Economy      04/09/2020

Riddles about geometric. Riddles about figures. Riddles about geometric shapes for children and preschoolers

It is rare to find a person without small dark marks on the body. Should we pay attention to these points? Only a doctor will distinguish between dangerous and normal moles - malignant melanoma or a harmless nevus - and will give recommendations on what to do with them. Is it worth worrying about the appearance of new formations, when an immediate appeal to specialists is required, what are the signs of cancer development - the answers to these questions remain to be known. No one is immune from trouble, and early diagnosis will save you from severe consequences.

What is a mole

The first tiny spots may appear in infants. A mole is a small formation on the skin - a nevus - which is considered benign, harmless. The basis of their appearance is melanocyte cells that accumulate the natural pigment melanin. Depending on its quantity, there is a difference in color. Available colors:

  • black;
  • pink;
  • brown;
  • blue;
  • violet;
  • transparent, colorless.

The form of neoplasms depends on the location, concentration of melanin. They can have a leg or be located under the skin, be flat and convex. The most common form is round, but there are exceptions. The development of neoplasms provokes ultraviolet radiation - natural from the sun, in a solarium. Hereditary factors have not been excluded. Common cause growth - a hormonal imbalance characteristic of periods:

  • puberty;
  • pregnancy;
  • menopause.

What are moles

One person can find very different neoplasms in himself. Types of moles are classified according to several criteria. This helps correct diagnosis in case of changes. They differ in:

  • origin- congenital, newly acquired;
  • structure- pigment, vascular;
  • place of education– in depth, on the surface, in the boundary layer;
  • elevation above the skin- flat - even, protruding by a hemisphere, on a leg, larger birthmarks;
  • potential threats- dangerous, degenerating into melanoma, non-dangerous.

Safe moles

Those with dark spots on the skin should be wary of their changes. In time, the detected signs of degeneration into melanoma contribute to the timely removal of the formation and the preservation of health. Safe moles are different:

  • the presence of a leg - it cannot be formed by malignant cells that grow randomly;
  • long-term state without change.

Spots that appear shortly after birth are not considered dangerous. It is important that they are small. Good - harmless - signs of neoplasms include:

  • flesh tone;
  • unchanged pattern of the skin of the nevus and adjacent tissues;
  • soft texture;
  • hair on the surface of the neoplasm - growing from the skin, indicates the absence of pathologies;
  • diameter not more than 5 mm;
  • symmetry;
  • in the form of a spot.

What moles are dangerous

Why do people with nevi on their bodies need to monitor their changes? There is always a threat of degeneration of non-dangerous neoplasms into a cancerous tumor. What moles are dangerous to health? Key signs to be aware of:

  • change in shades to the dark side, the appearance of multicolor;
  • rapid increase in size - more than two millimeters per year;
  • the occurrence of cracks;
  • the formation of asymmetry due to uneven growth;
  • lack of elasticity;
  • the appearance of itching, burning;
  • the presence of discomfort.

The appearance of dangerous moles requires an immediate visit to a specialist to clarify the nature of the changes, the likelihood of developing skin cancer. Pathological transformations provoke:

  • injury to the nevus by negligence;
  • self-deletion;
  • abuse of sun exposure, solarium use;
  • the location of the formation in places of frequent contact with clothing - on the neck, head, genitals, legs;
  • placement in the hair, on the face, palms - where there is a high probability of injury;
  • previously removed melanoma.

Why are moles dangerous?

Not a single person is protected from the sudden growth of cells of a harmless mole. Melanoma is an extremely serious disease. Changes not detected at the initial stage can be fatal. The provoking factor is the unsuccessful self-removal of neoplasms. Moles are dangerous with the ability:

  • go into an atypical - precancerous form;
  • grow to a large size;
  • turn into cancer
  • with small external changes, actively spread metastases throughout the body along the circulatory and lymphatic channels.

How quickly does melanoma develop from a mole?

The degeneration of a nevus into a cancerous formation can take place in different ways. The process depends on the stage of the disease, the type of tumor. Instant metastases are dangerous. Starts:

  • growth of cancerous (oncological) cells in the deep layers of the epidermis;
  • getting them into the blood, lymph;
  • penetration into the lungs, liver, kidneys;
  • growth in these organs;
  • complete damage to the body;
  • death.

The growth phases of pigment cells are observed, along which melanoma develops from a mole. There are varieties:

  • horizontal- there is a lesion of the upper layers of the skin, lasting up to 10 years, metastases do not appear;
  • vertical- accompanied by the spread of cancer cells throughout the organs, can last two years, has an unfavorable prognosis;
  • nodal- especially dangerous - it is characterized by deep spread within two months.

The first signs of melanoma

It is possible to help the patient only with the beginning of the detection of suspicious changes. The diagnostics, research, referral for surgical treatment save a person's life. The first signs of melanoma:

  • increase in the height of the neoplasm;
  • bleeding;
  • the appearance of secretions;
  • redness;
  • burning, itching;
  • tissue swelling;
  • softening of the nevus;
  • the appearance of a crust;
  • thickening;
  • hair loss;
  • expansion of pigmentation around the lesion.

With the further development of dangerous melanoma, the following are observed:

  • significant change in size;
  • the appearance of pain;
  • enlarged lymph nodes;
  • surface ulceration;
  • the formation of new foci;
  • bleeding from sites of pigmentation;
  • fluid separation;
  • skin thickening;
  • the appearance of an earthy shade;
  • signs of metastases - chronic cough, weight loss, convulsions, headaches.

How to distinguish a mole from melanoma

To recognize which moles are dangerous and non-dangerous, you need to know how they look. A person who has nevi, in order to exclude terrible consequences, must constantly monitor the appearance of new formations, the changes taking place. You can distinguish a mole from melanoma by signs. Non-hazardous neoplasm:

  • symmetrical;
  • with smooth edges;
  • uniform in color;
  • with dimensions not exceeding 6 mm.

Features of dangerous melanoma that require seeking help from dermatologists:

  • growth in a short time;
  • pronounced asymmetry of the form;
  • heterogeneity in color - the presence of inclusions of several shades;
  • lack of clear boundaries - the line along the contour is blurred, indented, in appearance resembles a coast on geographical map;
  • increased diameter over six millimeters;
  • variability of any parameters - color, size, shape.

What dangerous moles look like

How do nevi subject to pathological changes look? Only a doctor will help to correctly distinguish benign neoplasms. Dangerous formations look like this:

  • blue- seals under the skin with clear boundaries, with dimensions not exceeding 10 mm;
  • nodal– round, flat in shape, color – brown, black;
  • skin- often pale, prominent;
  • halo nevus- a pigment surrounded by a light or white rim;
  • Spitz- looks like a dome-shaped tumor of pink shades, with the possible presence of a hole through which blood and fluid leak;
  • connecting- connect separate formations into a whole.

Mole with jagged edges

One of the signs of the transformation of a non-dangerous formation into a dangerous one is a change in contours. Often it has blurry edges, scalloped borders. There are non-dangerous types of nevi - dysplastic. Only a specialist can make a correct diagnosis. A mole with uneven edges can be dangerous in case of additional signs of melanoma development:

  • accelerated sizing;
  • the presence of a clearly defined asymmetry;
  • the appearance of strongly indented boundaries.

Rough mole

Such a neoplasm is harmless if the diameter is not more than 5 mm and retains constant dimensions. Often its appearance signals a lack of vitamins, malnutrition. Doctors advise you to come for a consultation if you find that:

  • a smooth nevus turned into a rough one;
  • worried about burning, itching, tingling;
  • irregularities, seals appeared in the middle;
  • areas with different shades formed;
  • diameter has increased significantly.

A dangerous rough mole requires immediate examination in case of:

  • the appearance of bleeding;
  • development of the inflammatory process;
  • rapid resizing;
  • formation of asymmetry;
  • the formation of purulent secretions;
  • the occurrence of pain when touched;
  • the emergence of an irregular shape, blurred boundaries, along the edges of the neoplasm.

Large moles

Formations on the skin of a large size - age spots. When they remain unchanged, do not cause inconvenience - this is a non-dangerous phenomenon. It is important to constantly monitor their appearance, color, size. To eliminate anxiety, you need to consult a dermatologist. During the visit, the specialist will diagnose, give a forecast of the risk of developing a malignant neoplasm. Large moles become dangerous if they:

  • injured;
  • condensed;
  • began to itch;
  • were unsuccessfully self-removed;
  • changed in size, shape;
  • bleed.

What moles can be removed

Often, nevi cause trouble for women, being in a conspicuous place - the face, neck. Even if they don't bother, using delete would be the right solution - appearance will improve significantly. After the procedure, the doctor must send the tissues to the clinic to decide if the mole is malignant or not. If the neoplasm is harmless, does not bother, does not change size, then surgery is not required. What moles cannot be removed? Experts believe:

  • there are no contraindications;
  • it is important to choose the right method of excision.

You should be attentive to skin neoplasms, it is unacceptable to use their self-removal. Only the doctor will determine a dangerous or non-dangerous nevus, decide what to do with it. Can be removed if:

  • are injured from clothing - on the neck, in the groin, armpits;
  • cause pain when touched;
  • are under the hair on the head, can be damaged when combing, cutting;
  • change color, shape, shape;
  • significantly increase in size;
  • differ in the presence of burning, itching;
  • accompanied by inflammation and bleeding.

Discuss

How to distinguish between dangerous and non-dangerous moles on the human body

Today we would like to tell you about the appearance of moles on the human body. The article reveals such issues as the occurrence, spots and, during pregnancy. The material promotes awareness of the safety of moles for a number of reasons and calls for medical attention if moles begin to behave unusually.

Period of occurrence

Moles appear on the human body from birth, although initially a person is born without pigment spots at all. Their main number falls on adolescence, when the formation of the human body takes place. They can appear and disappear, manifest themselves in very different ways, grow or die, but every inexplicable behavior of a mole should be perceived as a danger, and a person who has noticed moles on the body or their change is simply obliged to consult a doctor in order to avoid more serious problems than itching or peeling of a mole.

All moles can be radically different from each other and be of different colors: brown, skin colors, black, pink, white and even purple, and also be of different shapes and structures: angular, flat, bumpy and sharp, or even subcutaneous and almost invisible .

Causes

Moles can appear on a person's body at any time in his life, despite the fact that the main period of their birth falls on the age of up to 25 years. Moles can appear anywhere and anytime, just as they can disappear. Let's look at the various causes of the appearance of moles:

The appearance of moles in humans may be due to hereditary type information that is embedded in DNA. Hence the name of moles, as they are often inherited and occur in the same place as in older relatives.

Emergence from the sun

One of the most effective on the occurrence and growth of moles is solar radiation. Under its action, a certain substance, melanin, is produced quite intensively in the human body. It forms the basis of moles. An excess of melanin, which is perfectly produced by the body under ultraviolet rays, forms huge colonies of new moles on the surface of the skin, which is already a risk factor. Each mole with an increased amount of sun exposure is able to transform into a tumor, moreover, a malignant one.

Moles due to trauma and some viruses

At the same time, a huge number of doctors are sure that moles appear due to radiation and x-ray radiation which each person takes at one time or another. Another opinion is a viral infection and insect bites, which leave their marks on human skin for a long time. With any of the above effects on the body, a process can be launched during which melanocytes are grouped and find an exit on the surface of the skin.

The opinion of alternative medicine

The appearance of moles on the body is due to the release of internal energy. Energy accumulates at the site of inflammation, concentrates over time and provokes the appearance of age spots and moles on the skin.

Hormonal surges

The release and formation of melanin, which in turn pigments the mole, is strongly affected by the hormone that is secreted by the pituitary gland. For this reason, any hormonal surge in the human body can lead to the multiple birth of new moles. This may explain moles during pregnancy or during puberty. Such surges can lead not only to the birth of moles, but also to their disappearance.

How to prevent the occurrence

After reading everything that is written above, and realizing the causes of moles on the body, you can try to prevent the growth in the number of moles. For this, according to experts, the most important thing is to refrain from increased consumption of solar radiation, that is, to sunbathe less during the solar zenith and minimize visits to the solarium, for a start this will be enough.

New moles

The appearance of new moles on the body is often explained by the influence of adverse factors on the body. It could be severe stress, hormonal surge, radiation exposure and adverse ecological situation. If the appearance of new moles worries precisely because of their number, then it is worth spending less time in the sun and protecting your skin with the help of special creams, lotions and protective filters. Often, moles are a normal behavior of the body, until such time as they begin to disturb the owner and bring physical or aesthetic discomfort. It is worth paying attention to the growth of a mole, an increase in the number of moles, peeling, itching and other factors of a sharp change in the structure and size of moles on the body.

Variety of birthmarks in children

The appearance of red or any other moles on the body of a child can be due to a wide variety of reasons and be both dangerous and safe. As a rule, only moles appear on the skin of babies belonging to one of the groups:
  • Light brown moles or skin color, not too noticeable, most often remain for life. If their number grows, then the concern is justified and it is necessary to consult a doctor for advice.
  • Birthmarks of dark color and different shapes. They can appear on any part of the body. They are also harmless and can last a lifetime, but with any changes in shape, color and quantity, a visit to a doctor is mandatory.
  • Red moles on the face, back of the head and neck of the child. in children due to the expansion of small vessels during childbirth. Single manifestations may remain for a long time, but identical and mirror red moles may disappear within the next year.
  • The spot is Mongoloid. Most often appears in children with swarthy skin, located on the buttocks of the child or in the lower back. The blue-lilac mark does not need any treatment and most often disappears by about 13-15 years.
  • Hemangiomas. They are convex or flat.
Convex - a clear indicator that moles are formed during pregnancy and appear during this period, and sometimes immediately after the birth of a child. They do not bring any negative sensations and may disappear in a few years, but if convex hemangiomas are located in places of risk, where they can be damaged, then it is advisable to cure them. It is necessary to start treatment even at a young age, since convex hemengiomas, if they have not disappeared, can grow, and up to 15 years can turn into a huge spot. And, finally, flat formations - they are not dangerous and do not grow over the years, and therefore it is not necessary to remove them at all.

Doctors' warnings

All doctors insist on the same thing. Removal of moles is not at all a mandatory process, since in themselves they do not bring any harm, except for moles on the face that can spoil a person’s appearance or those moles that are located on dangerous areas of the body: on the neck, on the eyelids, on the toes and so on. These moles can be easily damaged and cause some anxiety. In addition, the appearance of moles on the body most often does not cause any harm to the body, except in those cases if you notice signs of the transformation of a mole, which are the first reason to see a doctor:

  • compaction and a sharp change in the size of the mole;
  • discoloration of the mole, sharp pigmentation in a dark color;
  • cracks in the mole, peeling, bleeding;
  • inflammation of a mole or its areola;
  • violation of the structure of the skin pattern on the surface of moles.
Having considered all the warnings of doctors, the reasons for the appearance of moles, we can conclude that if the mole does not behave unusually and does not cause discomfort, then the only reason for its removal may be an aesthetic and emotional reason, and if this should happen, then we recommend laser removal of moles or removal with nitrogen.
The procedures are practically painless and do not leave visible marks on the skin, but should only be carried out by qualified specialists.

Video: How to recognize spots on the skin

Dot
Let our courtyard today, it’s gloomy outside the window.
I took a felt-tip pen, a pencil, decided to draw figures.
Before me is a sheet of paper, how white and clean it is.
With a felt-tip pen, poke in the middle of the leaflet.

And a dot is obtained on the sheet.

Line
Let there be a lot of points, I lead the way through them.
By connecting dot to dot, I drew a line-line.
The path, curving, winds, the path is called a line.

Straight line
My mother advised me to drive straight ahead.
How to make a straight line - does not work.
Is my felt-tip pen lame or does my hand go astray?
But with a ruler on a sheet, it’s so easy to draw a line.
Look, it's a straight line, it's a straight line.
Angle (acute, straight, obtuse)
My mother took the paper and turned down the corner.
An angle like this in adults is called right.
If the angle is already sharp, if it is wider, then it is obtuse.

***
I'm sharp - I want to draw, now I'll take it and draw it.
I lead two straight lines from a point, as if two beams,
And we see sharp corner we are like the edge of a sword.
And for the obtuse angle, we repeat everything again:
We draw two straight lines from a point, but we will separate them wider.
Look at my drawing, it's like scissors inside,
If we take two rings, we will push it to the end.

Triangle
The plane flies through the sky, delta wing
My bike has a triangular saddle
There is such an object - a square, and all this is a triangle.
Then my mother put three matches on the table
And she folded a triangle of matches for me.
And at this time I drew and watched my mother,
I connected three straight lines and did the same.

***
We will build a house with you
The roof will be a triangle,
The corners of the roof are sharp,
How many? One two Three!
Square
An older brother came from school, laid out a square from matches.
Mom gave me chocolate, I broke off a slice - a square.
And the table is a square, and the chair is a square, and the poster on the wall is a square.
The board where the chess stands, and each cell is a square,
There are horses and elephants, fighting figures.

***

Here are the four sides
And they are always equal.
And that figure, guys,
It's called a square.

Circle and circle
We live together with my brother, we have so much fun together,
We will put a mug on the sheet, circle it with a pencil.
It turned out what you need - the circle is called.
My drawing brother considers himself a master,
Everything inside the circle, he painted over with a felt-tip pen.
Here is a red circle, a circle, a blue rim around the edge.
circle - plate, wheel, circle - hoop, belt.
circle - the outline of a circle. I look at our sheet
He began to look for a corner near the circle, but could not find it.
Brother laughs - that's it! Yes, the circle has no corner,
You won’t find corners on a plate and a coin, they don’t exist.

***
Let's draw a circle:
Mouth in it and a couple of dots.
The sun is round and the ball
The circle has long been familiar to us, that means.
Trapeze
Trapeze, trapezoid, there is such a figure,
And I don't know her. Where do you live, trapeze,
America, China? Maybe behind the trapeze
Should you go to Greece? Mom says don't
Trapeze next to you. I will dispel your sadness
You wait a minute. And on the ironing board
Lays her skirt, irons it,
So as not to puff up the bag:
- Here's a trapeze for you, you shouldn't go to Greece.

Oval
How to draw an oval? I called my brother for help.
My brother took a felt-tip pen and skillfully drew an oval for me:
You slightly flatten the circle, you get an oval.
How many times have I seen him, in the bathroom the mirror is an oval!
An oval and a dish and an egg. Mom says face
Yours is oval. Let it be oval
As long as it's not sad. We laughed and in the oval
The face was drawn. Oval - elongated circle
And her face is surprised.

***

An oval hangs on the wall
I saw myself in him.
I like to look in the mirror
In front of him I like to spin
Cube
The postman brought us a box - a parcel for me and my brother.
The box is a cube, it has six sides, all sides are squares.
What is in the package? There are shavings and sawdust
Sweets and bagels, jars with jam.

Cylinder
- Cylinder, what is it? I asked my dad.
Father laughed: - Top hat, this is a hat.
To have a correct idea,
The cylinder, let's say, is a tin can.
The pipe of the steamer is a cylinder, the pipe on our roof is also,
All pipes are similar to a cylinder. And I gave an example like this -
My beloved kaleidoscope, you can't take your eyes off it,
It also looks like a cylinder.

Cone
Mom said: - And now
About the cone will be my story.
Stargazer in a high cap
Counts the stars all year round.
The cone is the stargazer's hat.
That's what he is. Understood? That's it.
Mom was at the table
She poured oil into bottles.
- Where is the funnel? No funnel.
Look. Don't stand on the sidelines.
- Mom, I'm not going to move,
Tell me more about the cone.
-The funnel is in the form of a watering can cone.
Come on, find me quickly.
I couldn't find the funnel
But mom made a bag,
Wrap cardboard around your finger
And deftly fastened with a paper clip.
Oil is pouring, mom is happy
The cone came out just right.

Pyramid
I saw the picture. In this picture
There is a pyramid in the sandy desert.
Everything in the pyramid is extraordinary,
There is some mystery and mystery in it.
The Spasskaya Tower on Red Square
Both children and adults are well known.
Look at the tower, ordinary in appearance,
What's on top of her? Pyramid!

Ball
Hit! Hit! Another hit!
A ball flies into the gate - a ball!
And this is a ball of watermelon green, round, delicious.
Look better - what a ball! It is made up of circles.
Cut the watermelon into circles and taste them.

figure painting
Holding scissors like a sword
I decided to create a picture,
colored paper
I cut like a web.
And in the corner of the album sheet
I glue a square at home.
The path is stretching like a ribbon…
From a square window
It can be seen from the forest edge
Come to your feeder
Without fear, animals can:
At the rectangular door
Food for them is left in a niche,
Eat boldly, only quietly:
There on the triangular roof
With a ringing song
The nightingale perched.
Clouds above him, and above,
Illuminating everything around
The circle of the sun is golden!
So as not to grieve in winter
For those who will live in the house
I really need a garden
Fruits vegetables. And so:
Finely cut strips
There will be boards for the fence,
And behind them for order
There will be an earthen bed.
Hey, who's in the house! Do not snooze!
Harvest!
Two circles - two watermelons,
Three ovals - corn
And a huge circle of turnips
To snatch - you need a friend,
Well, I glued it very tightly -
This is a fabulous turnip.
Where is parents? Look!
What I did, appreciate!
Mom and dad praise their son
His painting is a success!

Riddles about geometric shapes

Slightly flattened square
Invites to know:
Acute angle and obtuse
Forever bound by fate.
Guess what's the matter?
How do we name the shape? (Rhombus).
***

The wheel rolled
After all, it looks like
Like a visual nature
Only for a round figure.
Got it, dear friend?
Well, of course, this is ... (circle).
***

look at the figure
And in the album draw
Three corners. three sides
Connect with each other.
It turned out not a square,
And beautiful ... (triangle).
***

I am a figure - no matter where,
Always very smooth
All angles in me are equal
And four sides.
Cube is my favorite brother
Because I…. (square).
***

He looks like an egg
Or on your face.
Here is a circle -
Very strange appearance
The circle became flattened.
It turned out suddenly .... (oval).
***

triangle filed
And got the shape:
Two obtuse corners inside
And two sharp ones - look.
Not a square, not a triangle
It looks like a polygon. (Trapezoid).
***

Let's take a part from the line
And let's call the shape
Not a piece - it's too sharp,
And probably…. (segment).
***

In mathematics, she
Always nice:
No trailing comma
It seems simple to all of us.
And at the end, finishing the line,
We will put, brothers, .... (point).
***

Looks like a circle, but the thing is
What else do we call
Drawn circle.
What's the secret? Tell me friend!
This strange appearance
It's called…. (circle).
***

Fitted in shape
Very fine needle
Not a line and not a straight line
What kind of line is this?
alive in mathematics
This one is very smooth ... (beam).
***

Six obtuse corners inside
Look at the figure
And imagine that from a square
Got his brother.
Too many corners here
Are you ready to name it? (Polygon)
***

We stretched the square
And presented at a glance
Who did he look like
Or something very similar?
Not a brick, not a triangle -
It became a square ... (rectangle).
***

Under the ruler I draw
Very flat, simple
A notable trait for everyone.
How do I name the figure?
Don't rush, talk
And give me an answer soon. (Line).
***

All shapes repeat
And calmly call:
Here is an oval, a circle, a circle ...
Go on, dear friend!
***

Riddles about geometric bodies

How can we not turn it
There are exactly six equal faces.
We can play loto with him,
Let's just be careful:
He is neither kind nor rude
Because it is ... (cube).
***

We get down to business again
Studying the body again:
Can he become a ball
And fly a little.
Very round, not oval.
Guessed? This is ... (ball).
***

The Egyptians laid them down
And so cleverly made
What are they worth for centuries.
Guess yourself kids
What are these bodies?
Where is the peak visible to everyone?
Guessed? Because of the look
Everyone knows ... (pyramid).
***

Look, there is a bucket -
Lid on top, bottom on bottom.
Two circles connected
And got a figure.
What is the name of the body?
Gotta figure it out quickly. (Cylinder).
***

It looks like a bucket
But a completely different bottom:
Not a circle but a triangle
Or even a hexagon.
The body is very capricious
Because it is ... (prism).
***

Here is the cap on the head -
It's a clown on the grass.
But the cap is not a pyramid
You can see it right away, brothers:
Circle at the base of the cap.
What is his name then? (Cone).
***

Let's take a book with a ball -
Let's call them bodies.
Let's draw the figures
Circle with an oval and a beam.

We continue to draw
Rhombus, segment and square.
Guessed or not
Geometry secret?

drawn drawing
Can tell a lot
The main thing to remember
And the secret to understand it:

The body can be bulky
Round or tripartite.
You can take the body in your hands
And the figures to draw
Can we, or draw
And place on the plane.

How much is there in the world
Amazing secrets.
We all want to know
The world of wisdom to know


There are two triangles in front of you. The top one is completely divided into four fragments, painted in different colors. In the lower triangle, the same fragments are arranged in a different order. Attention, question. Where did the extra empty square come from? The answer is inside.


In reality this is not optical illusion, but an interesting task. The areas of the shaded figures, of course, are equal to each other (32 cells), however, what is visually observed as 13 × 5 triangles, in fact, is not, and has different areas (S13 × 5 = 32.5 cells). That is, the error masked in the condition of the problem is that the initial figure is named a triangle (in fact, it is a concave quad). This is clearly visible in the diagram below - the "hypotenuses" of the upper and lower figures pass through different points: (8,3) at the top and (5,2) at the bottom. The secret is in the properties of the blue and red triangles. This is easy to verify by calculations.


The ratios of the lengths of the corresponding sides of the blue and red triangles are not equal to each other (2/3 and 5/8), so these triangles are not similar, which means they have different angles at the corresponding vertices. Let's call the first figure, which is a concave quadrilateral, and the second figure, which is a concave octagon, pseudotriangles. If the bottom sides of these pseudo-triangles are parallel, then the hypotenuses in both 13×5 pseudo-triangles are actually broken lines(in the upper figure, a kink is created inward, and in the lower figure, outward). If we superimpose the upper and lower figures 13 × 5 on each other, then a parallelogram is formed between their “hypotenuses”, which contains the “extra” area. In the figure-scheme, this parallelogram is shown in the correct proportions.

The acute angle in this parallelogram is equal to arcctg 46 ≈ 0°1′18.2″. At this angle, the minute hand on a working watch moves in 12.45 s. It is by this amount that the obtuse angle in the parallelogram under consideration differs from the developed one. Visually, such an insignificant difference is imperceptible, but it is clearly visible on the animation.

According to Martin Gardner, this problem was invented by New York amateur illusionist Paul Currie in 1953. However, the principle behind it was known as early as the 1860s. You can see that the lengths of the sides of the figures from this problem (2, 3, 5, 8, 13) are consecutive Fibonacci numbers.