# IV International competition of research and creative work of students Start in science

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Always breathtaking when showing a mathematical focus. Mathematics is not just the most accurate science and logic in its purest form. She can take and show a mathematical focus. Great pride is not only a young mathematician who entertains the company in this way. All attention is riveted on a connoisseur of numbers, glowing with happiness. To begin, we offer 5 mathematical tricks.

## Make a number Let someone in the company guess the number. He’s in the ear so that you don’t hear him, he will tell him to his neighbor. This will be the controller. Suppose a child concocts the number 34. We must invite him to divide it into three and communicate the remainder aloud. It will look like this: 34: 3 = 11 (1). He said, "Unit." Then let him divide the conceived number into five. He will get this answer: Z4: 5 = 6 (4). He will name only the remainder: "Four." After that, he must divide his secret number by seven. Easily coping with the task in this way З4: 7 = 28 (6), he will mockingly say: “Six”. Everyone with a slight irony expects you to get the right result. And here you show a real mathematical focus, having done some calculations in your mind. The first remainder - a unit - must be multiplied by seventy and remember this easy answer. The second remainder of 4 is multiplied by 21, we get eighty-four and remember. The last spoken number, six, is multiplied by fifteen. Get ninety. Now add up the three numbers obtained: 70 + 84 + 90 = 244 and divide the sum by one hundred and five. This is convenient to do on paper in a column. You did not limit yourself in time at the beginning of the game: 244: 105 = (2) 34. What is the answer? Everyone is waiting. You give out the rest, that is, the desired number: thirty-four. Everyone is amazed. You are at the pinnacle of success.

## How to interest children in fifth grade

“This mathematician is coming again,” the children think with boredom and boredom. And she prepared a surprise. She came and said that now she’s guessing the birthday of the one who first raises his hand. Immediately, a forest of hands rose and requests fell: “Anna Nikolaevna, guess with me!” • 13x3 = 39,
• 39:9=4(3),
• 4x3 = 12,
• 3:3=1,
• 12+1= 13.

After such a warm-up, the lesson will be interesting, after which everyone will ask the teacher to explain how the correct answer came out. Then she will be able to offer students mathematical tricks with answers.

## Simple arithmetic

Children will understand that arithmetic, which is translated from Greek, means "the art of counting", this must be especially emphasized, the Greeks had a very respected science. She developed the ability to reason, which is important in life and in reflections on literature lessons when analyzing prose, poems, and works of great painters.

## Next trick: guess how old you are

Mathematical tricks for grade 5 should be quite simple and entertaining. Children can play numbers with guessing games. They will ask each other: “How old are you?” Two go to the board. In the hands of one of them is a leaflet with a hint that the teacher had prepared in advance. He reads the first question, in which you want to multiply his age by 5. Suppose that the one who answers is another eleven years old. Then he gets the number fifty five. The second student asks to add 8. The whole class considers and receives an answer - sixty-three. • 11x5 = 55,
• 55+8=63,
• 63x2 = 126,
• 126 – 6= 120,
• 120x10 = 1200,
• 1200 – 100= 1100,
• 1100: 100= 11.

He considered not only a student at the blackboard, but the whole class took a lively part in the calculations. So in a game form you can complete the lesson. Everyone was interested. Such math tricks for grade 5 with answers make the lessons very exciting.

##### The author of the work was awarded a diploma of the winner of the II degree

“The subject of mathematics is so serious that it’s useful not to miss the chance, to make it a little entertaining”

B. Pascal

When we first met in a math lesson, the teacher promised to guess the date of birth of each student in our class if we quickly and correctly perform the arithmetic operations she offers. First, we had to multiply our birthday by 2, add 5 to the resulting number, multiply the result by 50, and finally add the month of our birth. After we called the received number to the teacher, she, as promised, guessed the date of our birth and was mistaken only when we ourselves were to blame for the wrong calculations. I really liked this trick. I also wondered what lies at the heart of this focus. It was then that I decided that I would definitely investigate the issue of mathematical tricks, find out their secrets, make a selection of tricks and amaze and amuse my friends and acquaintances by demonstrating mathematical tricks in math classes, extracurricular activities, and even at home holidays.

In Internet sources, I read that mathematical tricks do not enjoy much attention either from mathematicians or from magicians. The former consider them simple fun, the latter - too boring.

But, in my opinion, this is not at all the case. Mathematical tricks have a deep meaning.

Mathematical tricks are experiments based on mathematical knowledge, on the properties of figures and numbers, convicted in an extravagant form. To understand the essence of an experiment is to understand even a small, but very important mathematical regularity.

The ability of a person to guess the numbers conceived by others seems surprising to the uninitiated. But if we learn the secrets of tricks, we can not only show them, but also come up with our new tricks. And the secret of focus becomes clear when we record the proposed actions in the form of a mathematical expression, transforming which we get the secret of guessing.

In my work, I want to prove that mathematical tricks help to develop memory, ingenuity, the ability to think logically, improve oral counting skills and, finally, simply increase the students' interest in mathematics, which should improve the quality of their knowledge.

Objective: explore math tricks.

Study the literature on the topic under study.

Demonstrate a few tricks.

Explain them in terms of mathematics.

Attract classmates to math.

Subject of study: math tricks

Object of study: "Secrets" of mathematical tricks

Research Methods: study and analysis of literature on entertaining mathematics, independent modeling of mathematical tricks.

Practical significance: the material can be used in mathematics and extracurricular classes, mathematical evenings and holidays, when conducting mathematical competitions.

Chapter 1. The history of the emergence of mathematical tricks.

Focus - a skillful trick based on optical illusion, attention with the help of dexterous and quick reception, movement (Ozhegov's dictionary)

The history of mathematical tricks.

The first document that mentions illusion art is Ancient Egyptian papyrus. It contains traditions relating to 2900 BC, the era of the reign of Pharaoh Cheops.

Initially, magicians and sorcerers used tricks. The priests of Babylon and Egypt created a huge number of unique tricks with the help of excellent knowledge of mathematics, physics, astronomy and chemistry. The list of miracles performed by priests can include: peals of thunder, sparkle of lightning, doors of temples themselves opening suddenly, statues of gods appearing from underground, sounding musical instruments themselves, voice.

In Ancient Hellas, the harmonious development of personality was not conceived without games. And the games of the ancients were not only sports. Our ancestors knew chess and checkers, puzzles and riddles were not alien to them. Such games at all times have not been alienated by scientists, thinkers, and educators. They created them. Since ancient times, the puzzles of Pythagoras and Archimedes, the Russian naval commander S.O. Makarov and the American S. Loyd have been known.

The first mention of mathematical tricks we meet in the book of the Russian mathematician Leonty Filippovich Magnitsky, published in 1703. We all know the great Russian poet M.Yu. Lermontov, but not everyone knows that he was a great lover of mathematics, especially he was attracted to mathematical tricks, which he knew a great many, and some of them he invented himself.

The enormous cognitive and educational value of intellectual games was repeatedly pointed out by K.D.Ushinsky, A.S. Makarenko, A.V. Lunacharsky. Among those who were fond of them were K.E. Tsiolkovsky, K.S. Stanislavsky, I.G. Erenburg and many other prominent people.

Separately, I would like to mention the American mathematician, magician, journalist, writer and popularizer of science Martin Gardner.

He was born on October 21, 1914. He graduated from the Faculty of Mathematics of the University of Chicago. Founder (mid-50s), author and presenter (until 1983) of the heading “Mathematical Games” of the journal “Scientific American” (“In the World of Science”). Gardner interprets entertainment as a synonym for fascinating, interesting in cognition, but alien to idle entertainment. Among Gardner's works there are philosophical essays, essays on the history of mathematics, mathematical tricks and "comics", popular science sketches, science fiction stories, tasks on quick wits.

Gardner's articles and books on entertaining mathematics are particularly popular. In our country, seven books by Martin Gardner have been published, which captivate the reader and encourage independent research. The "Gardner" style is characterized by clarity, brightness and persuasiveness of presentation, brilliance and paradox of thought, novelty and depth of scientific ideas.

Among our compatriots, I would like to name Y. I. Perelman. Yakov Isidorovich Perelman did not make any scientific discoveries, did not invent anything in the field of technology. He did not have any academic ranks and degrees. But he was devoted to science and for forty-three years brought people the joy of communicating with science. It is with his books that the journey begins in the fascinating world of mathematics, physics, astronomy. And it was his books that helped me write this work. Ignatiev E.I., Kordemsky B.A. made a huge contribution to the popularization of mathematics. and many other Russian scientists, educators, methodologists.

Mathematical tricks are interesting precisely because each trick is based on mathematical laws. Their meaning is to guess the numbers conceived by the audience. Millions of people in all parts of the world are keen on mathematical tricks. And this is not surprising. “Mind gymnastics” is useful at any age. And tricks train memory, sharpen quick-wittedness, develop perseverance, the ability to think logically, analyze and compare.

Chapter 2. Math Tricks

Focus “Guess the conceived number.”

Ask any student to think of a number.

Then the student must multiply this number by 2, add 8 to the result,

divide the result by 2

and take away the intended number.

As a result, the magician boldly calls the number 4.

The viewer conceived the number 7

1) 7●2 = 14 2) 14 + 8 = 22 3) 22/2 = 11 4) 11 – 7 = 4

The number X is guessed.

2) X ● 2 2) X ● 2 + 8 3) (X ● 2 + 8) / 2 4) (X ● 2 + 8) / 2 - X = X + 4 - X = 4

We got 4, regardless of the number originally requested

Focus “Magic table”.

You see a table in which numbers from 1 to 31 are written in a special way in five columns.

I invite those present to think of any number from this table and indicate in which columns of the table this number is located.

## Ah, that entertaining arithmetic!

In the lesson, children can briefly talk about the biographies of the founders of arithmetic in Hellas. For example, about Pythagoras, Euclid, Archimedes. Explain that they indicated numbers with letters. Ask the children who came up with the modern numbers that we use. This they should remember from the lessons learned in history. Tell why and by whom zero was invented.

## Adding multiple digits is a new focus

This is a competition in counting speed. Let the whole class watch curiously the notes on the board.

One student will write several three-digit numbers. Suppose 538, 784, 296, 429. The second one, which knows the secret, quickly adds to it its own series of numbers: 461, 215, 703, 570. In it, each number supplements the opponent’s number to 9. This series instantly adds up to the formula x * (10 ʸ - 1), where x is the number of written numbers, and y is the number of digits of each number. That is 4 * (10³ - 1) = 3 996.

## How Math Becomes Favorite

Not too complicated mathematical entertainments that amaze the imagination of not only peers in the yard, but also of parents who have not picked up entertaining tasks in arithmetic for a long time, will make you interested in mathematical puzzles. And then start looking for and reading books by Ya.I. Perelman, the greatest magician who can show complex things as a detective story with a sequel. These books I want to constantly re-read, they are written so lively, fun and interesting. For example, he has a story called “Bargain”. An old man came to the greedy rich millionaire and suggested that every day for a month he would bring one thousand rubles, and that, in turn, would pay for it. On the first day, 1 kopeck, on the second twice as much - 2 kopecks, on the third day - 4. So every day the amount of payment for a thousand rubles will double. “It's just great, I agree,” the rich man exclaimed. Everything went fine for him for two weeks, and then he began to notice that for 1000 rubles he pays significantly more. We will not retell the whole story with money. Let's just say one thing. ## Fifth focus

He is simple and entertaining. Let the two in the class go to the board. One, knowing the result in advance, will declare: “No matter what you do, whatever numbers you choose, you will have only five answers under my leadership.” Everyone will be amazed, but will carefully monitor the actions at the board. Anyone who does not know the secret will write any number, even a very long one. It will only be more difficult for him from this. Suppose he wrote two hundred and twenty-one. Now it is necessary to add the number following it, that is, two hundred twenty-two. They must be added, and the sum will be four hundred forty-three. To add nine more to it. It turned out four hundred and fifty-two. Next, it must be divided into two. From the quotient, which is the number two hundred twenty-six, it is necessary to subtract the very first number, two hundred twenty-one. The answer is five, as promised. Here's what it looks like:

You are interested? Then let's continue!

## Guess the crossed out number

Let someone think of a number, for example, 256. He must add all the numbers in the number. It will turn out 13. From the planned number should be subtracted the amount received: 256-13 = 243. In this difference, cross out any number and report the remaining ones. For example, crossed out four, and you immediately spoke about it. Everyone is amazed. How it's done? You were told numbers two and three. We are looking for a figure that, in total with the reported ones, will give the nearest number divisible by nine, without any remainder - in this case four (2 + 3 + 4 = 9). So we got the crossed out number four.

Why did this happen? Because if you subtract the sum of its digits from a number, you will surely get a number divisible by nine, that is, one whose sum of digits is nine.

We show this example on three-digit numbers. Conceived number seven hundred thirty eight. The sum of its numbers is eighteen. 738-18 = 720. Strikethrough seven. Stacked two and zero. The nearest number, which is divisible by nine, is missing seven. The answer is correctly guessed: seven.

## Favorite trick

Multiplying a two- or three-digit number by eleven is a remarkably easy, useful, and beautiful trick.

Our task is to multiply forty-five by eleven in the mind. It is enough to add both numbers, four and five, and then their sum, nine, put between four and five. We get the correct answer: 495. Check this mathematical focus on the calculator.

For a three-digit number, we give an example. Take the number 214. It must be multiplied by eleven. The answer will begin with the first digit and end with the last. And what's in the middle? It should be like this. Add the first digit to the second (2 + 1 = 3), and the second to the third (1 + 4 = 5), arrange them in the following sequence 2354. This is the answer. Recalculate on the calculator using other numbers.

Now you know the simple mathematical tricks and their secrets.

### The content of this mathematical focus.

Announce to the audience that you can guess the birthday of any stranger sitting in the hall.

• Call anyone and offer him to multiply by the 2nd day of his birthday
• Then let the viewer add up the resulting work and the number 5,
• Now let him multiply by 50 the amount received.
• To this result it is necessary to add the number of the month of birth (July - 7, January - 1)
• name the resulting number aloud.

After a second, you name the day and month of birth of the viewer.

### Focus content.

• Ask any viewer to think of a number,
• after this number he should multiply by 2,
• divide the result by 2 and
• subtract the intended number.

As a result, you boldly call the number 4.

### The secret of focus.

For example, the viewer conceived the number 7. 7x2 = 14 14+ 8 = 22 22: 2 = 11 11-7 = 4

## Guessed birthday

### The content of this mathematical focus.

Announce to the audience that you can guess the birthday of any stranger sitting in the hall.

• Call anyone and offer him to multiply by the 2nd day of his birthday
• Затем пусть зритель сложит получившееся произведение и число 5,
• теперь пусть умножит на 50 полученную сумму.
• К этому результату необходимо прибавить номер месяца рождения (июль — 7, январь — 1)
• вслух назвать полученное число.

Через секунду вы называете день и месяц рождения зрителя.

### Секрет этого математического фокуса.

Everything is very simple. В уме от того числа, которое назвал зритель, отнимите 250.

У вас должно выйти трехзначное или четырехзначное число. Первая и вторая цифры — день рождения, две последние — месяц.

## Number guessing trick

For this mathematical focus you will need:

• pre-prepared sheets of paper (by the number of spectators),
• pencils or pens (according to the number of viewers),
• calculators.

### Focus content.

Introduce yourself to the audience as a great mathematician, a trainer of numbers, reading other people's thoughts. Ask viewers to think of a number. You can ask absolutely any question, for example: how many days a week would you like to ride a bicycle, eat semolina, do not go to school, run through puddles. The whole point is not in the question, but in the number conceived by the audience.

Distribute pieces of paper and pens to the audience and give the task a written answer to your question. Let everyone write how many days a week he would like to eat carrots.

Now let everyone multiply this number by 2, then add 5 to the resulting number of carrots, and then multiply this amount by 50. Now, let everyone do the following: if it was already a birthday this year, add 1,750, if not - 1,749. Now everyone should subtract their year of birth from this number and add 7 to this number.

### The secret of focus.

For example, the viewer conceived the number 7. 7x2 = 14 14+ 8 = 22 22: 2 = 11 11-7 = 4

## Guessed birthday

### The content of this mathematical focus.

Announce to the audience that you can guess the birthday of any stranger sitting in the hall.

• Call anyone and offer him to multiply by the 2nd day of his birthday
• Then let the viewer add up the resulting work and the number 5,
• Now let him multiply by 50 the amount received.
• To this result it is necessary to add the number of the month of birth (July - 7, January - 1)
• name the resulting number aloud.

After a second, you name the day and month of birth of the viewer.

### The secret of this mathematical trick.

Everything is very simple. In the mind, subtract 250 from the number the viewer called.

You should get a three-digit or four-digit number. The first and second digits are the birthday, the last two are the month.

## Number guessing trick

For this mathematical focus you will need:

• pre-prepared sheets of paper (by the number of spectators),
• pencils or pens (according to the number of viewers),
• calculators.

### Focus content.

Introduce yourself to the audience as a great mathematician, a trainer of numbers, reading other people's thoughts. Ask viewers to think of a number. You can ask absolutely any question, for example: how many days a week would you like to ride a bicycle, eat semolina, do not go to school, run through puddles. The whole point is not in the question, but in the number conceived by the audience.

Distribute pieces of paper and pens to the audience and give the task a written answer to your question. Let everyone write how many days a week he would like to eat carrots.

Now let everyone multiply this number by 2, then add 5 to the resulting number of carrots, and then multiply this amount by 50. Now, let everyone do the following: if it was already a birthday this year, add 1,750, if not - 1,749. Now everyone should subtract their year of birth from this number and add 7 to this number.

## The guessed result of mathematical calculations

You will need: pre-prepared sheets of paper, pencils or pens, calculators.

## Focus content.

Invite viewers to think of a three-digit number and write it down on paper. When making a number, one condition must be met: the number of hundreds should not be equal to the number of units and should not be one less or more than it. If you are still confused in hundreds and units, then in the first place in three-digit numbers are hundreds, in the second tens, in the third units (for example, the number 531 will do).

• Now the audience must turn over the conceived number, i.e. write the numbers in reverse order (135).
• Then the audience should take these two numbers and subtract the smaller from the larger (531 - 135).
• The resulting difference again needs to be turned over (396, 693) and add these two numbers (396 + 693).
• Then one of the viewers should add 100 to the amount received, the second - 200, the third - 300, etc.
• Now you can guess what happened to each viewer, but provided that they add 1 089 to their last number. The first viewer, who added 100, will get 1 189, the second - 1 289, the third - 1 389.
• Now ask any of the viewers to name the resulting figure.
• You should get a two-digit or three-digit number. The first digit is the number of carrots, the rest is the person's age. The secret of focus. No matter how much they add or take away, these are all tricks of algebra. Only your viewers are not aware of this, all secret trick in the numbers that you make them add, subtract, divide.
• Here is how it looks. For example, you make 2 days a week to eat carrots.
• Now multiply 2 by 2, you get 4.
• Then add 4 to 4, it’s 9, then 9 times 50, it’s 450.

Let's say your birthday is July 18, 1997. For example, now is September-month and your birthday has already passed.

• So, add the number 1 750 to 450, you get 2,200.
• Now subtract the year of birth 1997 from the number 2,200, you get 203, add 7 to this number.
• The result is 210 (2 days and 10 years).

In the second case, subtract 1,997 from the number 2,199, get the number 202, add 7, get 209. So, 2 days of carrots and 9 years of mystery are made up.

Tip: Before performing this mathematical trick, give out the calculators to the audience so that they don’t make mistakes in the calculations, and for yourself, write down the order of operations with numbers on the card for the first time: what to multiply, what to add, what to subtract from.

### The secret of focus.

In order to find out what happened, you do not need to know the planned number. The main thing is to add to the number 1 089 that number (100, 200, 300, 400.), which they added at the very end. In order not to confuse who has what happened, at the very end of the focus you can give out cards with the numbers 100, 200, 300 and ask them to keep them while guessing the final result.

You will need: pre-prepared sheets of paper (according to the number of spectators), pencils or pens, calculators.

### The content of the mathematical focus.

• Invite your viewers to think of a two-digit number.
• Now let them multiply the number of its tens by 2,
• add the number 5 to this work,
• multiply this amount by 5,
• to the resulting product will be added 10 and the number of units of that number that was conceived.

Let any viewer say what he did. Subtract the number 35 from the result (it is better to do this in the mind or on the calculator, without dedicating the audience to your actions), and you can name the number conceived by the audience.

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