Mikhail Gershenzon Professor's puzzles Puzzles: a collection of riddles, tricks and entertaining tasks. Moss and sand landscapes

Professor Stewart's Math Puzzles is a collection of amazing stories, puzzles, curiosities, jokes and exciting math facts that inspire curiosity, teach, surprise and delight.
It turns out that tossing a coin is an unfair lot, and mussels are located on stones in a strictly defined way. The reader will find out why it seems to us that our friends have more friends than ourselves, why the bubbles in the glass with Guinness move down and not up, how to play poker by mail, and much, much more.
Sunday telegraph

What is the book "Math Puzzles by Professor Stewart" about

Why Professor Stewart's Math Puzzles is worth reading

• Ian Stewart introduces the reader to the world of mathematics by intriguingly talking about puzzles, problems, hypotheses and proof authors - famous mathematicians.
• The reader can test his strength and solve simple and complex problems. The correct answers are given at the end of the book.
• The book is in the top 100 of the Math Games section on Amazon.

about the author

Professor Ian Stewart — famous popularizer of mathematics awarded with a number of top international academic awards. In 2001 he became a member of the Royal Society of London. Professor Emeritus at the University of Warwick Institute of Mathematics, where he is engaged in both scientific research in the field of nonlinear dynamics and the popularization of mathematics.

Biography

Born in Odessa in the family of a pediatrician and his wife Bella Gershenzon. He graduated from high school in Odessa, studied for three years in the natural department. Carried away by literature, in 1921 he entered the. Upon graduation in 1925, he was left at the institute for scientific work, led a workshop on stanza, prepared for publication the books "Composition of poetic material" and "Theory of literary translation" (due to the closure of the VLHI, they remained unpublished). Member since 1934. He worked as an editor of children's literature in, then in Detgiz.

Creativity and translations of books by foreign authors

“The name of Mikhail Abramovich Gershenzon can be safely put on the cover of“ The Tales of Uncle Remus ”next to the name of Joel Harris.<…>It is impossible to imagine Brother Rabbit, Brother Fox, Mother Meadows, Brother Turtle and other characters in the fairy tale differently than they exist in Gershenzon. "

On the fronts of the Great Patriotic War

From July 1941 political worker, translator. On August 8, 1942, he was seriously wounded in battle and soon died in a hospital.

Gershenzon, who retold the ballads about Robin Hood in prose, was an admirer of Byron and Shelley as individuals. In 1942, he died in the Great Patriotic War, died like Robin Hood. He was a military translator, that is, a very little military man, and when the battalion commander was killed, he ran forward, picked up a revolver or whatever weapon was there, said: "Forward, follow me!", And the people followed him, the soldiers. He died as a hero, having managed to write to his wife the words that he was happy that he had accepted a dignified death. A totally Robingood romantic situation. (

From the editorial board

Fascinating undertakings and insidious logical problems, tricks and homemade products, interesting observations and unusual notes - all these are an integral part of the wonderful books prepared and written by Mikhail Abramovich Gershenzon (1900–1942). In an afterword to one of his collections, the author wrote: “This book contains various undertakings - both easy and difficult, choose what you want. If you can't do it yourself, ask your elders to help you. Adults, in truth, also love all sorts of ventures, only they forgot what they had fun with when they were children ... ".

These words, which were spoken many years ago, are actually relevant in our time. Solving riddles, untangling puzzles and quests that develop memory and logic, make you come up with non-standard solutions and broaden your horizons - an exciting pastime. How sometimes it is useful and interesting to smash your head over a difficult task, and how much joy and delight its correct solution brings!

M.A. Gershenzon was a highly educated person, worked as an editor of children's literature, translated, was fond of English songs and ballads. In his translation and processing, we first got acquainted with "The Tales of Uncle Remus", which are still loved by young readers. He also wrote the books "Merry Hour", "Just How Much", "Know-it-all Riddles" and others.

During his short life, Mikhail Abramovich managed to do a lot: edit, write, translate - and died like his beloved hero Robin Hood. During the Great Patriotic War M.A. Gershenzon was a front-line translator, a non-military man. But after the death of the battalion commander, he found the strength to lead the soldiers into the attack and died, struck down by a burst of machine-gun fire, having managed to strike a note that he was happy to accept such a dignified death.

Mikhail Abramovich wrote good books, gave us amazing translations, remained on the pages of his works in riddles and scientific experiments. His talented works still excite the minds of readers in our time and make them solve fascinating and so close to us puzzles.

Cheerful artist

Game of artists

Artists love to play this game. Each one takes a pencil and a sheet of paper and draws a sheet of 20 cells. The facilitator prepares a 20-word list in advance. All at the ready, pencils in hand. The facilitator calls out the words, counting to three after each word. While he is counting, all the players must have time to sketch this word with any pattern in one of the cells for memory. Let the drawing be incomprehensible to others - if only the player could then repeat in order all the named objects and concepts. Whoever has time to remember more won and is playing the next game. Sometimes one stroke is enough to memorize a word. Here is a sample list: window, fire, lamp, hare, man, sun, breath, grate, light, book, rain, smoke, rabbit, classroom, fire, joke, boat, magnet, steam, lightning. This is a difficult list. You can think of it easier.

"Mixes"

There is such a funny book - "Mixes". In this book, the drawings of the animals are confused, just like the words in the title "Confusion". Imagine an animal with the head of a crocodile and the legs of a rooster. Or the head of an elephant is set on a fish tail. It's a lot of fun to play these "confusions".

Take each strip of paper, fold each strip into three parts.

So I drew the head of a giraffe, folded up a third of the strip so that it was not visible whose head I drew, and outlined with two dots in the second third where the neck ends. Then I give it to my neighbor. He draws someone's torso, folds it in and passes it to a neighbor. He draws on the legs.

This is how everyone passes on to each other the drawings they have begun, and immediately then, as we unfold the stripes, we have the funniest zoo in the world ready.

Miraculous transformations

Do you want to amuse your little brother or sister? Cut three identical strips of thick paper. Draw on two strips whatever you like: for example, a giraffe on one, a girl on the other. Fold both strips in half. The lower half of the first picture is glued to the upper half of the second, free halves are glued to the third strip.

You hold the focus card in your hand so that the giraffe is visible. Then, with a quick movement, you flip the middle flap - and suddenly the giraffe turns into a girl.

If you get the hang of it, it will be completely imperceptible how one drawing is replaced by another.

Five points

Here's a piece of paper with five dots on it. I put them in no order at all. And you have to draw the little man so that two points fall on his hands, two points on his legs. And a person's nose should be where the fifth point is.

There are several clowns in the pictures drawn in this way. The small cells show how the points were located. It can be seen that points can be used in different ways.

You don't have to be an artist at all to play this game. But artists love to play with five points.

It is very fun when several people play five points. One of the players puts together several pieces of paper and pierces the entire stack in five places with a pin. Then on all sheets of paper the points will be located in the same way. And the "artists" have no right to look at each other - that is why different positions of the figures are obtained. It is even possible to give out awards to those who make better use of the location of the points.

Fascinating undertakings and insidious logical problems, tricks and homemade products, interesting observations and unusual notes - all these are an integral part of the wonderful books prepared and written by Mikhail Abramovich Gershenzon (1900–1942). In an afterword to one of his collections, the author wrote: “This book contains various undertakings - both easy and difficult, choose what you want. If you can't do it yourself, ask your elders to help you. Adults, in truth, also love all sorts of ventures, only they forgot what they had fun with when they were children ... ".

These words, which were spoken many years ago, are actually relevant in our time. Solving riddles, untangling puzzles and quests that develop memory and logic, make you come up with non-standard solutions and broaden your horizons - an exciting pastime. How sometimes it is useful and interesting to smash your head over a difficult task, and how much joy and delight its correct solution brings!

M.A. Gershenzon was a highly educated person, worked as an editor of children's literature, translated, was fond of English songs and ballads. In his translation and processing, we first got acquainted with "The Tales of Uncle Remus", which are still loved by young readers. He also wrote the books "Merry Hour", "Just How Much", "Know-it-all Riddles" and others.

During his short life, Mikhail Abramovich managed to do a lot: edit, write, translate - and died like his beloved hero Robin Hood. During the Great Patriotic War M.A. Gershenzon was a front-line translator, a non-military man. But after the death of the battalion commander, he found the strength to lead the soldiers into the attack and died, struck down by a burst of machine-gun fire, having managed to strike a note that he was happy to accept such a dignified death.

Mikhail Abramovich wrote good books, gave us amazing translations, remained on the pages of his works in riddles and scientific experiments. His talented works still excite the minds of readers in our time and make them solve fascinating and so close to us puzzles.

There is such a funny book - "Mixes". In this book, the drawings of the animals are confused, just like the words in the title "Confusion". Imagine an animal with the head of a crocodile and the legs of a rooster. Or the head of an elephant is set on a fish tail. It's a lot of fun to play these "confusions".

Take each strip of paper, fold each strip into three parts.

So I drew the head of a giraffe, folded up a third of the strip so that it was not visible whose head I drew, and outlined with two dots in the second third where the neck ends. Then I give it to my neighbor. He draws someone's torso, folds it in and passes it to a neighbor. He draws on the legs.

This is how everyone passes on to each other the drawings they have begun, and immediately then, as we unfold the stripes, we have the funniest zoo in the world ready.

Do you want to amuse your little brother or sister? Cut three identical strips of thick paper. Draw on two strips whatever you like: for example, a giraffe on one, a girl on the other. Fold both strips in half. The lower half of the first picture is glued to the upper half of the second, free halves are glued to the third strip.

You hold the focus card in your hand so that the giraffe is visible. Then, with a quick movement, you flip the middle flap - and suddenly the giraffe turns into a girl.

If you get the hang of it, it will be completely imperceptible how one drawing is replaced by another.

Here's a piece of paper with five dots on it. I put them in no order at all. And you have to draw the little man so that two points fall on his hands, two points on his legs. And a person's nose should be where the fifth point is.

There are several clowns in the pictures drawn in this way. The small cells show how the points were located. It can be seen that points can be used in different ways.

You don't have to be an artist at all to play this game. But artists love to play with five points.

It is very fun when several people play five points. One of the players puts together several pieces of paper and pierces the entire stack in five places with a pin. Then on all sheets of paper the points will be located in the same way. And the "artists" have no right to look at each other - that is why different positions of the figures are obtained. It is even possible to give out awards to those who make better use of the location of the points.

This is a very simple game. All sit in a circle. One of the players is in the middle. He points to someone and speaks.

From the editorial board

Fascinating undertakings and insidious logical problems, tricks and homemade products, interesting observations and unusual notes - all these are an integral part of the wonderful books prepared and written by Mikhail Abramovich Gershenzon (1900–1942). In an afterword to one of his collections, the author wrote: “This book contains various undertakings - both easy and difficult, choose what you want. If you can't do it yourself, ask your elders to help you. Adults, in truth, also love all sorts of ventures, only they forgot what they had fun with when they were children ... ".

These words, which were spoken many years ago, are actually relevant in our time. Solving riddles, untangling puzzles and quests that develop memory and logic, make you come up with non-standard solutions and broaden your horizons - an exciting pastime. How sometimes it is useful and interesting to smash your head over a difficult task, and how much joy and delight its correct solution brings!

M.A. Gershenzon was a highly educated person, worked as an editor of children's literature, translated, was fond of English songs and ballads. In his translation and processing, we first got acquainted with "The Tales of Uncle Remus", which are still loved by young readers. He also wrote the books "Merry Hour", "Just How Much", "Know-it-all Riddles" and others.

During his short life, Mikhail Abramovich managed to do a lot: edit, write, translate - and died like his beloved hero Robin Hood. During the Great Patriotic War M.A. Gershenzon was a front-line translator, a non-military man. But after the death of the battalion commander, he found the strength to lead the soldiers into the attack and died, struck down by a burst of machine-gun fire, having managed to strike a note that he was happy to accept such a dignified death.

Mikhail Abramovich wrote good books, gave us amazing translations, remained on the pages of his works in riddles and scientific experiments. His talented works still excite the minds of readers in our time and make them solve fascinating and so close to us puzzles.

Cheerful artist

Game of artists

Artists love to play this game. Each one takes a pencil and a sheet of paper and draws a sheet of 20 cells. The facilitator prepares a 20-word list in advance. All at the ready, pencils in hand. The facilitator calls out the words, counting to three after each word. While he is counting, all the players must have time to sketch this word with any pattern in one of the cells for memory. Let the drawing be incomprehensible to others - if only the player could then repeat in order all the named objects and concepts. Whoever has time to remember more won and is playing the next game. Sometimes one stroke is enough to memorize a word. Here is a sample list: window, fire, lamp, hare, man, sun, breath, grate, light, book, rain, smoke, rabbit, classroom, fire, joke, boat, magnet, steam, lightning. This is a difficult list. You can think of it easier.

"Mixes"

There is such a funny book - "Mixes". In this book, the drawings of the animals are confused, just like the words in the title "Confusion". Imagine an animal with the head of a crocodile and the legs of a rooster. Or the head of an elephant is set on a fish tail. It's a lot of fun to play these "confusions".

Take each strip of paper, fold each strip into three parts.

So I drew the head of a giraffe, folded up a third of the strip so that it was not visible whose head I drew, and outlined with two dots in the second third where the neck ends. Then I give it to my neighbor. He draws someone's torso, folds it in and passes it to a neighbor. He draws on the legs.

This is how everyone passes on to each other the drawings they have begun, and immediately then, as we unfold the stripes, we have the funniest zoo in the world ready.

Miraculous transformations

Do you want to amuse your little brother or sister? Cut three identical strips of thick paper. Draw on two strips whatever you like: for example, a giraffe on one, a girl on the other. Fold both strips in half. The lower half of the first picture is glued to the upper half of the second, free halves are glued to the third strip.

You hold the focus card in your hand so that the giraffe is visible. Then, with a quick movement, you flip the middle flap - and suddenly the giraffe turns into a girl.

If you get the hang of it, it will be completely imperceptible how one drawing is replaced by another.

Five points

Here's a piece of paper with five dots on it. I put them in no order at all. And you have to draw the little man so that two points fall on his hands, two points on his legs. And a person's nose should be where the fifth point is.

There are several clowns in the pictures drawn in this way. The small cells show how the points were located. It can be seen that points can be used in different ways.

You don't have to be an artist at all to play this game. But artists love to play with five points.

It is very fun when several people play five points. One of the players puts together several pieces of paper and pierces the entire stack in five places with a pin. Then on all sheets of paper the points will be located in the same way. And the "artists" have no right to look at each other - that is why different positions of the figures are obtained. It is even possible to give out awards to those who make better use of the location of the points.

Another game

This is a very simple game. All sit in a circle. One of the players is in the middle. He points to someone and says:

- Name five yellow items!

- Name five items in blue!

So he goes through different colors. Anyone who cannot remember five objects of the named color in one minute leaves the game.

Tangram

Tangram is an old game. Cut three squares out of different colored paper. Stick them on cardboard and cut each square along a ruler as shown in the figure. Here are some figurines from the wonderful land of Tangram. Make them from your colorful pieces. You will probably come up with many new figures that are not in this picture.

Chur, no mistakes!

All day I sat and drew, and I had only one piece of paper. Do you think you can draw little on one piece of paper? I put twenty different things here. Here, try, find them all, just, mind you, no mistakes!

With one stroke

These funny drawings are drawn with a single stroke of the pen. Can you draw like that?

One line

Draw each of these shapes with one continuous line, without lifting your pencil from the paper or drawing lines twice.

Two dogs

Two sick dogs are drawn here. Draw four strokes so that the dogs immediately recover and run.

Three rabbits

Here are drawn three earless rabbits and three ears. Redraw them onto a piece of paper, cut and fold them so that each rabbit has two ears.

Three lizards

Three lizards basked in the sand. I drew them. And then I thought: what is the greater distance - from the upper nose to the middle nose or from the middle nose to the lower nose?

First answer at random, by eye, and then take a ruler and check.

I know in advance: you will think that there is some kind of trick. But that won't help you. You'd be wrong anyway.

Deceptions of sight

You can't do anything here without a ruler. We'll have to measure, measure, measure. Otherwise, how can you check if the eyes are telling the truth?

To tell you a secret, the eyes are lying pretty well.

Are these lines straight or curved?

Which of the lower lines is a continuation of the upper one?

Is this figure much larger in height than in width?

Which segment of the straight line is longer - upper or lower?

Which shape is bigger - top or bottom?

Is this telegraph pole obscuring the correct gate?

Look from a distance at this drawing - it will seem to you that these are real honeycombs - regular hexagons. But these are the right circles!

Sea lions

It would be more correct to call these animals "sea jugglers", because they are easy to learn to juggle with balls. You've probably seen these amazing animals in the circus. They are thrown with balls, and catch them on the tip of the nose, and crawl with the ball on the nose from one stand to another.

With quick, short movements, rotate this drawing in front of your eyes. The ball will immediately spin quickly on the bow of this sea juggler.

Are they laughing or angry?

These clowns are always fighting. You will never understand with them - they are friends, then they are not. You can only hear:

- I'm not messing with you!

- Let's make it up!

- Well, you don't run around, and you don't need to.

- Will you be with me?

- Water flows in the river.

So I drew them like this: look at the drawing - they are found, turn them upside down - they quarreled again.

Upside down

When you look at these letters and numbers, it seems that their upper and lower halves are exactly the same.

Well, turn this line upside down!

Where is the horse?

The horse is not a needle, but will go into the thicket - it is not easy to find it. Come on, try, find her!

Ten or nine?

Draw ten sticks of the same size on a piece of paper, at the same distance from each other - as in the picture. Place a ruler on this drawing and draw a straight line so that it goes through the top end of the first stick and through the bottom end of the last one. Cut the sheet along this line and slide the halves as shown in the picture. How many sticks are there now? Where did the tenth go?

The figure shows eleven sticks arranged in a circle. Redraw this drawing with care, cut out the circle and stick a pin in the center of it. Turn the circle a little - instead of eleven there will be ten sticks.

Snake and mouse

There is only one mouse, but there are many snakes. How can she get to her burrow? She is afraid to catch the eyes of the snakes. We have to choose such a road so that the snakes do not notice it. A mouse is running quietly - which road?

Double drawings

Look below - you will see a strange drawing. You can't make out what is drawn here. Either a squirrel or a bird.

Cut out exactly the lattice shown in the picture out of thick paper. Put the grate on the picture. Move the grate down a little - you will see a squirrel. Move it a little more - you will see a bird.

Using this grid, you yourself can draw such double pictures.

Living shadows

Of course you have a table lamp. And there is a wall if you are sitting at home. And nothing else is needed to arrange a game of living shadows.

If there is dark wallpaper on the wall, pin up a sheet of white paper or a sheet. Sit between the lamp and the wall and show the shadows. The lamp should be low - at the height of your hands.

Choose any animal, any bird. Learn to fold your fingers, as in the picture on the right, and on a white screen you will get the shadow of a bunny, a goat, a pig, a wolf, a rooster, a man.

Move your fingers and the shadows will come to life. The hare will bend its ears, the dog will open its mouth. The dog swallows a tidbit: at first it opened its mouth, then slammed it shut. And on the next page, a swan swims, flaps its wings, bends its head to the water.

If you keep your hand closer to the lamp, the shadow will be larger; if you keep your hand farther from the lamp, closer to the screen, the shadow is clearer, blacker and smaller.

This is great to use. Watch the rider ride. The left hand is farther from the screen, its shadow is large. And the right one is closer to the screen, and the rider seems small. The piece of cardboard between the fingers is the rider's hat. The lace is a bridle.

Here, for example, is a chef who tastes a freshly cooked soup. A chef's cap, pan, bottle are cut out of cardboard. The right hand is far from the screen - a large head is obtained in the shadow. The left one is right next to the screen. Things are on a narrow table or on a board. Cut the cap about the size of your palm. The chef opens his mouth, you carry the spoon past your right hand, and in the shadow it turns out as if the chef tasted his soup. Move the little and ring fingers of your right hand so that everyone can see that the cook likes the treat.

And now we will show how a fisherman catches fish.

Cut out such figures from thick paper: a hat, a saucepan, a shoe and a fish. Make a fishing rod out of wire. Roll the wire into a ring at one end; the ring should be large enough to fit snugly on your thumb. Tie a string with a pin hook to the other end of the rod. Make a boat out of a board or a strip of cardboard. Lean her against a book on the edge of the table.

Fold the fingers of your right hand as shown in the figure so that the fisherman appears in the shadow; pinch the hat between the middle and forefinger. Place the rod over your thumb. Move this finger so that the fisherman does not sit idle. Your left hand is free. You can put it on the hook under the table, then a saucepan, then a shoe, then a fish.

Well, what did you get there? Fish? The fisherman is happy. You move your fingers a little, and everyone can see how happy he is with good prey. But how sad he shakes his head when he sees that there is a shoe on the hook!

Shadow theater

We were already playing in the shadow theater when the cook was cooking dinner or the fisherman was fishing. But there our hands were the main actors. Only to help our hands, we took cardboard boxes and all sorts of small things.

But it is possible that all the actors were made of cardboard.

Stretch the sheet in the doorway between the two rooms; block the lower part of the door with a sheet of plywood. Spectators are sitting in a dark room. The shadow theater works in the bright one. The lamp is three to four steps from the screen.

Figures can be made movable. We cut out their parts from cardboard, connect them with rivets from nylon fishing line (read how to do this on p. 105). We will attach wires or threads to the parts of our actors so that we can control their movements.

For example, figurines of geese that shake their heads: they peck at grain. The body of the goose is fixed on a block. The head, together with the neck, rotates on a rivet. The wire is attached to the neck; if you pull the wire, the goose will bend its head.

Here are the figures of people: a violinist is playing the violin, an old man is feeding his dog, a hammer is raising and lowering a hammer. Try to put on a whole shadow show. You need to start with the simplest. It is very easy to stage a fairy tale "The Turnip" in a shadow theater.

Shadow portraits

Sit a comrade between the lamp and the wall so that a clear, correct shadow falls on the wall. You can find such a position so that the dark silhouette on the wall is completely similar to your comrade: a comrade has a snub nose - and a snub in the shadow; a comrade has a swirl on his forehead - and a swirl on the shadow. Pin a large piece of paper to the wall and sketch out this silhouette. Of course, the comrade must sit still for two minutes.

If you cut out your drawing, stick it on a different color of paper, you'll get a great portrait.

Moss and sand landscapes

Pick up and dry between old newspapers, pressed against books, more moss of different types. You can glue very beautiful landscapes out of thin branches of moss on a cardboard. Prepare more sand of different colors: run glue over the cardboard, sprinkle it with sand - you get a sandy path between the trees.

"On a horse!"

Redraw these two dull horses on one piece of paper, and these riders on the other. Do not rotate the figures: redraw them exactly as they are located here.

The command is given: "On a horse!"

Put the riders on their horses, so that the horses gallop dashingly.

Bird in a cage

Draw an empty cage on a piece of paper, and a bird a few millimeters from the cage. How to put this bird in a cage?

Take half of a postcard, place it between the bird and the cage, perpendicular to the piece of paper. Touch the edge of the postcard with your nose and look with one eye at the bird, the other at the cage; in a moment it will seem to you that the bird has moved from its place and entered the cage.

However, you don't even need to draw anything. We have both a cage and a bird here. Take a postcard and look. Just stand in front of the light so that the shadow from the postcard does not fall on the drawing.

Cyclist and steam locomotive

Have you ever seen a cyclist chasing a steam locomotive? Look at the wheels, rotate this page in front of your eyes with light jerks - such a race will begin, just hold on!

Don't get hit by the wheels!

Optical illusion

Take three equal length strips of white paper; one of them should be twice as narrow as the others.

Cross two wide strips in the form of the letter "X", and at the intersection of them put a vertical narrow one. It will appear longer than wide stripes.

This experience will be especially effective if the white stripes are placed on black paper or cloth.

Now try to expand the strips in the form of the letter "I" so that the narrow strip lies obliquely between the two wide ones. This time she will seem shorter to you than her neighbors.

Second optical illusion

Turn a sheet of thick white paper into the lattice shown in the figure.

Cut out a narrow strip from thin cardboard with strictly straight edges and secure it with a pin, as on an axis, in one of the corners of the lattice.

If you rotate the strip so that it is almost perpendicular to the bars of the grill, it will still appear to be bounded by two straight lines. But if you tilt it lower, these lines break, and it seems to us that the segments that are visible through the cuts are not at all a continuation of each other.

In our drawing, the third, lower strip seems so broken that I want to take a ruler and check if its edges are really straight lines!

Third optical illusion

Look at the strip shown on the next page on the left, placing it in front of you at a distance of at least 3 m. This strip is colored so that black gradually turns into white; in shape it is an elongated rectangle. Despite the fact that the edges of this strip are strictly parallel, it will seem to you widened in its white part and narrowed in black. Instead of a rectangle, it looks like a trapezoid.

Now we put this strip on another, wide strip, colored in the same way, but put it so that the white part of the narrow strip lies on the dark part of the wide one. The optical illusion will instantly disappear, and the narrow strip will turn back into a regular rectangle.

Try to prepare such strips of paper in an enlarged size, then the optical illusion will be even more striking.

Compass or eye?

Take a compass and draw some concentric circles. But press the pencil to the paper not along the entire length of the circles, but only on a small part of each circle, so that these arcs, located on different circles, lay "tiles", slightly covering one another.

When you look at such a drawing, it seems that if we continue our arcs, their extensions will intersect at one point.

Take a compass, check it - make sure that your eyes have deceived you.

Put down the compass - again it is hard to believe that these arcs are parallel.

Who is right, a compass or an eye?

This optical illusion will be even brighter if you draw our simple, larger drawing on a large sheet of paper.

Live picture

When a picture hangs on the wall, it soon gets boring: nothing in it changes, it is always the same.

We will make a live picture. If we want it, everything in her will change.

Draw with colored pencils several figures, such as in the picture here, and several objects - a hoop, an umbrella, a stick, a ball. Cut out the juggler acrobats and the things they juggle with.

Paste over the board with a piece of cloth - it is best to take a piece of paper or cloth for this; but you can stick it on cardboard and just a strip of skin (sandpaper).

Lay out the cut out figures on matter - as you like. Cover them with glass on top and tie the board and glass with two laces. Hang on the wall.

If you get tired of this picture, arrange the figures differently: the acrobats will show a new number. They will give a new performance every day.

Spiral? No, not a spiral

Without a compass, you will not believe that these are the right circles. It appears to be a spiral, just like the curls on a snail shell. Take a compass and check - you will see how your eyes deceive you. And here the correct circles are drawn, both here and here.

It is not the artist's fault that the circles are crumpled, flattened and crooked. It's only your eyes that are to blame - ask the compass, he will tell you.

Don't believe your eyes. These circles were drawn by a good draftsman, only very cunning.

Who is higher?

Of the three people pictured on the next page, who is higher? If you believe your eyes - No. 3, isn't it?

Take a ruler, measure all three, and you will be convinced that you are deceived by a visual illusion. No. 1 above everyone else. It is 2 mm higher than No. 3, which steps in front.

This picture was drawn in violation of the rules of perspective. Our eye is accustomed to the fact that objects, moving away, decrease; that's why we decided that No. 3, the farthest from us, should be larger than No. 2 and No. 1, which are in the foreground.

Strange city

A very cunning artist painted this picture. All buildings and towers were lopsided as if the earthquake had passed. Just about to fall apart.

But if you take a strip of thick paper, cut it out exactly as in the picture, make a window in it the size of a pea and put it in the place indicated in the picture, and then look through this window, the walls will straighten, and you will see a great view of the medieval cities. You will see that the houses are not flat, as always in the picture, but three-dimensional, as in a stereoscope.

Bend the strip to make a stand. The stand must lie exactly on the spot indicated in red.

At the end of the sections of this book, you can find answers to some of the more difficult problems. Just do not rush to give up right away, try to think carefully yourself.