult future.

Place your child at a table and give him a black pencil and a piece of paper. Turn on some nice music and ask him to close his eyes.Let the child draw, interweave random lines among themselves, 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, using colored pencils, highlight shapes similar to animals, objects, and all kinds of images. Let the child himself be the source of ideas.

Tip: puzzles with matches (sticks) will be an excellent workout for the mind. Such small puzzles will be interesting for both children and adults. They are available to everyone!

Exercises to develop creative thinking

Stand up. Take any book from the shelf. On two different pages, blindly select a few words.Now try to find everything in common that they might have. For example, the word “carpet” and “tree”: they both lie on the ground, their images are found in fairy tales (a flying carpet, a tree along which the learned cat walks), etc.

If you play with a child, choose simpler words: cat-dog, tomato-pear, table-chair.Write a dozen of any nouns on a piece of paper: “strawberry”, “fish”, “water”, etc. Now imagine that this sheet is the customer’s requirements, and you yourself are a builder-architect.

Build a house using these as basic requirements.For example, the wallpaper will be red “strawberry” in color, and the walls of the house will shine in the sun like fish scales. Let the house itself stand on the top of a mountain, where the sky is simply infinitely blue, like water, etc.

While sitting in a room, find an object within your sight that is familiar and interesting to you. For example, “apple”.

Come up with five adjectives that go perfectly with the subject:

1. Green
2. Sour
3. Delicious
4. Soft
5. Juicy

Now we complicate the task and come up with five more adjectives, but those that are absolutely inappropriate in meaning: prickly, rough, plush, tin, slender.Some words are not so easy to work with, but that makes the task more interesting: well, train, wind, wall.

Take a pencil in your hands and draw a column of crosses in a checkered notebook.The width and height are arbitrary, just make sure they are far enough apart from each other.

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 short stories with action.

Make up a whole story! It's not as difficult as it might seem at first glance.

Tip: read the words backwards: fairy tale - akzaks, bottle - aklytub, spoon - akzhol. This is certainly a useful activity that will help pass the time waiting in line or on public transport.

Online Games to Improve Creative Thinking

IQ-ball

You are a small, round, living ball with a suction cup flying out of your body.The goal is to get candy at each level, overcoming all kinds of obstacles. You will be hampered by fixed and moving elements, time limits, and inertia.

It is not possible to push off or cling with your paw from all surfaces. You will have to think quickly, the reach of the goal depends on this.

"Black cat"

In front of you is a field created from circles. A black cat sits in the middle of it. With a mouse click, you can fill a mini-area through which the cat can no longer pass.

One move is made by you, the next move is made by a cunning animal.Your task is to prevent him from running out of the edge of the playing field, because this means losing. Here you will have to use all your intellect and creative thinking, and most importantly, choose the right battle strategy.

In this case, we can advise you not to rush, but to think through your move in advance, marking the circles after one.In this case, you will always have time to block the path for 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.

The same “doodle” can carry several meanings at the same time:

1. Facade
2. Wedge
3. Cloak
4. Diamond, etc.

The positive result of the game is the acceleration of thought processes, the diversified development of imagination and creativity. Such simple fun can captivate you for a long time.

"Memory Matrix"

Many adults and children are familiar with this game. In front of you is a field that is filled with squares for a few seconds.Then they disappear. Your task is to find these very figures “from memory”.

With each subsequent level the field grows and the task becomes more complicated. The game develops memory, creativity and the ability to quickly concentrate.

Advice: try playing Lines 98. At the same time, it develops logical thinking.

Tricky, educational 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 jewel hunter who is beyond this land. The arsenal consists of only two planks, the length of each is slightly less than the width of the ditch.

It is impossible to jump over or fly over it, there is no rope to fasten the boards together, as well as nails, and each one individually is easy to fall into the abyss.

The goal is to get to the treasure. The answer to this riddle is based on the principles of geometry: “place” the first board on the corner of the ditch 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.Place a bold 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 point itself.

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 is it that only round pizza is made all over the world, but delivered 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-shaped dishes.

In the case of a box, several factors are important:

1. This makes it easier to get food out of it
2. Square boxes are much cheaper and easier to produce than round ones.
3. The pizza seems more impressive in it

Advice: train your brain with small tasks at least several times a week and very soon you will feel that it has become much easier for you to find innovative solutions at work and in life, to think outside the strict framework of logical thinking.

It is difficult to captivate modern children with anything. They love watching cartoons and playing computer games. But smart parents are always able to interest their child. For example, they might ask him to find a way to draw an envelope without lifting his hand. Read below about some of the tricks of this task.

Warm-up

Before you start tormenting your child with logical tasks, you need to do preparatory work with him. Why is it needed? So that the child does not cheat when he starts to puzzle over the question of how to draw an envelope without lifting his hand. After all, the most interesting thing in this problem is that the line must go from point to point continuously.

What tasks can be offered to a child as a warm-up? Of course, the first thing should be eights. Drawing this number relieves stress, cleanses the brain, and trains the hand. In general, a useful exercise. After this, you can move on to drawing rounded shapes. These can be curls or any other squiggles, the main thing is that during the drawing process the child does not lift the pencil and depicts everything in one smooth line.

How to draw a closed envelope

Many parents themselves spent more than one hour before offering such a task to their child. You can try it too. But we can immediately disappoint you - it is simply impossible to complete such a task without cheating a little. Therefore, we will tell you a method that will help you and your child go a little beyond ordinary logic in order to understand how to draw a closed envelope without lifting your hand.

Take a sheet of paper and bend its edge. We bend it back. Now our task is to draw the top edge of the closed envelope just on the fold line. To make it easier to understand, let's place dots at the ends of the rectangle. Let's number them starting from the upper left corner. The number one will appear here and further clockwise. From the number 4 to 1 we draw a line, now we connect 1 to 2 and now we draw a diagonal to 4. From 4 to 3 we draw a straight line, and then again a diagonal to 1.

Now let's get to the fun part. We bend the edge of our sheet and draw a zigzag, which forms, as it were, the head of our envelope. It will go from 1 to 2. All that remains is to connect 2 and 3 with a straight line - and the puzzle is solved. Bend part of the sheet back. The riddle of how to draw an envelope without lifting your hand can be offered not only to children, but also to friends or colleagues.

How to draw an open envelope

Those who carefully read the previous paragraph and created their own drawing based on the description already understood how to answer the question posed above. After all, the solution to the riddle of how to draw an open envelope without lifting your hand will be similar to that written in the previous paragraph. Only here you won’t have to bend and bend parts of the sheet. The entire image will be made with one line according to the same pattern.

But if you don’t want to repeat yourself, then we offer another method that will lead to the same result. How to draw an envelope without taking your hands off using the second method? To begin with, we again draw a rectangle with dots and number it again, as in the previous paragraph. From the number 4 to 2 we draw a diagonal, from 2 to 3 we draw a straight line, and from 3 to 1 we draw a diagonal again. Next you need to draw a corner. From 1 to 2 we draw a zigzag, which marks the top of the envelope. From 2 we return to 1 with a straight line and complete our construction by alternately drawing straight lines from 1 to 4 and from 4 to 3.

Why are such tasks needed?

These should be done not only for children, but also for adults. Thanks to them, the human brain tenses up and begins to work. If you train yourself to perform a similar task every day, after a month you will notice that in critical situations, solutions are generated faster and less effort is spent on it. It is especially useful for schoolchildren to study logic problems. In this way, they train creativity and learn to approach standard issues in an unconventional way.