Math tricks (1-3)

In this section, we will give a free tutorial on magic tricks, with which you will surely surprise your comrades, friends, relatives, and we will start this section with mathematical tricks.

The main theme of mathematical tricks is guessing the intended numbers or the results of actions on them. The whole "secret" of these tricks is that the "guesser" knows and knows how to use the special properties of numbers, while the "thinker" does not know these properties).

Mathematical tricks are interesting in that each trick has its own mathematical interest and consists in "revealing" its theoretical foundations, which in most cases are quite simple, but sometimes they are cleverly disguised.

You can check the feasibility of each trick on any example, but to justify most arithmetic tricks, it is most convenient to resort to algebra. At first, you can omit the "evidence" of tricks and limit yourself to just digesting their content to show your friends. But the proofs will not make it difficult for those who like to think and are familiar with the rudiments of algebra.

Only the basic framework of mathematical tricks is given here, since their practical arrangement may vary according to conditions and place, as well as to your taste, wit and invention.

Guessing the intended number (7 tricks)

Focus 1 .

First math trick with numbers.
Think of a number. Subtract 1. Double the remainder and add the originally conceived number. Tell the result. I'll guess the number.

Guessing method.
Add 2 to the result, and divide the sum by 3. The quotient is the intended number.
Example.
Conceived 18; 18-1=17; 17x2 = 34; 34 + 18=52. Guess: 52 + 2 = 54; 54:3=18.
Proof. Let's denote the given number as x. We perform the required actions:

x-1; 2(x-1); 2(x-1) + x;

Result

2x - 2 + x = 3x - 2.

Adding 2, we get 3x, and dividing by 3, we get the intended number x.

Focus 2.

The second trick from the series "math tricks".
Have your friend think of a number. Then have him alternately multiply and divide the number he has in mind several times by various numbers you assign arbitrarily. Let him not tell you the result of actions.

After a few multiplications and divisions, stop and invite the person who thought of the number to divide the result he got by the number that he thought of, then add the number he thought to the last quotient and tell you the result. From this result, you immediately guess the number your friend thought of.

The secret is very simple. The guesser himself also needs to think of an arbitrary number (for example, 1) and perform on it all the multiplications and divisions assigned to him, up to the division by the originally conceived number. Then, in the quotient, he will get the same number as the other thinker, even if the originally conceived numbers were different for them. After that, the guesser must subtract his result from the result reported to him. The difference will be the desired number.

Example. The number 7 is conceived. Multiplied by 12. The result (84) is divided by 2. The resulting number (42) is multiplied by 5. The result (210) is divided by 3. It turned out 70, and after dividing by the conceived number and adding the conceived number -17.

At the same time, you “inwardly” thought of the number 1. Multiply by 12, it turns out 12. Divide by 2, it turns out 6. Multiply by 5, it turns out 30. Divide by 3, it turns out 10. Subtracting 10 from 17, you get the desired number 7.

Note 1. To enhance the effect, you can allow the person who thought of the number himself to assign the numbers by which he would like to multiply and divide the resulting results, if only he would tell you these numbers each time.

Remark 2. It is not necessary to alternate multiplications and divisions. You can assign multiple multiplications first and then multiple divisions, or vice versa.

Prove this arithmetic trick, i.e. show "in letters" that the trick succeeds for any conceived number.

Focus 3.

Let's continue the free magic trick training and show an interesting mathematical trick with numbers.
To teach this trick, we will accept or agree to call the majority of an odd number that part of it that is 1 more than the other. So, for the number 13, the majority is 7, for the number 21, the majority is 11.

Think of a number. Add half of it to it, or, if it is odd, then most of it. To this amount, add half of it or, if it is odd, then most of it. Divide the resulting number by 9, tell the quotient, and if you get the remainder, then say whether it is greater than, equal to or less than five. Depending on the answer to the question, the conceived number is equal to:

Quadruple quotient if there is no remainder;
- quadruple quotient +1 if the remainder is less than five;
- quadruple quotient + 2 if the remainder is five;
- quadruple quotient + 3 if the remainder is greater than five;

Example. Conceived 15. Performing the required actions, we have:

15 + 8 = 23, 23 + 12 = 35, 35: 9 = 3 (remainder 8). Reported: "quotient three, remainder greater than five".

We guess: 3 4 + 3 = 15. 15 is planned.

Prove this mathematical trick as well. When thinking about the proof, I advise you to take into account that any integer (hence, k conceived) can be represented in one of the following forms:

4n, 4n + 1, 4n + 2, 4n + 3,

where the letter n can be given values: 0, 1, 2, 3, 4, ...

Continued Free Trick Training:

The number in the envelope

simple arithmetic

1. Write down how many days a week you want to make love.
2. Multiply this number by 2.
3. Add 5 to the resulting number.
4. Multiply the amount by 50.
5. If you already had a birthday this year, add 1750, if not - 1749.
6. Subtract your year of birth from the resulting number.
7. Add 7 to the resulting number.
The first digit of the resulting number is the number of days per week on which you want to make love. The last two are your age.

Guess the crossed out number

You stand with your back to the board. The participant writes down any six-digit number on the board. You ask him to write a new number from the digits of the original number rearranged in any order. Then the smaller number is subtracted from the larger number. The resulting difference is multiplied by any number. In the resulting product, one digit that is not equal to zero is arbitrarily crossed out. Then the participant must tell you in random order all the uncrossed numbers. You guess the crossed out one.

Focus secret . If the numbers are rearranged and the smaller one is subtracted from the larger one, then the resulting difference is divided by 9. It is clear that the product must also be divisible by 9. The sum of the digits of this product must also be divisible by 9. When you are called the numbers, you mentally add them. After all the numbers are called to you, you must figure out which number to add to your sum so that the resulting number is divisible by 9. In the course of the steps, you can always add the numbers of the resulting intermediate amount to facilitate the calculation. For example, if you have a sum of 25 and must add 6, then you can add 6 not to 25, but to 7 (2 + 5). As a result, you can get not 13, but 4 (1 + 3).

Mysterious squares

The demonstrator stands with his back to the audience, and one of them selects any month on the monthly table calendar and marks on it some square containing 9 numbers. Now it is enough for the viewer to name the smallest of them, so that the demonstrator immediately, after a quick count, announces the sum of these nine numbers.

Explanation. The demonstrator needs to add 8 to the named number and multiply the result by 9

Guess the date of birth

So, first you need to choose a "victim", then ask her to count to yourself:
1. Multiply your birthday (to yourself) by two.
2. Add 5 to the result.
3. Multiply the result by 50.
4. Add the number of the month in which you were born.

Ask the person to say the number. Then just subtract 250 from the resulting one, and you're done. Get 4 or 3 digits. The first 2 (maybe one digit) is the day, and the last two are the month .

sly leaf

You choose 5 participants among the spectators and give them the same leaflets. Let the first of them write any two-digit number on a piece of paper and show this number to the second. The second participant must add the same number to the right and left of this number and divide this number by 3. He writes the result on a piece of paper (only the result!), Shows it to the third participant, then folds the piece of paper and passes it to you. The third viewer divides the number he sees by 7, writes the result on a piece of paper, shows it to the fourth viewer, folds the piece of paper and passes it to you. The fourth viewer divides the number by 13, writes the result on a piece of paper, shows it to the fifth viewer, folds the piece of paper and passes it to you. The fifth viewer divides the number by 37, writes the result on a piece of paper, adds it up and passes it to you. You take the same piece of paper, without looking at the received pieces of paper, write the original number, fold your piece of paper, approach the first viewer and show his piece of paper to the rest of the audience. Then you take out your leaflet, unfold it and, having called the number to the audience, show it.

Focus secret. If the same number is added to the left and right of any two-digit number, then the result will be a number 10,101 times greater than the original. 3 7 13 37 \u003d 10 101. Therefore, the number written on the piece of paper by the fifth participant coincides with the number written by the first participant. You show this leaflet to the audience (anything can be written on your leaflet).

The number in the envelope

The magician writes the number 1089 on a piece of paper, puts the piece of paper in an envelope and seals it. Offers someone, giving him this envelope, to write on it a three-digit number such that the extreme digits in it are different and differ from each other by more than 1.

Let him then swap the extreme numbers and subtract the smaller number from the larger three-digit number. As a result, let him rearrange the extreme numbers again and add the resulting three-digit number to the difference of the first two. When he receives the amount, the magician invites him to open the envelope. There he will find a piece of paper with the number 1089, which he did.

Mathematical tricks from simple to complex: dive into the tempting world of numbers.

Focus 1: "Familiar numbers"

Write the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in sequence on a piece of paper. Ask a student to add in their mind any three numbers that follow one after the other. And the result - to name. For example, he will choose 5, 6 and 7. In this case, the sum will be 18. After that, the teacher immediately calls the intended numbers.

Focus secret:

Introduction

Learning tricks, a person develops artistry, creativity. Math tricks direct children's attention to the math lesson, thanks to the entertaining essence of the trick, combined with the mathematical nature of the secret (once having shown the trick, the child can be stimulated to be active in the lesson under the pretext of revealing the secret). The whole point of viewing the trick is to find a clue and enjoy the "magical actions."

Event goals

To arouse in students an interest in mathematics, to instill a love for it. Raise the spirits of the students. Explain what mathematical tricks are, why they are needed, teach children a few of them.

Event progress

To begin with, the teacher says a few words about mathematical tricks, asks the children a few questions: “Do you like tricks? .. And what tricks do you know, can you show? .. Do you want to learn new tricks?” - etc. After a short discussion, it is worth showing a presentation in mathematics on the topic of mathematical tricks.

After being shown , you should transcend to demonstrate tricks. There are many mathematical tricks of various kinds, we will give just a few examples.

Focuses:

Day of the week in the palm of your hand
We number each day of the week (Monday - 1, Tuesday - 2, etc.). Any student can guess one of the days (a number from 1 to 7), the teacher suggests multiplying the guessed number by 2, then adding 5, multiplying the amount by 5, at the end add zero. The class is told the result, from which 250 is subtracted. As a result, the number of hundreds will correspond to the day

Focus secret: Substitute instead of the number of the day "x":

((2x+5)*5)*10=(10x+25)*10=100x+250

100x+250-250=100x. Therefore, the number of hundreds always corresponds to the number of the day.

Note: Tricks of this kind are the most common of all mathematical tricks, so do not fill the event with only them.

phenomenal memory

The teacher writes on a piece of paper a very long series of numbers (22-26 numbers) and declares that he can list all the numbers in the series from memory in the same order. Having done, you can repeat the trick to prove that the number series is absolutely arbitrary (there really should not be any pattern in it).

Focus secret: All numbers in the row are just well-known phone numbers (you can take the last 4-7 numbers from each number).

Note: As you can see from the example, in some mathematical tricks an ordinary trick is used.

Intuition, or magic nine

One student (or all at once) writes a number from 3 different digits, and next to it - a number from the same digits, but in reverse order. The smaller number is subtracted from the larger number. Not seeing the result, the teacher says that in the middle of the answer received is nine (if the answer is a two-digit number, then write it as 0 ...). And indeed, the nine stands, where it was predicted by the teacher.

Focus secret: Since only 1 and 3 digits are interchanged, then the larger number, the digit in the units digit will always be less, which means that you will need to take 1 from the tens digit, and when you need to subtract tens - from the hundreds digit (to understand - try to solve in a column) . For example, 653-356=297.

Note: The secrets of the most interesting mathematical tricks usually cannot be guessed at first glance, and the trick itself is difficult to attribute to any subgroup.

Conclusion

Math tricks are a great way to make children fall in love with the subject being studied, to understand all the splendor of its properties and rules.

Math tricks 4-7
Guessing the intended number

Focus 4.

The fourth trick in the seriesMath trickssection Let's start as in the previous trick, that is, offer to think of a number and add half or most of it to it, then again add half of the resulting amount or most of it.

But now, instead of the requirement to divide the result by 9, offer to name all the digits of the resulting result, except for one, by digits, so long as this unknown figure is not zero.

It is also necessary that the person who thought of the number should say the rank of the number that is hidden from him, and in what cases (in the first, in the second, or in the first and second, or not even once) did he have to add most of the number.

After that, in order to find out the intended number, you need to add up all the numbers that are named and add:

- 0 if you never had to add most of the number;
- 6, if only in the first case it was necessary to add most of the number;
- 4, if only in the second case it was necessary to add most of the number;
- 1 if in both cases it was necessary to add most of the number.

Further, in all cases, the resulting sum must be supplemented to the nearest multiple of nine. This addition will be a hidden figure. Now, knowing all the digits of the result, and hence the whole result, it is not difficult to find the intended number. To do this, you need to divide the result by 9, multiply the quotient by 4 and, depending on the size of the remainder, add 1, 2 or 3 to the product.

Example 1 The number 28 was conceived. After the required actions were completed, it turned out 63. They hid the number 3. Then the guesser complements the number of tens reported to him 6 to 9 and receives the number of units 3. The result 63 is found. The desired number is (63:9)x4 = 28.

Example 2 The number 125 was conceived. After performing all the required actions, it turned out to be 282. Let's say, the number of hundreds is hidden 2. It is reported: the digits of tens and units are 8 and 2, respectively, and most of the number was added only in the first case.

Guess: 8+2+6=16. The closest multiple of nine is 18. So the hidden hundreds digit is 18-16 = 2.

We determine (guess) the intended number: 282:9 = 31 (remainder 3); 31x4+1 = 125.

Example 3 Let the thinker of the number say that the last result he got consists of three digits, the first digit being 1, and the last 7, and most of the number had to be added in two cases.

We guess the intended number: 1+7+1=9. The complement to a multiple of nine is zero or nine, but zero cannot be hidden by condition, therefore, the hidden number is 9 and the whole result is 197. Divide 197 by 9; 197:9 = 21 (remainder 8). The intended number is 21 4+3 = 87.

Prove your focus. This is not difficult, especially for those who have understood the essence of the proof of the previous trick.

Focus 5.

We continuemath tricksto guess the given number. Fifth math trick. Think of a number (less than a hundred, so as not to complicate the calculations) and square it. Add any number to the planned number (just tell me which one) and square the resulting amount too. Find the difference between the resulting squares and report the result.

To guess the conceived number, it is enough to divide half of this result by the number added to the conceived one, and subtract half of the divisor from the quotient.

Example. Conceived 53; 53 squared \u003d 53x53 \u003d 2809. 6 was added to the intended number:

53 + 6 = 59, 59x59 = 3481, 3481 -2809 = 672.

This result has been reported.
Guessing:

072:12 = 60, 0:2 = 3, 50 - 3 = 53.

The intended number is 53.
Find proof.

Focus 6.

Sixth math trick. Invite your friend to think of any number within the range from 6 to 60. Now let him divide the conceived number first by 3, then divide it by 4, and then by 5 and report the remainder of the divisions. From these remainders, using the key formula, you will find the intended number.

Let the remainders R 1 , R2 and R3 . Now remember this formula:

S=40R1 +45R2 +36 R3 .

If it turns out S=0, then the number 60 is conceived; if S is not equal to zero, then the remainder of dividing S by 60 will give you the intended number. It will not be so easy for your friend who has conceived a number to guess the secret of guessing that you own.

Example. Conceived 14. Remains reported: R1 =2, R2 =2, R3 =4.

Guessing:

S \u003d 40x2 + 45x2 + 36x4 \u003d 314;
314:60 = 5

and the remainder is 14.
The intended number is 14.

Do not blindly believe the formula proposed without a conclusion. First make sure that it works flawlessly in all cases allowed by the focus condition, and then demonstrate the focus.

Focus 7.

The seventh math trick in the seriesmathematical tricks for guessing the intended number. Having understood the mathematical basis of the tricks presented here, you can modify them in every possible way, come up with other rules for guessing numbers, and diversify the proposed questions.

Here, for example, is such a topic. In the previous trick, guessing the intended number by its remainders from division were proposed as divisors of the numbers 3, 4 and 5. Let's replace them with other divisors, for example, such as 3, 5, 7, and expand the limits for the intended numbers from 7 to 100. Factors in the key formula, of course, will also change. Match them to a new key formula suitable for the occasion.

Answer.
S=70R1 +21R2 +15R3 , where R1 , R2 and R3 - respectively, the remainders from dividing the intended number by 3, 5 and 7. Guess the intended number. It is equal to the remainder of dividing S by 105 (if S = 0, then 105 is intended).

Focus on Rhinoceros

(cool trick .. for showing non-believers in tricks, but EVERYTHING who knows :)))

Think of a number from 1 to 10. Guessed?

You've got a two-digit number.

Add the first digit of this two-digit number to the second. Example: if the number is 21, then you need to add 2 + 1. .Next: folded?

Subtract 4 from the result.

Now think of a letter for this number alphabetically. That is, if you get 1, then this is the letter A; 2-letter B; 3-B; 4-G, etc.

Now you have guessed and keep a letter in your head, remember this letter and think of a European country.

See answer below...

Answer: There are no rhinoceroses in Denmark!!! Ha-ha-ha...

After all the mathematical calculations, you get 9, then 5. This is the letter D. There is one country for the letter D - Denmark.

The rest must be brought
play! You can as if I can read minds, etc.

In order to surprise your friends and loved ones by performing magic tricks, you do not need to have super dexterous hands and mysterious magical props. It is enough to know the secrets of interesting tricks based on mathematics.

Math Tricks: Secrets and Solutions

1. NINE

On a table in the form of a nine (see picture), you need to lay out 12-20 coins. Twelve is the minimum. From those present, a person is selected who will guess. In order to avoid errors in the calculations, it is possible to organize collegial guessing from several, or even all those present. You stand with your back to the audience.

Rice. 3 Nine

The guesser thinks of a number that is greater than the number of coins that make up the "leg" of the nine. The maximum value of the number is theoretically unlimited, but it should still be based on common sense. To avoid possible jokes, its value can be limited in advance. After that, the guesser counts as many coins as he conceived as follows: starting from the “leg” from the bottom up, and then further, counterclockwise around the ring. After he counts the intended number of coins, the count is repeated. You should start with exactly the coin on which the previous account stopped. But now the guesser counts the coins from one to the intended number along the ring in a clockwise direction. Under the coin, the account on which ended, the guesser hides, for example, a small inconspicuous piece of paper.

You turn to the audience, make "magic passes" over the table while looking at the audience, and pick up the hidden coin.

FOCUS SECRET. Everything is very simple. The fact is that no matter what number is conceived, the account ends in any case in the same place. To get started, do this trick in your mind with any number, and you will know what kind of coin it will be. If you are asked to repeat the trick, the nine should be modified by removing or adding a few coins to the leg. This technique will allow you to change the position of the "hidden" coin.

2 . Heads or tails?

Another trick with coins is based on the difference between heads and tails. A handful of little things are laid out on the table. You ask someone in the audience to turn over the coins at random, one at a time. Each inversion should be accompanied by the word "is." These actions should be done behind your back. The same coin can be flipped multiple times. At the end, the guesser covers one of the coins with his hand. You turn around and say exactly how the coin lies - “heads” or “tails” up.

FOCUS SECRET. The whole point of the focus is in your preparation. After the coins are scattered, it is necessary to count the number of "eagles". With each "is" you need to add one to this number. It all depends on the final number. If it turned out to be even, then the number of “eagles” in the final combination is even, if the sum is odd, then the number of “eagles” is odd. The position of the hidden coin will be "speak" open.

This trick can be done with any of the same items that can be placed in one of two possible ways.

As you already understood, the above tricks, like all mathematical tricks, are based on the properties of figures and numbers, and their secrets are in the exact reflection of a certain mathematical pattern.

It sounds like magic...but it's actually math! Do you want to become a magician? Thanks to this book, you will always have mathematical tricks in your arsenal. With pencil and paper, you can do the most incredible things. For example, correctly guessing a person's age, reading someone's mind, making accurate predictions, demonstrating your amazing memory. This book will allow you to acquire "sleight of hand", will teach you everything that is listed above, and even more. In it you will find tips on how to prepare the audience for a particular focus. And, best of all, you will learn the secrets of these amazing magic tricks. Dare!

Focus with marked dates

Focus starts like this. The viewer is offered to open the monthly report card for any month and circle one date in each of the five columns of his choice. (In the case when the numbers are arranged in six columns, which is very rare, the sixth column is not taken into account.) In this case, the demonstrator stands with his back to those present.

Still not turning around, he asks, "How many Mondays do you circle?", then, "How many Tuesdays?" and so on, going through all the days of the week. After the seventh and final question, the demonstrator announces the sum of the circled numbers.

Focus secret. The sum of the numbers in a string that starts on the first of the month is always 75 (except in non-leap year February). Each marked number in the next line increases this sum by 1, in the next line by 2, and so on; each marked number in the previous line reduces the mentioned sum by 1, in the previous line by 2, etc. Let, for example, the first day of the month falls on Thursday and one Monday, one Thursday and three Saturdays are circled; the demonstrator performs the calculation in his mind:

75 + 3 * 2 - 1 * 3 = 78

and announces the result.

Of course, the viewer must know in advance what day the first day of the month chosen by the viewer falls on.

1. By the principle of mathematical focus.

(Einstein as a mathematician magician).

Tricks are based on deceiving people in the hope that this deception will not be immediately noticed. They are harmless in that the magician does not even assume that they will unconditionally believe him. The only hope is that the essence of his trick will not be immediately revealed. Focus is a kind of entertainment, nothing more.

It is very difficult to understand whether Einstein considered himself a magician. It is possible that he believed in his genius and absolutely did not have the gift of self-criticism. After all, even his best friend at that time, he himself tried, without the support of the Academies of Sciences, to put him in a psychiatric hospital - for criticizing his article. This is instead of checking for the hundredth time if there is an error in it. It is not known if he checked his article at least once after it was published. But, as you know, finding your own mistake is much more difficult.

The disadvantage of Einstein's critics is that they usually refute the conclusions of the "theory of relativity", instead of looking for an error in the work itself, which is much easier. I have already done this kind of work once, but this time I decided to approach Einstein's "work" from a different angle. There is no need to do any math at all. Einstein's mistakes, of course, are not mathematical, but logical.

What is "math trick"? I will give an example familiar to me from the school bench, although the text that I am citing may be somewhat different.

Guess the number

Ask someone to think of any number, then subtract 1 from it, multiply the result by 2, subtract the intended number from the product and tell you the result. By adding the number 2 to it, you will guess what was intended.

Guess the date of birth

Multiply your birth date by 2, add 5, multiply by 50, and add the number of the month. From the number that turned out, subtract 250 and get the birthday and month.

Guess the result of operations on an unknown number

Someone thought of a number. You ask to multiply it by 2, then add 12 to the product, divide the sum in half and subtract the intended number from it. Whatever number is intended, the result will always be 6.

Today I want to offer you a mathematical focus from the series "Entertaining tasks". With this trick, you can surprise your friends. If you don't know when your friends' birthday is, you can guess their birthday using some simple math.calculations. You can, of course, just ask any person when his birthday is. But it is much more interesting to surprise a person, entertain, amuse or simply impress with the help of mathematics.

Surprise a friend by guessing his date of birth without asking her!

What needs to be done?

So:

Tell your friend to multiply his date of birth by two, but don't say the result of his calculations out loud.

Now ask him to add five to the number that he got.

Next step: the last result obtained, have your friend multiply by 50. If multiplication is difficult, you can take a calculator. To ensure that there is no error. It is very important!

And finally, ask your friend to add the ordinal number of the month in which he was born to the last result obtained.

All!

Now ask him to voice the result that he got after all the calculations.

Now you subtract 250 from the voiced number. You will get a 3-4 digit number as a result.

The first 1-2 digits from the left in this number is the date of birth, and the next two are the month of your friend's birth.

Shine with this trick in the circle of your friends, acquaintances and relatives!

Wish you luck!

This math trick with phone numbershowed me a brunette. Her reaction was quite emotional: "The removal of the brain! How can this be ?!". Indeed, the impression is that shamans with tambourines are dancing around the calculator. Here is a description of this mathematical trick with a phone number. I will clarify right away that the focus is designed for a city seven-digit phone number.

The text of the work is placed without images and formulas.
The full version of the work is available in the "Job Files" tab in PDF format

"The subject of mathematics is so serious that it is useful to seize the opportunity, to make it a little entertaining"

B. Pascal

When we first met in a mathematics 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 proposed. First, we had to multiply our birthday by 2, add 5 to the resulting number, multiply the result by 50 and, finally, add the number of the month of our birth to what we got. 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 incorrect calculations. I really liked this trick. I also wondered what underlies this focus. It was then that I decided that I would definitely research the issue of mathematical tricks, learn their secrets, make a selection of tricks and surprise and entertain my friends and acquaintances by demonstrating mathematical tricks in math classes, extracurricular activities and even at home holidays.

I have read in Internet sources that mathematical tricks do not receive special attention from either mathematicians or magicians. The first consider them simple fun, the second - too boring.

But, in my opinion, this is not so at all. Mathematical tricks have their own deep meaning.

Mathematical tricks are experiments based on mathematical knowledge, on the properties of figures and numbers, exposed in an extravagant form. To understand the essence of this or that experiment means 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 own new tricks. And the secret of focus becomes clear when we write down 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 develop memory, quick wits, the ability to think logically, improve mental counting skills and, finally, simply increase students' interest in mathematics, which should improve the quality of their knowledge.

Objective: explore math tricks.

Tasks:

Study the literature on the topic under study.

Demonstrate multiple tricks.

Explain them in terms of mathematics.

To draw the attention of classmates to the study of mathematics.

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 lessons and extracurricular activities, at mathematical evenings and holidays, during mathematical competitions.

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

Focus- a skillful trick based on deception of vision, attention with the help of a deft and quick technique, movement (Ozhegov's dictionary)

The history of the emergence of mathematical tricks.

The first document that mentions illusionary art is an ancient Egyptian papyrus. It contains legends relating to 2900 BC, the era of the reign of Pharaoh Cheops.

Initially, tricks were used by sorcerers and healers. 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: thunder, flashing lightning, temple doors opening by themselves, statues of gods suddenly appearing from under the ground, musical instruments themselves, voice.

In Ancient Hellas, without games, the harmonious development of the personality was not conceived. 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 were not alienated by scientists, thinkers, teachers. 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, he was especially attracted to mathematical tricks, which he knew a great many, and some of them he invented himself.

K.D.Ushinsky, A.S.Makarenko, A.V.Lunacharsky repeatedly pointed out the enormous cognitive and educational value of intellectual games. 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 (Gardner).

He was born on October 21, 1914. Graduated from the Department of Mathematics at the University of Chicago. Founder (mid-1950s), author and presenter (until 1983) of the Mathematical Games column of Scientific American (In the World of Science). Gardner interprets entertainment as a synonym for fascinating, interesting in knowledge, but alien to idle entertainment. Among the works of Gardner are philosophical essays, essays on the history of mathematics, mathematical tricks and "comics", popular science studies, science fiction stories, quick wits.

Particularly popular were Gardner's articles and books on entertaining mathematics. Seven books by Martin Gardner have been published in our country, which captivate the reader and encourage independent research. "Gardner" style is characterized by intelligibility, brightness and persuasiveness of presentation, brilliance and paradoxical thought, novelty and depth of scientific ideas.

Among our compatriots, I would like to name the name of Ya.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 titles or 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 into the fascinating world of mathematics, physics, and astronomy begins. 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, teachers, 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 addicted to mathematical tricks. And this is not surprising. “Mind gymnastics” is useful at any age. And tricks train memory, sharpen intelligence, develop perseverance, the ability to think logically, analyze and compare.

Chapter 2

Focus "Guess the intended 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 subtract the intended number.

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

Focus clue:

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 original number

Focus "Magic table".

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

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

After that, I will name the number you have planned

Focus clue:

This table is compiled as follows: each column corresponds to a certain number, having calculated the sum of which the magician guesses the number you have chosen

For example: You thought of the number 27.

This number is in the 1st, 2nd, 4th and 5th columns.

It is enough to add the numbers located in the first row of the table in the corresponding columns, and we will get the intended number. (1+2+8+16=27).

Focus "Favorite number".

Any of those present conceives their favorite number.

I suggest that he multiply the number 15873 by his favorite number multiplied by 7.

Focus clue:

1) 15873 * 7 \u003d 111111. Thus, multiplying 15873 by 7 and by your favorite digit, we get a number written only by your favorite digit.

For example, favorite number is 5

1) 15873 *(7*5) 2) 15873 *35 = 555555.

4. Focus "Guess the planned day of the week."

We number all the days of the week: Monday - the first, Tuesday - the second, etc.

Have someone think of any day of the week. I suggest you the following actions: multiply the number of the planned day by 2, add 5 to the product, multiply the resulting amount by 5, add 0 to the resulting number, tell the magician the result.

Focus clue:

Let's say Thursday is conceived, that is, the 4th day.

Let's do the following: ((4×2+5)*5)*10 = 650,

650 - 250 = 400.

The number of hundreds and shows the hidden day of the week.

By the way, the trick that our teacher showed us at the beginning of the school year to guess the date of birth has the same secret.

Let my birthday (and this is a one-digit or two-digit number) X, and the number of the month of my birth at then we have:

(2 · X+ 5) 50 + at= 100 X + 250 + y. If we now subtract 250 from the result, we get a three or four digit number, the last two digits of which indicate the month number, and the first one or two digits indicate the birthday.

5. Focus "Familiar numbers"

After that, the magician immediately calls the intended numbers.

Trick clue:

6. Focus

2. Ask a friend to write down a number from 100 to 999. The only condition! The difference between the first and last digits must be greater than one. For example, the number 346 is suitable, since 6 - 3 = 3, and 3 is greater than 1. But the number 344 is not suitable, since 4 - 3 = 1.

3. Suppose your friend has already chosen a number and wrote it down. Your task is to rewrite this number in reverse order (346, and you write 643).

4. Now subtract the smaller number from the larger number (643 - 346 = 297).

6. Add both numbers (297+792).

Focus clue:

100a+10b+c; a - c > 1.

100a + 10b + c - 100c - 10b - a = 99a - 99c = 99(a - c).

a - c \u003d 2, 99 * 2 \u003d 198, 198 + 891 \u003d 1089,

a - c \u003d 3, 99 * 3 \u003d 297, 297 + 792 \u003d 1089,

a - c \u003d 4, 99 * 4 \u003d 396, 396 + 693 \u003d 1089,

a - c \u003d 9, 99 * 9 \u003d 891, 891 + 198 \u003d 1089.

7. Focus

A circle of comrades who are not initiated into the mathematical secret of the Scheherazade number can be struck by the following trick.

Have someone write on a piece of paper - secret from the magician - a three-digit number, then have them add the same number to it again. The result is a six-digit number consisting of three repeated digits.

The magician offers the same comrade or his neighbor to divide - secretly from him - this number by 7: at the same time, he warns that there will be no remainder. The result is passed to another neighbor who divides it by 11, there should be no remainder. The result is passed to the next neighbor, who is asked to divide the number by 13 (again without a remainder).

The result of the third division is transmitted to the first comrade with the words:

Here is the number you have in mind.

Focus clue:

This beautiful arithmetic trick, which gives the impression of magic to the uninitiated, is explained very simply. To attribute it to a three-digit number itself means to multiply it by 1001 (Scheherazade's number), that is, by the product of 71113. It is clear that if the intended number is first multiplied by 1001, and then divided by 1001, then you will get it yourself.

This focus can be changed. Suggest dividing by 7, then by 11, and then by the intended number. Then we can say with confidence what will happen as a result of 13.

8. Focus "Guess the result of calculations without asking anything"

Let's write some number between 1 and 50 on a piece of paper and hide it without showing it to the participants of the trick.

In turn, have each participant write what he wishes, a number greater than 50 but greater than 100, and without showing you, perform the following actions:

add 99 - x to its number, where x is the number you wrote on a piece of paper (you will calculate this difference in your mind and tell the participants of the focus the finished result);

cross out the leftmost figure in the resulting sum and add the same figure to the remaining number;

the resulting number will be subtracted from the number originally written by him.

As a result, all participants will get the same number, exactly the one that you wrote down and hidden.

Focus clue:

my number X , where " X" greater than 1 but less than 50.

Conceived number at , where " at" greater than 50 but less than or equal to 100.

y - (y + 99 - x - 100 + 1) = y - y - 99 + x + 100 - 1 = x.

9. Focus, modeled by myself.

Guessing the number of the house and apartment of the focus participant.

Add 8 to the house number, multiply the result by 8, multiply the result by 125, add the apartment number to the result. Tell me how much you got, and I'll tell you the number of your house and apartment number.

Focus secret:

(X + 8) * 8 * 125 + Y - 8000 = 1000X + 8000 + Y - 8000 = 1000X + Y.

The last one, two, three digits are the apartment number, the first 1 - 2 digits are the house number.

Conclusions.

Previously, I did not understand the significance of mathematical tricks, because I understood little about them. I learned that the secret to many magic tricks is equations. While doing research, I was convinced that mathematical tricks are interesting for schoolchildren.

Thanks to the work, I increased my knowledge, and also realized that tricks sharpen the ability to think logically, analyze and compare.

In addition, I realized that my current knowledge is not enough to understand the nature of many tricks that I encountered while researching the topic. This applies to knowledge of algebra and geometry. Therefore, I will continue to study mathematical tricks in the next classes.

Conclusion

There is an interesting story.

“A long time ago there was an old man who, dying, left 19 camels to his three sons. He bequeathed half 1/2 to his eldest son, a fourth to his middle son, and a fifth to his youngest. Unable to find solutions on their own (after all, the problem in “whole camels” has no solution), the brothers turned to the sage.

O wise one! - said the elder brother, - my father left us 19 camels and ordered us to divide among ourselves: the elder - half, the middle - a quarter, the youngest - a fifth, but 19 is not divisible by either 2, or 4, or five. Can you, venerable one, help our grief, for we want to fulfill the will of the father?

“There is nothing easier,” the sage answered them. Take my camel and go home.

The brothers at home easily divided the 20 camels in half, into 4 and 5. The older brother received 10 camels, the middle brother 5, and the younger 4 camels. At the same time, one camel (10 + 4 + 5 = 19) remained superfluous. The brothers returned to the sage and complained:

Oh, sage, again we did not fulfill the will of the father! This camel is superfluous. - Not superfluous, - answered the sage, - this is my camel. Return it and go home. "There are no unsolvable problems. There is always a way out" (folk wisdom)

Mathematical tricks are varied. In many mathematical tricks, numbers are veiled by objects related to numbers. They develop skills in quick mental counting, calculation skills, as you can think of small and large numbers, awaken the imagination, surprise, fascinate, develop the creative beginnings of the individual, artistic abilities, stimulate the need for creative self-expression. Mathematical tricks contribute to concentration. The magic of focus can wake up the sleepy, stir up the lazy, make the slow-witted think. After all, without unraveling the secret of the focus, it is impossible to understand and appreciate all its charms. And the secret of focus most often has a mathematical nature.

Literature

Perelman, Ya.I. Entertaining arithmetic. Numbers and tricks / Ya.I. Perelman. - M.: OLMA Media Group, 2013

Perelman, Ya.I. "Live Mathematics", D .: VAP, 1994

Kordemsky, B.A. Mathematical ingenuity. - M.: Science. Ch. ed. Phys.-Math. lit., 1991

Ignatiev E.I. In the realm of ingenuity - M .: Nauka. Ch. ed. Phys.-Math. lit., 1984

M. Gardner "Mathematical miracles and secrets" - Moscow: "Nauka", 1988

Application

Focus 1: "Familiar numbers"

Write the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in sequence on a piece of paper. Ask a student to add in their mind any three numbers that follow one after the other. And name the result.

For example, he will choose 5, 6 and 7. In this case, the sum will be 18.

After that, I immediately called the planned numbers.

Focus secret:

It only takes a little ingenuity to do this trick.

When they call the sum (5 + 6 + 7) \u003d 18, in your mind divide it by 3. In our case, you get 6. This is the desired average figure. The number before it is 5, and after it is 7. The whole effect of this trick is in a lightning-fast response.

Focus 2

1. Write the number 1089 on a piece of paper and temporarily put it aside (without showing it to anyone).

2. Ask a friend to write down a number from 100 to 999. The only condition! The difference between the first and last digits must be greater than one. For example, the number 346 is suitable, since 6-3=3, and 3 is greater than 1. But the number 344, for example, is not suitable, since 4-3=1. Clear? If not, please read first.

3. Suppose your friend has already chosen a number and wrote it down. Your task is to rewrite this number in reverse order (346, and you write 643). Ready?

4. Now subtract the smaller number from the larger number (643-346=297).

5. Now write down the resulting answer in reverse order (it was 297, it will become 792).

6. Add both numbers (297+792).

7. Voila! Show your leaf with the magic number 1089. You knew in advance what answer you would get! Indeed, 297+792=1089! Focus-pocus!!! The most interesting thing is that this algorithm always works!

INTRODUCTION

Like many other subjects that are at the intersection of two disciplines, mathematical tricks do not receive special attention from either mathematicians or magicians. The first tend to regard them as empty fun, the second neglect them as too boring. Mathematical tricks, to put it bluntly, do not belong to the category of tricks that can keep an audience of non-mathematicians spellbound; such tricks usually take a lot of time, and they are not very effective; on the other hand, there is hardly a person who is going to draw deep mathematical truths from their contemplation.

And yet, mathematical tricks, like chess, have their own special charm. Chess combines the elegance of mathematical construction with the pleasure that the game can deliver. In mathematical tricks, the elegance of mathematical constructions is combined with amusement. It is not surprising, therefore, that they bring the greatest pleasure to those who are simultaneously familiar with both of these areas.

Objective: study of mathematical tricks.

Tasks:

1. Study the literature on this issue and Internet resources.

2. Select and summarize the most interesting, fascinating mathematical tricks.

3. Conduct selected math tricks in class.

4. Find out what is the secret of mathematical tricks.

Object of study:mathematical tricks based on the properties of numbers, actions, mathematical laws, equations.

Research methods

Study, analysis, practical application of the acquired knowledge.

Relevance of the topic:is as follows: mathematical tricks are rarely considered and applied in teaching mathematics.

Hypothesis: It can be assumed that if you draw the attention of students to mathematical tricks, then it will be possible to interest them in studying the subject of mathematics, to promote the development of oral counting skills to demonstrate mathematical tricks.

Chapter 1. Theoretical part.

1.1. Illusionists and conjurers of the world.

The history of hocus pocus.

The art of illusion has its roots in ancient times, when the techniques and techniques for manipulating people's minds began to be used not only to control them (as shamans and priests did), but also for entertainment (fakirs' performances). In the Middle Ages, more professional artists appeared: puppeteers, magicians using various mechanisms, as well as card players and cheaters.

In the XV century. The girl was executed for witchcraft. It was in Germany. Her fault was only that she performed a trick with a handkerchief: they tore it into pieces, and then connected them, turning them into a whole handkerchief. Passed down from generation to generation, tricks for several hundred years served not only for entertainment, but also made the poor rich, the rich poor, and also brought joy to one and meant collapse to another.

Simultaneously with the development of magic tricks, there was an active development of deceptive tricks, which does not quite adorn the trick business. However, the true talent and skill of the "correct" magicians can bring all dishonest tricks to naught. The first mention of magicians came to us from the distant XVII century. The inhabitants of Germany and Holland were indelibly impressed by the "wizard" Ohes Vohes (the magician borrowed this name from the mysterious magician-demon from Norwegian legends).

During his magical sessions, the magician used to say: “Focus pocus tonus talonus, vade celeriter ubeo. The spectators, however, disassembled from all this muttering only the mysterious “hocus pocus”. Therefore, the wizard received the nickname of the same name. These magic words seemed funny to other representatives of the profession, they picked them up, and soon all illusionists and tricksters began to call their performances tricks.

At the end of the XVIII - beginning of the XIX century. with the development of mechanical engineering, mechanical illusory automatic toys appear. Three such mechanical dolls, which depicted human figures, were invented by Friedrich von Claus, director of the physico-mathematical office of the Vienna Imperial Palace. His figures could write on paper.

Designer Jacques de Vaux-Canun made functioning mechanical figures of a flutist and drummer in full human growth and a duck that could quack, peck food and flap its wings. The Hungarian Wolfgang von Kempelen invented the "chess player" piece, with which one could play a game of chess. But in fact, only the hand of the puppet moving the chess pieces on the board was mechanical, but it was controlled by a chess player - a man sitting inside.

In the XVIII century. The performances of magicians were perfected by the Italian Giuseppe Pinetti. It was he who first began to show tricks not in the marketplaces, but on a real theater stage. He made it an art for a sophisticated audience, furnishing tricks with lush scenery, intricate plots. In the English newspapers of that time, there were notes about his performances in London in 1784. Pinetti surprised the audience with his abilities: he read texts with his eyes closed, distinguished objects in closed boxes.

The magician even attracted the attention of the monarch of England, George III, who invited Pinetti to perform in front of members of the royal family at Windsor Castle. The magician did not lose face, he brought with him a huge number of assistants, exotic animals, complex mechanisms, large mirrors.

After such a performance, Pinetti went on an international tour of Europe, Portugal, France, Germany and even Russia were on his way. In St. Petersburg, he held several performances and was even invited to the palace of Emperor Paul I. When Pinetti was leaving Russia, Tsar Paul I asked him to surprise everyone with some kind of magic. At that time, it was possible to leave St. Petersburg through 15 gates. Pinetti promised the tsar that he would pass through all 15 outposts at the same time, and he kept his word. The tsar was brought 15 reports from 15 outposts that Pinetti left precisely through each outpost. In 1800, Giuseppe died at the age of 50.

Giuseppe adored his tricks, he lived an illusion and created it in his daily life. It was said that while walking along the street, the magician could buy a hot bun from the stall and, in front of a crowd of onlookers, breaking it in half, pulled out a gold coin. In a second, this coin turned into a medallion with the magician's initials.

The famous magician Ben Ali often showed such a trick to the fair. He approached any merchant, bought pies from him, in front of the assembled people he broke them in half, and a coin was found in each pie. The surprised merchant could not believe in this miracle and began to “check” all his other pies, in which, of course, there was nothing. The audience laughed. When food was brought to Ben Ali in a restaurant, he covered the entire table with a blanket, and when he took it off, instead of food, there was a shoe on the table. The shoe was covered again and the food returned.

Among the well-known illusionists of that time, one can safely rank two other famous Italians: Giacomo Casanova (1725-1798) and Count Alessandro Cagliostro (1743-1795). Numerous legends circulated and circulate about their magic tricks, it is difficult to distinguish what is true in them and what is fiction of an enthusiastic crowd.

At the end of the XVIII - beginning of the XIX century. In Europe, the industrial revolution begins, steam engines, a steamboat, spinning machines and many technical innovations appear. Tricks become more technical and complex, magicians become professionals - the inventors of complex mechanical tricks.

The place of "wizards", "magicians" and "sorcerers" is occupied by "doctors" and "professors", giving the tricks "scientific" and "seriousness". These are such "scholarly magicians" as Jean-Eugene-Robert Houdin, who is called the "father of modern magic." Modern magicians still use the mechanisms of Jean-Eugène-Robert Houdin.

1.2. Math tricks.

Numbers surround us everywhere: in stores, on the street, at work, at home. It is not surprising that in the entire history of mankind, many tricks were invented with them, which later began to turn into tricks. Tricks with numbers can be demonstrated anywhere, in front of any audience; sleight of hand is not needed here, but only a good memory and knowledge of the system of actions is required.

1. Focus “Phenomenal memory”.

To carry out this trick, it is necessary to prepare many cards, on each of which put its number (two-digit number) and write down a seven-digit number according to a special algorithm. The “magician” distributes cards to the participants and announces that he has memorized the numbers written on each card. Any participant calls the number of the card, and the magician, after a little thought, says what number is written on this card. The solution to this trick is simple: in order to name the number, the "magician" does the following - adds the number 5 to the card number, flips the digits of the resulting two-digit number, then each next digit is obtained by adding the last two, if a two-digit number is obtained, then the unit digit is taken. For example: card number - 46. Add 5, get 51, rearrange the numbers - get 15, add the numbers, the next - 6, then 5 + 6 = 11, i.e. take 1, then 6 + 1 = 7, then the numbers 8, 5. The number on the card: 1561785.

2. Focus "Guess the intended number."

The magician invites one of the students to write any three-digit number on a piece of paper. Then add the same number to it again. Get a six-digit number. Pass the sheet to a neighbor, let him divide this number by 7. Pass the sheet further, let the next student divide the resulting number by 11. Pass the result further again, let the next student divide the resulting number by 13. Then pass the sheet to the “magician”. He can name a given number. Focus clue:

When we assigned the same number to a three-digit number, we thereby multiplied it by 1001, and then, dividing it sequentially by 7, 11, 13, we divided it by 1001, that is, we got the intended three-digit number.

3. Focus "Magic Table".

There is a table on the board or screen, in which, in a known way, numbers from 1 to 31 are written in five columns. The magician invites those present to think of any number from this table and indicate in which columns of the table this number is located. After that, he calls the number you conceived.

Focus clue:

For example, you thought of the number 27. This number is in the 1st, 2nd, 4th and 5th columns. It is enough to add the numbers located in the last row of the table in the corresponding columns, and we will get the intended number. (1+2+8+16=27).

4. Focus "Guess the crossed out number."

Let someone think of some multi-digit number, for example, the number 847. Ask him to find the sum of the digits of this number (8+4+7=19) and subtract it from the intended number. It turns out: 847-19=828. including what happens, let him cross out the number - it doesn’t matter which one, and tell you all the rest. You will immediately tell him the crossed out figure, although you do not know the intended number and did not see what was done with it.

This is done very simply: a digit is searched for, which, together with the sum of the digits communicated to you, would be the nearest number divisible by 9 without a remainder. If, for example, in the number 828 the first digit (8) was crossed out and you were told the numbers 2 and 8, then by adding 2 + 8, you realize that up to the nearest number divisible by 9, i.e. up to 18, is not enough 8. This is the crossed out number.

Why is it so?

Because if we subtract the sum of its digits from any number, then there will remain a number that is divisible by 9 without a remainder, in other words, one whose sum of digits is divisible by 9. Indeed, let in the intended number a be the number of hundreds, and b be the number tens, s is the units digit. So in total in this number of units 100a + 10b + s. Subtracting from this number the sum of the digits (a+b+c), we get: 100a+10b+c-(a+b+c)=99a+9b=9(11a+c), a number divisible by 9. When performing the trick, it may happen that the sum of the numbers given to you is itself divisible by 9, for example 4 and 5. This shows that the crossed out number is either 0 or 9. Then you should answer: 0 or 9.

5. Focus "Who has what card?".

An assistant is needed to perform the trick.

There are three cards with ratings on the table: “3”, “4”, “5”. Three people come up to the table and each takes one of the cards and shows it to the magician's assistant. The magician, without looking, must guess who took what. The assistant tells him: “Guess” and the “magician” calls who has which card.

Focus clue:

Consider the possible options. Cards can be arranged as follows: 3, 4, 5 4, 3, 5 5, 3, 4

3, 5, 4 4, 5, 3 5, 4, 3

Since the assistant sees which card each person took, he will help the “magician”. To do this, you need to remember 6 signals. We number six cases:

First - 3, 4, 5

Second - 3, 5, 4

Third - 4, 3, 5

Fourth - 4, 5, 3

Fifth - 5, 3, 4

Sixth - 5, 4, 3

If the case is the first, then the assistant says: “Done!”

If the case is the second, then: “So, it’s ready!”

If the third case - then: "Guess!"

If the fourth - then: "So, guess!"

If the fifth - then: "Guess!"

If the sixth - then: "So, guess!".

Thus, if the option starts with the number 3, then “Done!”, If with the number 4, then “Guess!”, If with the number 5, then “Guess!”, And the students take the cards in turn.

6. Focus “Who took what?”

To perform this witty trick, you need to prepare some three small things that fit in your pocket, for example, a pencil, a key and an eraser, and a plate of 24 nuts. The magician invites three students to hide a pencil, key or eraser in their pocket during their absence, and he will guess who took what. The guessing procedure is carried out as follows. Returning to the room after things are hidden in the pockets, the magician hands them nuts from the plate to keep. Gives one nut to the first, two to the second, three to the third. Then he leaves the room again, leaving the following instruction: everyone should take more nuts from the plate, namely: the owner of the pencil takes as many nuts as he was given; the owner of the key takes twice as many nuts as he was given; the owner of the eraser takes four times the number of nuts that was handed to him. Other nuts remain on the plate. When all this is done, the “magician” enters the room, glances at the plate and announces who has what thing in his pocket. The key to the trick is as follows: each way of distributing things in pockets corresponds to a certain number of remaining nuts. Let's designate the names of the focus participants - Vladimir, Alexander and Svyatoslav. We also denote things with letters: a pencil - K, a key - KL, an eraser - L. How can three things be located between three participants? Six ways:

There can be no other cases. Let us now see what remainders correspond to each of these cases:

You see that the balance of nuts is different in all cases, therefore, knowing the remainder, it is easy to establish what is the distribution of things between the participants. The magician again - for the third time - leaves the room and looks there in his notebook with the last tablet (there is no need to memorize it). According to the plate, he determines who has what thing. For example, if there are 5 nuts left on the plate, then this means a case (KL, L, K), that is: Vladimir has the key, Alexander has the eraser, Svyatoslav has the pencil.

7. Focus “Favorite number”.

Any of those present conceives their favorite number. The magician invites him to multiply the number 15873 by his favorite number multiplied by 7. For example, if the favorite number is 5, then let him multiply by 35. You will get a work written only with your favorite number. The second option is also possible: multiply the number 12345679 by your favorite number multiplied by 9, in our case this is the number 45. The explanation for this trick is quite simple: if you multiply 15873 by 7, you get 111111, and if you multiply 12345679 by 9, you get 111111111.

8. Focus "Guess the intended number without asking anything."

The magician offers students the following actions:

The first student thinks of some two-digit number, the second one assigns the same number to it on the right and left, the third one divides the six-digit number received by 7, the fourth one by 3, the fifth one by 13, the sixth one by 37 and gives his answer to the thinker, who sees that his number has returned to him. The secret of the trick: if you assign the same number to the right and left of any two-digit number, then the two-digit number will increase by 10101 times. The number 10101 is equal to the product of the numbers 3, 7, 13 and 37, so after dividing we get the intended number.

9. Focus "Number in an envelope."

The magician writes the number 1089 on a piece of paper, puts the piece of paper in an envelope and seals it. Offers someone, giving him this envelope, to write on it a three-digit number such that the extreme digits in it are different and differ from each other by more than 1. Let him then swap the extreme digits and subtract the smaller one from the larger three-digit number . As a result, let him rearrange the extreme numbers again and add the resulting three-digit number to the difference of the first two. When he receives the amount, the magician invites him to open the envelope. There he will find a piece of paper with the number 1089, which he did.

10. Focus "Guessing the day, month and year of birth."

The magician asks students to do the following: “Multiply the number of the month in which you were born by 100, then add your birthday, multiply the result by 2, add 2 to the resulting number, multiply the result by 5, add 1 to the resulting number, attribute to the result 0, add another 1 to the resulting number, and finally add the number of your years. After that, tell me what number you got. Now the “magician” has to subtract 111 from the named number, and then split the remainder into three sides from right to left, two digits each. The middle two digits represent birthday , the first two or one - month number , and the last two digits are number of years , knowing the number of years, the magician determines the year of birth.

11. Focus "Guess the planned day of the week."

We number all the days of the week: Monday is the first, Tuesday is the second, etc. Let someone think of any day of the week. The magician offers him the following actions: multiply the number of the planned day by 2, add 5 to the product, multiply the resulting amount by 5, add 0 to the resulting number, and tell the magician the result. From this number he subtracts 250 and the number of hundreds will be the number of the planned day. The clue to the trick: let's say Thursday is conceived, that is, the 4th day. Let's perform the following actions: ((4*2+5)*5)*10=650, 650 - 250=400.

12. Focus "Guess the age."

The magician invites one of the students to multiply the number of his years by 10, then multiply any single-digit number by 9, subtract the second from the first product and report the resulting difference. In this number, the "magician" must add the number of units with the number of tens - the number of years will be obtained.

13. Focus "On the remainder of the division."

Invite the viewer to think of any number from 0 to 60. Ask them to divide this number by 3, then by 4, and finally by 5, and then name the remainder in order. This is quite enough to guess the intended number.
Secret of the trick: To guess the number, you need to multiply the first remainder by 40, the second by 45 and the third by 36. If you add up all the products and divide the sum by 60, then the remainder will be the intended number.
For example: the intended number is 10. After division, the remainders are 1, 2, 0. With them, you perform the following actions: 1 × 40 = 40,

2 × 45 = 90, 0 × 36 = 0, 40 + 90 + 0 = 130, 130: 60 = 2. Here, after dividing 130 by 60, the intended number 10 is obtained in the remainder.

14. Focus "Who is older?"

Tell two viewers that you can, without knowing their age, determine how much one is older than the other. Invite the youngest to subtract the number of his years from 99. And then let the elder add the number of his years to this difference and announce the result.
To determine the difference in age, you need to subtract 100 from the resulting number and add one to the result.
For example, the age of the younger viewer is 9 years old, and the older one is 14. Subtract 9 from 99 and get 90; 90 plus 14 equals 104. Subtract 100 from 104 and add one. We get 5 - this will be the age difference.

15. Focus "Six suitable numbers."
On six pieces of paper so that the audience cannot see, write six different numbers. Tell the audience that no matter what number they name from 1 to 60 now, you will add it from the numbers that are written on the sheets.
Whatever number the audience calls after that, lay out these or those sheets, and their sum will correspond to the named number, although adding sixty from six numbers seems like an impossible task.
Focus secret: In fact, the task is quite doable. On six sheets of paper you wrote the numbers: 1, 32, 4, 8, 16, 2. Whatever number from 1 to 60 the audience names now, it will be easy for you to lay out the required number. They called, for example, 51. Lay out sheets 32, 16, 1, 2, it will turn out 51. Or, for example, they will call 27: 1 + 8 + 16 + 2 = 27, etc.

16. Focus "Shifting cards."

Write numbers from 1 to 16 on 16 identical cards. Invite one of the spectators to guess one of the written numbers. Gather the cards in a pile face down, and then, opening the cards one at a time, stack them face up alternately in two piles. Ask the spectator who is thinking of the number what pile it is in.
Then lay the pile, which does not contain the intended number, on the pile indicated by the viewer, and, turning the resulting pile of 16 cards numbers down, arrange the cards again into two piles, as indicated above. This procedure with the decomposition of cards should be done only four times. After the fourth answer, you can easily find a card with the intended number.
Secret of the trick: The card with the intended number will be the bottom one in the stack of 8 cards last indicated. This is easy to understand if you imagine where the card with the intended number will fall each time the cards are laid out.
After the cards have been arranged into two piles for the first time, then again put into one pile, as indicated in the trick condition, the card with the intended number is among the eight lower cards. These eight cards will be distributed evenly between the two piles the next time they are laid out.
This means that after the cards are collected in one pile for the second time, the card with the intended number will be among the four lower cards. The third time, it will be among the two bottom cards, and, finally, after the fourth unfolding of the cards, the hidden card will be the bottom one in one of the piles.

17. Focus "Exact date".

Ask someone to think of an important date in their life, whether it's a birthday, a public holiday, or even a completely made-up day. Let's take March 25 as an example.
Without looking at the date, ask him to do the following operations on the calculator:
month number (January - 1st, December - 12th) = 3;
multiply by 5 = 15;
add 6 = 21;
multiply by 4 = 84;
add 9 = 93;
multiply by 5 = 465;
add day number = 490;
add 700 = 1190.
Ask what the calculator shows, then quickly subtract 865. The resulting number is the exact date: the last two digits are the day of the month, and the first number (or numbers) is the number of the month. In this case, 1190 - 865 = 325, that is, March (3rd month), 25th day.

18. Focus "All roads lead to zero."

The viewer thinks of a two-digit number, performs certain actions, and as a result, he gets zero.
Focus secret:
The viewer guesses any two-digit number. For example, 45. Then he must swap the numbers, it will turn out 54. The result obtained is recorded 4 times in a row. 54545454. The viewer removes the 1st and last digit of this number 454545. The resulting number is multiplied by 3. In this case, the answer is 1363635. The resulting number is divided by 7 (it turns out 194805). We divide this number by 9 (it turns out 21645). We divide the number by 13 (it turns out 1665). The resulting number is divided by the originally conceived (45) answer 37. Please note that 37 is always obtained for any originally conceived numbers. So, to get it, it remains to subtract 37 by any options.
This trick can surprise even strong mathematicians.

2. Conclusion.

Mathematical tricks are varied. In many mathematical tricks, numbers are veiled by objects related to numbers. They develop skills in rapid mental counting, calculation skills as viewers can guess both small and large numbers. Mathematical tricks with numbers are based on the ability to deal with numbers and the laws of exact science, while such tricks in no way diminish its importance.

Tricks using mathematics can not only entertain a person who is experienced in the exact sciences, but also attract attention and develop interest in the “queen of sciences” among those who are just getting to know her.

With our research work, we tried to prove to our viewers that mathematics is a very interesting and informative subject, and not dry and boring as it might seem at first glance.

After working with theoretical material and applying it in practice, we made the following conclusions:

1. Learning to unravel the secrets of mathematical tricks is quite simple, the main thing is to understand the essence of the ongoing mathematical transformations, and you can easily surprise others.

2. In order to effectively speak to the audience, you need to train attention, memory, as well as the ability to quickly and correctly count in your mind.

By studying tricks, you can learn to think rationally and look at the root. Arrange small performances at home, at school and with friends, and your life will become more interesting and brighter! A five-minute intellectual exercise in the form of a math trick can make math your favorite subject!

3. List of used literature.

1. Akopyan A.A. A big book of tricks and tricks from the repertoire of Harutyun and Hmayak Akopyan. -M.: Eksmo, 2008. -400s.
2. Vadimov A.A. The art of focus, M., 1959.
3. Gardner M. Mathematical miracles and secrets: mathematical tricks and puzzles / per. from English. V.S. Berman. – M.: Nauka, 1978. -128s.
4. Cowlan A. Focuses. Become a real wizard!/Translated from English. M. Polyakova. - M.: Egmont Russia Ltd., - 2007. -64p.
5. The best tricks and experiments. –M.:
6. Nagibin F.F., Kanin E.S. Math Box: A Student's Guide. - M.: Enlightenment, 1984. -160s.
7. Ozhegov S.I. Dictionary of the Russian language. - M.: Russian language, 1983. - 816s.
8. Samoilenko I. Amazing tricks and tricks. Mastery Secrets. Tricks and tricks for beginners. Wizard's Desk Book. - Rostov-on-Don: Vladis: M.: RIPOL classic, 2008. -416p.
9. Peter Eldin. Children's encyclopedia. Focuses. M.: Astrel, 2001. - 64s.
10. Chkanikov I. Games and entertainment. - M .: State. publishing house of children's literature, -1957. -512s.

Focus “Phenomenal memory”.

To carry out this trick, it is necessary to prepare many cards, on each of which put its number (two-digit number) and write down a seven-digit number according to a special algorithm. The “magician” distributes cards to the participants and announces that he has memorized the numbers written on each card. Any participant calls the number of the card, and the magician, after a little thought, says what number is written on this card. The solution to this trick is simple: in order to name the number, the “magician” does the following - adds the number 5 to the card number, flips the digits of the resulting two-digit number, then each next digit is obtained by adding the last two, if a two-digit number is obtained, then the unit digit is taken. For example: card number - 46. Add 5, get 51, rearrange the numbers - get 15, add the numbers, the next - 6, then 5 + 6 = 11, i.e. take 1, then 6 + 1 = 7, then the numbers 8, 5. The number on the card: 1561785.

Focus "Guess the intended number."

The magician invites one of the students to write any three-digit number on a piece of paper. Then add the same number to it again. Get a six-digit number. Pass the sheet to a neighbor, let him divide this number by 7. Pass the sheet further, let the next student divide the resulting number by 11. Pass the result further again, let the next student divide the resulting number by 13. Then pass the sheet to the “magician”. He can name a given number. Focus clue:

When we assigned the same number to a three-digit number, we thereby multiplied it by 1001, and then, dividing it sequentially by 7, 11, 13, we divided it by 1001, that is, we got the intended three-digit number.

Focus “Guess the crossed out number”.

Let someone think of some multi-digit number, for example, the number 847. Ask him to find the sum of the digits of this number (8+4+7=19) and subtract it from the intended number. It turns out: 847-19=828. including what happens, let him cross out the number - it doesn't matter which one, and tell you all the rest. You will immediately tell him the crossed out figure, although you do not know the intended number and did not see what was done with it.

This is done very simply: a digit is searched for, which, together with the sum of the digits communicated to you, would be the nearest number divisible by 9 without a remainder. If, for example, in the number 828 the first digit (8) was crossed out and you were told the numbers 2 and 8, then by adding 2 + 8, you realize that up to the nearest number divisible by 9, i.e. up to 18 - not enough 8. This is the crossed out number.

Why is it so?

Because if we subtract the sum of its digits from any number, then there will remain a number that is divisible by 9 without a remainder, in other words, one whose sum of digits is divisible by 9. Indeed, let in the intended number a be the number of hundreds, and b be the number tens, s is the units digit. So in total in this number of units 100a + 10b + s. Subtracting from this number the sum of the digits (a + b + c), we get: 100a + 10b + c- (a + b + c) \u003d 99a + 9b \u003d 9 (11a + c), i.e. a number divisible by 9 When performing a trick, it may happen that the sum of the numbers given to you is itself divisible by 9, for example 4 and 5. This shows that the crossed out number is either 0 or 9. Then you should answer: 0 or 9.

Focus "Favorite number".

Any of those present conceives their favorite number. The magician invites him to multiply the number 15873 by his favorite number multiplied by 7. For example, if the favorite number is 5, then let him multiply by 35. You will get a work written only with your favorite number. The second option is also possible: multiply the number 12345679 by your favorite number multiplied by 9, in our case this is the number 45. The explanation for this trick is quite simple: if you multiply 15873 by 7, you get 111111, and if you multiply 12345679 by 9, you get 111111111.

Focus "Guess the intended number without asking anything."

The magician offers students the following actions:

The first student thinks of some two-digit number, the second assigns the same number to the right and left, the third divides the six-digit number received by 7, the fourth by 3, the fifth by 13, the sixth by 37 and passes on his answer to the thinker, who sees that his number has returned to him. The secret of the trick: if you assign the same number to the right and left of any two-digit number, then the two-digit number will increase by 10101 times. The number 10101 is equal to the product of the numbers 3, 7, 13 and 37, so after dividing we get the intended number.

Fan Contest - “Merry Score”. A representative is invited from each team. There are two tables on the board, on which numbers from 1 to 25 are marked in disorder. At the signal of the leader, the students must find all the numbers on the table in order, whoever does this faster wins.

Focus “Number in an envelope”

The magician writes the number 1089 on a piece of paper, puts the piece of paper in an envelope and seals it. Offers someone, giving him this envelope, to write on it a three-digit number such that the extreme digits in it are different and differ from each other by more than 1. Let him then swap the extreme digits and subtract the smaller one from the larger three-digit number . As a result, let him rearrange the extreme numbers again and add the resulting three-digit number to the difference of the first two. When he receives the amount, the magician invites him to open the envelope. There he will find a piece of paper with the number 1089, which he did.

Focus “Guessing the day, month and year of birth”

The magician asks students to do the following: “Multiply the number of the month in which you were born by 100, then add your birthday, multiply the result by 2, add 2 to the resulting number, multiply the result by 5, add 1 to the resulting number, attribute to the result 0, add another 1 to the resulting number, and finally add the number of your years. After that, tell me what number you got. Now the “magician” has to subtract 111 from the named number, and then split the remainder into three sides from right to left, two digits each. The middle two digits indicate the birthday, the first two or one - the number of the month, and the last two digits - the number of years, knowing the number of years, the magician determines the year of birth.

Focus “Guess the planned day of the week”.

We number all the days of the week: Monday is the first, Tuesday is the second, etc. Let someone think of any day of the week. The magician offers him the following actions: multiply the number of the planned day by 2, add 5 to the product, multiply the resulting amount by 5, add 0 to the resulting number, and tell the magician the result. From this number he subtracts 250 and the number of hundreds will be the number of the planned day. The clue to the trick: let's say Thursday is conceived, that is, the 4th day. Let's do the following: ((4×2+5)*5)*10=650, 650 - 250=400.

Focus “Guess the age”.

The magician invites one of the students to multiply the number of his years by 10, then multiply any single-digit number by 9, subtract the second from the first product and report the resulting difference. In this number, the "magician" must add the number of units with the number of tens - the number of years will be obtained.