Can the scientist cat equality be true? Mathematical puzzles. Mathematical puzzles for tutor work

The scientist proved the equality of the classes P and NP, for the solution of which the Clay Mathematical Institute awarded a prize of one million US dollars.

Anatoly Vasilyevich Panyukov spent about 30 years searching for a solution to one of the most difficult problems of the millennium. Mathematicians all over the world have been trying for many years to prove or disprove the existence of the equality of the classes P and NP; there are about a hundred solutions, but none of them have yet been recognized. On this topic related to this problem, the head of the SUSU department defended his candidate and doctoral dissertations, but, as it seems to him, he only found the correct answer now.

The problem with the equality P = NP is this: if the positive answer to a question can be quickly verified (in polynomial time), then is it true that the answer to this question can be quickly found (in polynomial time and using polynomial memory)? In other words, is it really not easier to check the solution to a problem than to find it?
For example, is it true that among the numbers (−2, −3, 15, 14, 7, −10, ...) there are some such that their sum is 0 (problem on sums of subsets)? The answer is yes, because −2 −3 + 15 −10 = 0 can easily be verified with a few additions (the information needed to verify a positive answer is called a certificate). Does it follow that it is just as easy to pick up these numbers? Is checking a certificate as easy as finding it? It seems that the numbers are more difficult to come by, but this has not been proven.
The relationship between the classes P and NP is considered in computational complexity theory (a branch of computational theory), which studies the resources required to solve some problem. The most common resources are time (how many steps you need to take) and memory (how much memory you need to solve the problem).

“I discussed the results of my work at a number of inter-district conferences and among professionals. The results were presented at the Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences and in the journal “Automation and Mechanics”, published by the Russian Academy of Sciences, Doctor of Physical and Mathematical Sciences Anatoly Panyukov told Good News. – The longer professionals cannot find a refutation, the more correct the result is considered.

The equality of classes P and NP in the mathematical world is considered one of the pressing problems of the millennium. And the point is that if the equality is true, then most of the current optimization problems can be solved in an acceptable time, for example, in business or production. Nowadays, the exact solution of such problems is based on brute force, and can take more than a year.

“Most scientists are inclined to the hypothesis that the classes P and NP do not coincide, but if there is no error in the evidence presented, then this is not so,” noted Anatoly Panyukov.

If the Chelyabinsk scientist’s proof turns out to be correct, it will greatly influence the development of mathematics, economics and technical sciences. Optimization problems in business will be solved more accurately, hence there will be more profits and fewer costs for a company that uses special software to solve such problems.

The next step to recognize the work of the Chelyabinsk scientist will be the publication of the proof at the Clay Mathematical Institute, which announced a million-dollar prize for solving each of the millennium problems.

Currently, only one of the seven millennium problems (Poincaré's conjecture) has been solved. The Fields Medal for its solution was awarded to Grigory Perelman, who refused it.

For reference: Anatoly Vasilievich Panyukov (born in 1951) Doctor of Physical and Mathematical Sciences, Professor, Head of the Department of Economic and Mathematical Methods and Statistics at the Faculty of Computational Mathematics and Informatics, Member of the Association of Mathematical Programming, Scientific Secretary of the Scientific and Methodological Council for Mathematics Ministry of Education and Science of the Russian Federation (Chelyabinsk branch), member of the Scientific and Methodological Council of the Territorial Body of the Federal State Statistics Service for the Chelyabinsk Region, member of dissertation councils at South Ural and Perm State Universities. Author of more than 200 scientific and educational publications and more than 20 inventions. Head of the scientific seminar “Probative computing in economics, technology, natural sciences”, the work of which was supported by grants from the Russian Foundation for Basic Research, the Ministry of Education and the International Science and Technology Center. He trained seven candidates and two doctors of science. He has the titles “Honored Worker of Higher School of the Russian Federation” (2007), “Honored Worker of Higher Professional Education” (2001), “Inventor of the USSR” (1979), awarded a medal of the USSR Ministry of Higher Education (1979) and a Certificate of Honor from the Governor of the Chelyabinsk Region.

Ten days ago, Indian mathematician Vinay Deolalikar posted an article online in which, according to him, he proved one of the most important inequalities in mathematics - the inequality of complexity classes P and NP. This message caused an unprecedented resonance among Deolalikar’s colleagues - scientists abandoned their main work and began to read and discuss the article en masse. Almost immediately, experts discovered flaws in the proof, and a week later the mathematical community came to the conclusion that Deolalikar had failed to cope with the task.

Application for a million

The problem of the inequality of classes P and NP is one of the most intriguing in mathematics, even though most specialists are already confident that they are not equal (all scientists admit that until the basis of confidence is not based on a strict evidentiary foundation, it will remain in the field of intuition, not science). The significance of this problem, which the Clay Institute of Mathematics included in its list of the Seven Millennium Challenges, is enormous and extends not only to “speculative” mathematics, but also to computer science and computational theory.

Briefly, the problem of inequality of complexity classes P and NP is formulated as follows: “If a positive answer to a certain question can be quickly verified, then is it true that the answer to this question can be quickly found.” Problems for which this problem is relevant belong to the NP complexity class (problems of the P complexity class can be called simpler - in the sense that their solution can definitely be found in a reasonable time).

One example of problems of NP complexity class is breaking a cipher. Currently, the only way to solve this problem is to try all possible combinations. This process can take an incredibly long time. But when the correct code is found, the attacker will instantly understand that the problem has been solved (that is, the solution can be verified in a reasonable time). In the event that the complexity classes P and NP are still not equal (that is, problems whose solution cannot be found in a reasonable time cannot be reduced to simpler problems that can be solved quickly), then all criminals in the world will always have to break ciphers brute force. But if it suddenly turns out that inequality is actually equality (that is, complex problems of class NP can be reduced to simpler problems of class P), then brainy thieves will theoretically be able to come up with a more convenient algorithm that will allow them to crack any ciphers much faster.

Simplifying greatly, we can say that a rigorous proof of the inequality of the complexity classes P and NP will finally and irrevocably deprive humanity of the hope of solving complex problems (problems of the NP complexity class) otherwise than by a stupid search of all feasible solution options.

As always happens with problems of particular importance, attempts are made regularly to prove rigorously that the classes P and NP are equal or unequal. Typically, applications for solving the Millennium Challenge are made by people whose reputation in the scientific world is, to put it mildly, questionable, or even by amateurs who do not have a special education, but are fascinated by the scale of the challenge. None of the truly recognized specialists takes such work seriously, just as physicists do not take seriously the periodic attempts to prove that the general theory of relativity or Newton’s laws are fundamentally wrong.

But in this case, the author of the work, simply titled “P is not equal to NP,” was not a pseudo-scientific madman, but a working scientist, and working in a very respected place - Hewlett-Packard Research Laboratories in Palo Alto. Moreover, one of the authors of the Millennium Problem on the P and NP Inequality, Stephen Cook, gave a positive review of his article. In a cover letter that Cook sent to colleagues along with the paper (Cook was one of several leading mathematicians to whom the Indian sent his work for review), he wrote that Deolalikar's work was "a relatively serious bid to prove the inequality of the classes P and NP."

It is not known whether the recommendation of a luminary in the field of complexity theory (it is this area of mathematics that deals with the inequality P and NP) played a role, or the importance of the problem itself, but many mathematicians from different countries turned away from their main work and began to understand Deolalikar’s calculations . People who know about the inequality of complexity classes P and NP, but are not directly involved in this topic, also took an active part in the discussion. For example, they bombarded computer scientist Scott Aaronson of the Massachusetts Institute of Technology (MIT) with questions about the proof.

Aaronson was on vacation at the time Deolalikar's article appeared and could not immediately understand the evidence. However, to emphasize its importance, he stated that he would give the Indian $200,000 if the mathematical community and the Clay Institute found him correct. For this extravagant act, many colleagues condemned Aaronson, saying that a true scientist should rely only on facts, and not shock the public with beautiful gestures.

Shoals

Already in the first days of “sucking up” Deolalikar’s article, experts discovered several serious shortcomings in it. One of the first to publicly declare this was, oddly enough (or, conversely, not at all strange), it was Aaronson. In response to criticism from readers of his blog for publishing hasty conclusions, Aaronson shared several techniques he used to quickly assess the Indian's performance.

Aaronson, firstly, did not like the fact that Deolalikar did not present his paper in the classical lemma-theorem-proof structure for mathematicians. The scientist explains that this quibble is not caused by his innate conservatism, but by the fact that with this structure of work it is easier to catch “fleas”. Secondly, Aaronson noted that the summary of the paper, which should explain what the essence of the proof is and how the author managed to overcome the difficulties that have prevented the problem from being solved until now, is written extremely vaguely. Finally, the main point that confused Aaronson was the absence in Deolalikar's proof of an explanation of how it could be applied to the solution of some important particular problems associated with complexity theory.

A few days later, Neil Immerman of the University of Massachusetts said he had discovered a "very serious gap" in the Indian's work. Immerman's thoughts were published on the blog of University of Georgia computer scientist Richard Lipton, where the main discussion about the P and NP inequality took place. The scientist appealed to the fact that Deolalikar incorrectly defined problems that fall into the complexity class NP, but not P, and therefore all his other arguments are also invalid.

Immerman's conclusions forced even the most loyal experts to change their assessment of the Indian's work from "it is possible that yes" to "almost definitely no." Moreover, mathematicians even doubted that Deolalikar's work could yield significant insights that could be useful in further attempts to understand inequality. The verdict of the mathematical community (in English and with an abundance of mathematical terms) can be read.

Deolalikar himself responded to the criticism of his colleagues that he would try to take into account all the comments in the final version of the article, which will be prepared in the near future (since August 6, when the Indian sent out the first version of his work, he has already made changes to it once). If the mathematician’s assurances turn out to be true and the final version of the proof does see the light of day, one must think that experts will once again study the arguments presented by Deolalikar. But today the scientific community has already decided on its assessment.

New stage?

Even if we ignore the importance of the Millennium Challenges themselves, there is another interesting side to this story. The colossal scope of the discussion of Deolalikar's work is in itself an absolutely amazing event. Hundreds of mathematicians and computer scientists dropped everything they were doing and concentrated on studying the more than 100-page ( sic!) Indian labor. Judging by the speed with which scientists discovered errors, they must have spent many hours of their free - and perhaps even working - time diligently reading the article "P is not equal to NP". On one of the Wikipedia-like sites, a page was urgently created where everyone could express their thoughts on the evidence provided.

All this frantic activity suggests that through Deolalikar's work we are witnessing the birth of a new way of writing scientific papers. Making preprints available to the public before official publication has been practiced in the exact and natural sciences for a long time, but in this case, a new result - albeit negative - was the result of a brainstorming session conducted by dozens of specialists from around the world.

Of course, this method of obtaining scientific data still raises many questions (the most obvious is the question of the authorship of the results and the priority of discoveries), but, in the end, most new undertakings initially faced doubts and opposition. The survival of such undertakings is determined not by the attitude of society, but by the extent to which they are in demand. And if brainstorming and obtaining results is more effective than traditional methods of scientific work, then it may very well be that in the future such a practice will become generally accepted.

6th grade club

Head Evgeniy Aleksandrovich Astashov
2012/2013 academic year

Lesson 1. Problems for getting to know each other

Teachers have collected written work and are counting them before checking. Irina Sergeevna stacked them in stacks of one hundred works. Daniil Alekseevich can count five works in two seconds. In what shortest time can he count out 75 papers for checking? a) Offer a set of three weights, each of which weighs an integer number of grams, so that with their help on a cup scale without divisions one can weigh any integer weight from 1 to 7 grams. b) Would a set of some two weights (not necessarily with integer masses) be sufficient for this purpose?

Solution. Those interested only in mathematics are four times more likely to be interested in both subjects; those interested only in biology are three times more likely to be interested in both subjects. This means that the number of those who are interested in at least one of the two subjects should be divided by 8 (all of them together are 8 times more than those interested in both subjects). 8 and 16 are not enough, since 16 + 2 = 18< 20 (не забудем посчитать Олега и Пашу); 32, 40 и т.д. — много; 24 подходит. Итак, в классе 24 человека, которые интересуются математикой или биологией (а может быть, и тем, и другим), а ещё есть Олег и Паша. Таким обраом, всего в классе 24 + 2 = 26 человек.

The method to cut off all the heads and tails of the Snake in 9 blows is given in the answer. Now we will prove that this cannot be done in fewer strokes.

Ivan Tsarevich can use three types of attacks:
A) cut off two tails, one head will grow;
B) cut off two heads;
C) cut off one tail, two tails will grow (in fact, just add one tail).
It is useless to chop off one head, so we will not use such blows.

1. The number of type A strikes must be odd. In fact, only with such shots does the parity of the number of goals change. And the parity of the number of goals should change: at first there were 3 of them, and at the end there should be 0. If an even number of such shots are made, the number of goals will remain odd (and therefore will not be equal to zero).
2. Since only type A blows can reduce the number of tails, one such blow will not be enough. Therefore, there should be at least two such strikes, and taking into account the previous point, there should be at least three.
3. After three Type A hits, three new heads will grow, and a total of 6 heads will need to be cut off. This will require at least 3 Type B hits.
4. To chop off two tails 3 times with type A blows, you need to have 6 tails. To do this, you need to “grow” three additional tails by making 3 type C hits.
So, you need to make at least three strikes of each of the indicated types; in total - at least 9 blows.

Every student in our schools studies mathematics. Most of them find the subject difficult, which is true. Teachers and parents do a lot to ensure that students do not give up when overcoming learning difficulties and are not passive in the classroom... but the problems that arise in this process do not decrease. Therefore, it is necessary to develop interest in mathematics, using even the slightest inclinations of the student. For this purpose, we have made a selection of competitions that can be used to a greater extent in extracurricular work in mathematics (mathematics weeks, KVNs, evenings, etc.), but creatively working teachers find a place for some of them in the classroom.

< Рисунок 1> .

I. AUNCION

a) Auction of proverbs and sayings with numbers.

By drawing lots, the first team to name the proverb is determined; after the leader hits the hammer, a member of the second team names the proverb, etc. The last person to name the proverb wins.

Note that you can limit yourself to a specific number. Name proverbs and sayings where the word seven appears. For example: “Measure seven times, cut once”, “Seven do not wait for one”, “Seven nannies have a child without an eye”, “One with a fry, seven with a spoon”, “Seven troubles - one answer”, “Behind seven locks” ”, “Seven Fridays a week”, etc.

b) Auction of films with a number in the title.

c) Auction of songs that have a number.

It is enough to name the line with this number or sing it.

d) Auction charades.

Charade is a special riddle. You have to guess the word in it, but in parts. You can alternate between charades that have a mathematical element and those that don’t.

The first is a round object,
The second is something that does not exist in this world,
But what scares people?
Third - union. (Answer: charade).

To the name of the animal
Put one of the measures.
You'll get a full
A river in the former USSR. (Answer: Volga).

You will find the first syllable among the notes,
And the bull carries the second one.
So look for him along the way,
Do you want to find the whole thing? (Answer: road).

You suddenly insert a note behind the measure

And you will find everything among your friends. (Answer: Galya).

e) Auction on a given topic. Assignments on any topic that are communicated to students in advance are put up for auction. Let, for example, the topic be “Actions with algebraic fractions.”

4-5 teams participate in the competition. Lot No. 1 is projected onto the screen - five tasks for reducing fractions. The first team selects a task and assigns a price from 1 to 5 points. If the price of this team is higher than what others give, it receives this task and completes it, the remaining tasks must be bought by other teams. If the task is solved correctly, the team is awarded points - the price of this task; if incorrect, then these points (or part of them) are removed. Pay attention to one of the advantages of this competition: when choosing an example, students compare all five examples and mentally “scroll” in their heads the process of solving them.

II. CHAIN OF WORDS

The presenter says one word. The first captain (if this happens at KVN) repeats this word and adds his own. The second captain repeats the first two words and adds his own, and so on. One of the judges watches the game, writing down the words in order. The one who can name the most words to create a complete sentence wins.

A). Triangles are equilateral if all angles are equal or all sides are equal.

b). However, there are isosceles ones, which means that the angles at the base are then forty-five degrees.

III. EACH HAND HAS ITS BUSINESS

The players are given a sheet of paper and a pencil in each hand. Assignment: draw 3 triangles with your left hand and 3 circles with your right hand; or the left one writes even numbers (0, 2, 4, 6, 8), the right one writes odd numbers (1, 3, 5, 7, 9).

IV. STEP – THINK

Participants in this competition stand next to the presenter. Everyone takes their first steps, at which time the leader names a number, for example 7. During the next steps, the guys must name numbers that are multiples of 7: 14, 21, 28, etc. For each step - a number. The leader keeps pace with them, not allowing them to slow down. Once someone makes a mistake, he remains in place until the end of the other's movement. Other topics: multiplication table review; raising numbers to powers; square root extraction; finding a part of a number.

V. YOU – TO ME, I – TO YOU

< Рисунок 2>

The essence of the competition is clear from the name. Here is an example of problems that captains exchanged at KVN.

1. The wolf solved the example: 4872? 895 = 4360340 and started checking by division. The hare looked at this equation and said: “Don’t do extra work! And it’s clear that you were mistaken.” The wolf was surprised: “How do you see this?” What did the hare answer?

(Answer: one of the factors is a multiple of three, but the product is not).

2. In September, Petya and Styopa went to music lessons: Petya - in numbers divisible by 4, and Styopa - in numbers divisible by 5. Both went to the sports section in numbers divisible by 7. The rest of the days were spent fishing. How many days did the guys spend fishing?

(Answer: 15).

3. “What time is it?” - the Wolf asks the Hare. “The given time is a multiple of 5, and the time of day in hours is a multiple of the given,” answered the Hare. “This can’t happen!” - the Wolf was indignant. And what do you think?

(Answer: 15).

4. Vova claimed that this year there will be a month with five Sundays and five Wednesdays. Is he right?

Solution. Let's consider the most favorable case, when there are 31 days in a month.

31 = 4 * 7 + 3 and among three consecutive days of the week cannot be both Sunday and Wednesday, but only one of these days, then this month can have either 5 Sundays and 4 Wednesdays, or 4 Sundays and 5 Wednesdays. Therefore, Vova is wrong.

5. Three boxes contain cereal, vermicelli and sugar. On one of them it is written “Grains”, on the other – “Vermicelli”, on the third – “Grains or sugar”. Which box contains what if the contents of each box do not match the label?

(Answer. In the box with the inscription “Grains or sugar” there is vermicelli, with the inscription “Vermicelli” - cereals, with the inscription “Grains” - sugar).

6. The picture shows the houses in which Igor, Pavlik, Andrey and Gleb live. Igor's house and Pavlik's house are the same color, Pavlik's house and Andrey's house are the same height. Who's in which house< Рисунок 3>

VI. RACE FOR THE LEADER

< Рисунок 4>

So that the guys leave the event not upset by defeat, you can hold this competition and try to make a draw. Due to the current situation, by this time, answers to the tasks proposed below can be given by team members or their fans.

What an acrobat figure!
If it gets on your head,
It will be exactly three less. (Answer: number 9).

I am a number less than 10.
It's easy for you to find me
But if you command the letter “I”
Stand next to me, - I am everything!
Father and grandfather, and you and mother. (Answer: family).

I am an arithmetic sign
In the problem book you will find me in many lines,
Only “o” you insert, knowing how,
And I am a geographical point. (Answer: plus-pole.)

Zero turned his back on his brother,
He climbed up slowly.
Brothers have become a new number,
We will not find the end of it.
You can turn it around
Place your head down.
The number will still be the same
Well, think?
Say so! (Answer: number 8).

He turned tens into hundreds,
Or it can turn into millions.
He is equal among numbers,
But it cannot be divided into. (Answer: number 0).

Note that the assignments are not given in the form of problems, as in the competition “You are for me, and I am for you,” but in poetry for a reason. Before this competition, the guys had already worked hard. We need to try to change the intensity of passions, to capture the attention of the majority, which may have already dissipated. And a poem that appears, for example, on a portable board, prepared in advance, can help with this. If the question posed there is correctly answered (task 5), the presenters present this answer with a colorful drawing something like this:

< Рисунок 5>

Another possible approach is to use team artists. Based on the model, they will quickly make drawings on the board. You can easily find them from various sources. For example, see the list of references.

VII. A DARK HORSE

< Рисунок 6>

For this competition, we selected tasks in which it is necessary to find out whether an answer to the question posed is possible.

1. Multiply both sides of the inequality 9>5 by a 4. Can we say that the inequality 9a 4 >5a 4 is true?

(Answer: no. For a=0 we get 9a 4 =5a 4 since 0=0).

2. Can equality be true?

(Answer: yes, it can. For example, when x=y=1).

3. Is it possible to cut a triangle to make three quadrilaterals? (Answer: yes).

For example:

< Рисунок 7>

4. Having drawn 2 straight lines, is it possible to divide the triangle into a) two triangles and one quadrilateral, b) two triangles, two quadrilaterals and one pentagon.

A)< рисунок 8>

b)< рисунок 9>

VIII. PORTRAIT COMPETITION

The team is shown a portrait of a mathematician. You need to say his last name. You can make the competition more difficult by asking to name your area of activity.

IX. ERUDITE COMPETITION

a) An erudite participant of one team names the last name of a mathematician, and the other one names a mathematician whose last name begins with the last letter of the first scientist, etc.

Or the erudite of the second team names the surname of a mathematician, starting with any letter in the surname of the first scientist, etc.

b) Two students each participate in the erudite competition: A and B.

Questions are asked to each participant in the struggle for the title of erudite.

A. 5 2 =?; 7 2 =?, and what is the angle in a square? (Answer: 25; 49; 90 0).

B. Seven sparrows were sitting in the garden bed. A cat crept up to them and grabbed one. How many sparrows are left in the garden? (Answer: one).

A. What did the word “mathematics” originally mean? (Answer: knowledge, science).

B. What word does the name zero come from? (Answer: from the Latin word “nulla” - empty).

A. Calculate:(-2)? (-1)…3=? (Answer: 0.)

B. Calculate: (-3)+(-2)+…+3+4=? (Answer: 4.)

A; B. Name the ancient Russian measures of length one by one. (Answer: fathom, span, quarter...)

X. HISTORIAN COMPETITION

You need to tell an interesting story from the life of a famous mathematician, or highlight the essence of a fact, clearly presented in the form of a skit. Example: An old man bent over a drawing, and behind him was a warrior with a dagger.

Legend. It was only because of treason that Syracuse was taken by the Romans. “At that hour, Archimedes carefully examined some drawing and did not notice either the Roman invasion or the capture of the city. When suddenly a warrior stood in front of him and announced that Marcellus was calling him, Archimedes refused to follow him until he completed the task and found the proof. The warrior got angry, pulled out his sword and killed Archimedes.”

Archimedes was born in 287 BC. in the city of Syracuse on the island of Sicily, part of what is now Italy. Archimedes began to be interested in mathematics, astronomy, and mechanics at an early age. Archimedes' ideas were almost 2 millennia ahead of their time. Archimedes died during the capture of Syracuse in 212 BC.

XI. KNOW-ALL COMPETITION

Participants in this competition provide answers to the following questions:

a) about mathematicians;

b) about terms;

c) about formulas;

d) solve crosswords and puzzles.

Example of a rebus:

< Рисунок 10>

(Answer: fraction).

To prepare students and conduct competitions for scholars, historians, and know-it-alls, it is useful to adopt an encyclopedia for children. She will answer all your questions. You will find about two hundred mathematicians in the “Index of Names” section, where there are links to the pages of this book: what important things they have done.

Literature

Alexandrova E.B. Traveling around Karlikania and Al-Jebra / E.B. Alesandrova, V.A. Levshin. – M.: Children's literature, 1967. – 256 p.
Gritsaenko, N.P. Well, decide!: book. for students / N.P. Gritsaenko. – M: Education, 1998. – 192 p.
Lanina I.Ya. Not just a lesson: Developing interest in physics. - M.: Education, 1991.-223 p.
Mirakova T.N. Developmental tasks in mathematics lessons in grades V-VIII: a manual for teachers.
Petrovskaya N.A. Evening of the cheerful and savvy in the fourth grade / “Mathematics at school.” - 1988. - No. 3. - P. 56.
Samoilik G. Educational games.-2002.-No. 24.
Encyclopedia for children. T.11. Mathematics / Ch. ed. M.D. Aksenova. – M.: Avanta +, 2002. – 688 p.

On this page I post puzzles intended for Olympiad classes in grades 5-6. If your math tutor has given you an original puzzle and you don’t know how to solve it, send it to me by email or leave a corresponding entry in the feedback box. It may be useful to other mathematics tutors, as well as teachers of clubs and electives. I look through Olympiad problems on different sites, sorting them into classes and difficulty levels for posting on the site. This page contains a collection of entertaining puzzles collected over the years of tutoring. The page will gradually fill up. The wording of the tasks is standard. The same letters represent the same numbers, and different letters represent different numbers. You need to restore the records in accordance with this order. I use puzzles when preparing for the Kurchatov school in the 4th grade, also to awaken a love for mathematics.

Mathematical puzzles for tutor work

1)Number multiplication puzzle with repeating letters A, B, and C Identical letters in the multiplication example must be replaced with identical numbers.

2) Rebus mathematics Replace the same letters in the word “mathematics” with the same numbers so that all five actions received have equal answers.

3) Rebus Chai-Ai. Indicate some solution to the rebus (according to tradition, identical letters hide identical numbers, and different ones hide different ones).

4) Mathematical puzzle "scientist cat". Can the indicated equality become true if instead of its letters we put the numbers from 0 to 9? Different to different, same to same.

math tutor's note: The letter O does not have to correspond to the number O.

5) An interesting rebus was offered to my student at the last Internet Olympiad in mathematics for 4th grade.