Worst-case proof problems. Proof problems based on the Dirichlet principle


In a work that tells about events peasant war 1773-1774, Pushkin managed to harmoniously conduct love line. The image and characterization of Masha Mironova in “The Captain's Daughter” will prove to the reader that love can inspire under any circumstances. At the very scary times when there is danger everywhere, death of loved ones, fear for own life, mutual feelings will help overcome this.

Acquaintance. Will Shvabrin's words be confirmed?

At the first meeting, Peter did not yet understand what the commandant’s daughter really was like. Shvabrin described Masha as “a complete fool,” not with the best side. The eighteen-year-old young lady is very silent.

“Chubby, with light brown, smoothly combed hair.”

She behaves too modestly, rarely enters into conversation. So on the first day of meeting new residents,

“The girl sat in the corner, did not carry on the conversation, but started sewing.”

About marriage and respect for parents

Vasilisa Egorovna says that it’s time for her daughter to get married.

“What is her dowry? A comb, a broom, and an altyn of money.”

Maria became embarrassed, lowered her head, and tears flowed from her eyes. This indicates excessive modesty and obedience. She did not argue with her mother, did not contradict her, was not indignant. At that moment, Grinev looked at the Mironovs’ daughter with great respect.

Loyalty to sincere feelings

Masha will tell Peter that Shvabrin called her to be his wife. Having been refused, the arrogant officer harbored a grudge. She was not attracted by gifts, despite her parents' poverty. The girl does not have prudence. She cannot imagine how you can kiss a person under the aisle without having reciprocity towards him. She loves Peter sincerely and is ready to do anything for him.

Masha did not leave Petya’s side when he lay delirious after being wounded in a duel. She cared for the patient with all her might. When Grinev came to his senses and began to talk, she asked him to take care of himself.

“You should save yourself for me.”

Her actions and similar words prove how much she values ​​a person.

Respect for Grinev leads to the desire to receive a blessing for marriage from the beloved’s relatives. When the boy’s father sent a letter of refusal, the girl did not resist. She respects other people’s opinions and will not go against the will of Peter’s loved ones, to the detriment of her feelings. This may characterize her as a weak person, unable to defend herself. Education and respect for elders do not allow one to resist circumstances in this situation. In others life situations The girl will still show strength of character.

Mary's courage, loyalty to moral principles

When Shvabrin, having gone over to the side of the rebel Pugachev, holds Masha captive in the fortress, she will not submit to him, and will not be afraid to give Peter a letter asking for help. In such a dangerous situation, when her life is in danger of death, she will take risks. Without a drop of fear, Marya will tell Pugachev that she will not become Shvabrin’s wife.

“I will never be his wife! It’s better to decide to die.”


Your boundless love and devotion to the commandant's daughter Belogorsk fortress He will show it when he leaves for St. Petersburg to receive the Tsarina to ask her to pardon her beloved. The girl’s honesty and openness will so amaze the empress that she will fulfill her request. Soon Maria will become the wife of Peter Grinev. They will have children. They will live in Simbirsk province.

Respect and love of loved ones

In his diary of memories, the younger Grinev writes that his beloved was

Municipal educational institution

Beloyarskaya average comprehensive school

Literature section

Maria Sudakova Vladimirovna

Head: Luzanova Elena Valentinovna

teacher of Russian language and literature

Bely Yar, 2010

code___________________

Literature section

The image of Masha Mironova in the story by A.S. Pushkin “ Captain's daughter»

introduction

1. The image of the captain's daughter

2. The character of Masha Mironova

3. Evolution of the image of Masha Mironova

conclusion

Bibliography

Introduction. About Pushkin's story "The Captain's Daughter"

Historical works of fiction are one of the ways to master the specific historical content of a particular era. All sorts of things historical work informative. But the main purpose of historical prose is not so much a reconstruction of the past, but an attempt to connect the past and the present, to “embrace” the movement of history, and to look into the future.

Our work is relevant, because interest in Pushkin’s work has not waned for more than two hundred years, and each time researchers find new sources for the creation of one or another literary image.

Writers different eras By various reasons turned to the past. For example, romantics who did not find an ideal in the present looked for it in the past. Realist writers in the past tried to find answers to modern questions. And this method of searching for truth remains relevant to this day. Modern man I am still concerned about philosophical problems: what is good and evil?, how does the past affect the future?, what is the meaning human life? Therefore, the appeal of the modern reader to historical prose naturally.

A work that arouses not only interest in a certain historical era, but also love for the work of A.S. Pushkin in general is his novel “The Captain's Daughter”, where the main historical event is the uprising of Emelyan Pugachev.

The idea of ​​a historical story from Pugachev uprising arose from Pushkin under the influence social situation early 1830s But why did he call it that? famous writer your story? After all, the story is based on historical facts, and, according to many researchers, occupies central place development of relations between Grinev and Pugachev, a nobleman and a peasant king. Throughout the story, the path of development of P.A. is shown. Grineva. We see how it changes main character, there is an opening inner man in man. But what or who influences these changes inner world hero? Undoubtedly this is historical events and the first sincere love awakened by a simple girl, the captain's daughter. Who is she? Who is this captain's daughter? And here we would like to dwell in more detail on the image of Masha Mironova.

Goal of the work: trace all the changes that occurred with Masha Mironova, explain their reason.

Job Objectives: 1. Refer to the content of A.S. Pushkin’s story “The Captain’s Daughter”, and specifically, to the image of Masha Mironova.

2. Study critics’ reviews of Misha Mironova as a literary heroine.

This topic has not been sufficiently studied in the critical literature, which is why the idea arose to develop this topic.

The research material was the story by A.S. Pushkin “The Captain's Daughter”

We assume that the image of Masha Mironova has undergone significant changes throughout the story.

2. The image of the captain's daughter.

Pushkin uses laconicism when depicting main character. “Then a girl of about eighteen came in, chubby, ruddy, with light brown hair, smoothly combed behind her ears, which were on fire,” is how Pushkin describes the daughter of Captain Mironov. If you think about it, she wasn't a beauty, but she wasn't ugly either. We can note that the heroine is shy, modest, blushes every minute and is always silent. We can say that Masha “at first I don’t like” and “doesn’t make any impression” on Grinev. But one cannot judge by first impressions, especially since Grinev’s opinion about Masha soon changes. “Marya Ivanovna soon stopped being shy with me. We met. I found in it prudent And sensitive girl,” we read from Pushkin. What do the highlighted words mean? “Prudence is prudence, thoughtfulness in actions. Sensitive - highly susceptible to external influences“, we read in Ozhegov’s dictionary.

The reader guesses that some kind of feeling is awakening in Grinev’s soul... And only in Chapter 5 does Pushkin openly name this feeling to us - love. Let us pay attention to Masha’s care towards Grinev during her illness after the fight with Shvabrin. The simplicity and integrity of her feelings, the naturalness of its manifestation remain unnoticed, and for modern young people they are not understandable: after all, Masha and Grinev are connected only spiritual connection. During his illness, Grinev realizes that he loves Masha and proposes marriage. But the girl does not promise him anything, but chastely makes it clear that she also loves Pyotr Andreevich. As you know, Grinev’s parents do not give consent to their son’s marriage to the captain’s daughter, and Marya Ivanovna refuses to marry Grinev, sacrificing her love for the sake of her beloved. According to researcher A.S. Degozhskaya, the heroine of the story was “raised in patriarchal conditions: in the old days, marriage without parental consent was considered a sin.” The daughter of Captain Mironov knows “that Pyotr Grinev’s father is a man of tough character,” and he will not forgive his son for marrying against his will. Masha does not want to hurt her loved one, interfere with his happiness and harmony with his parents. This is how the strength of her character and sacrifice are demonstrated. We have no doubt that it’s hard for Masha, but for the sake of her beloved she is ready to give up her happiness.

2. The character of Masha Mironova

After hostilities and the death of her parents, Masha is left alone in the Belogorsk fortress. This is where the firmness, decisiveness of character, and the inflexibility of her will are revealed to us. The villain Shvabrin puts the girl in a punishment cell, not letting anyone in to see the prisoner, giving her only bread and water. All these tortures were necessary to obtain consent to marriage, since Marya Ivanovna did not agree voluntarily. There was and is only one person in her heart - Grinev. And in days of trials, in days of losing hope of uniting with Petrusha and in the face of danger, and perhaps even death itself, Marya Ivanovna retains her presence of mind and unshakable fortitude, she does not lose the power of faith. Before us is no longer a shy coward who is afraid of everything, but a brave girl, firm in her convictions. She faces death, but she hates Shvabrin. Who would have thought that Masha, the former quiet girl, could utter such words: “I will never be his wife: I better decided to die and will die if they don’t deliver me.”

Masha is a person strong will. She faces difficult trials, and she endures them with honor. And here's one more thing. Grinev is taken to prison. And this modest, shy girl, left without parents, considers it her moral duty to save Grinev. Marya Ivanovna goes to St. Petersburg. In a conversation with the empress, she admits: “I came to ask for mercy, not justice.” According to D. Blagoy, during Masha’s meeting with the Empress, “the character of the captain’s daughter, a simple Russian girl, essentially without any education, who, however, found in herself at the necessary moment enough “mind and heart”, is truly revealed to us.” firmness of spirit and unyielding determination in order to achieve the acquittal of his innocent fiancé.”

Masha Mironova is one of those heroes of The Captain's Daughter, in whom, according to Gogol, “simple greatness” was embodied ordinary people" Despite the fact that Masha Mironova bears the stamp of a different time, a different environment, the outback where she grew up and was formed, in Pushkin she became the bearer of those character traits that are organic to the indigenous nature of a Russian woman. Characters like hers are free from enthusiastic fervor, from ambitious impulses towards self-sacrifice, but always serve man and the triumph of truth and humanity. “Delight is short-lived, fickle, and therefore does not have the power to produce true great perfection,” wrote Pushkin. Thus, the captain’s daughter - Masha Mironova - in Pushkin’s work deserves to take a place next to Tatyana Larina, who became the embodiment of simple, but distinctively natural traits of the national female character.

Pushkin reveals the complex contradictions that arise between political and ethical conflicts in the destinies of his heroes. What is fair from the point of view of the laws of the noble state turns out to be inhumane. But also the ethics of the peasant uprising of the 18th century. revealed itself to Pushkin from a very cruel side. The complexity of Pushkin's thought is reflected in the construction of the novel. The composition of the novel is constructed exclusively symmetrically. At first Masha finds herself in trouble: harsh laws The peasant revolution destroys her family and threatens her happiness. Grinev goes to the peasant king and saves his bride. Then Grinev finds himself in trouble, the reason for which this time lies in the laws of noble statehood. Masha goes to the noble queen and saves the life of her fiancé.

4. The evolution of the character of Masha Mironova

At the beginning of the work, we are presented with a timid, timid girl, about whom her mother says that she is a “coward.” A homeless woman who only has “a fine comb, a broom, and an altyn of money.” Over time, the character of Marya Ivanovna, “a prudent and sensitive girl,” is revealed to readers. She is capable of deep and sincere love, but her innate nobility does not allow her to sacrifice her principles. She is ready to give up personal happiness because she does not have the blessing of her parents. “No, Pyotr Andreich,” answered Masha, “I will not marry you without the blessing of your parents. Without their blessing you will not be happy. Let us submit to the will of God.” But it changes dramatically surrounding life, “rebels of the villain Pugachev” come to the fortress, and Masha’s position changes. From the captain's daughter, she becomes Shvabrin's prisoner. It would seem that a weak and timid girl should submit to the will of her tormentor. But Masha shows here traits that were still living latently in her. She is ready to die, just not to become the wife of Alexei Ivanovich.

Saved by Pugachev and Grinev, Marya Ivanovna gradually regains her lost balance. But here’s a new test: Grinev is put on trial as a traitor. Only she can prove his innocence. Marya Ivanovna finds the strength and determination to go to the empress’s court to seek protection. Now in these fragile hands the fate of a loved one, the guarantee of future happiness. And we see that this girl had enough determination, resourcefulness and intelligence to save Grinev and restore justice.

Thus, throughout the novel, the character of this girl gradually changes.

conclusions

The composition of the novel is constructed exclusively symmetrically. At first, Masha finds herself in trouble: the harsh laws of the peasant revolution are destroying her family and threatening her happiness. Grinev goes to the peasant king and saves his bride. Then Grinev finds himself in trouble, the reason for which this time lies in the laws of noble statehood. Masha goes to the noble queen and saves the life of her fiancé.

Masha Mironova is one of those heroes of The Captain's Daughter, in whom, according to Gogol, the “simple greatness of ordinary people” was embodied. Masha is a person of strong will. From a timid, dumb “coward” she grows into a brave and determined heroine, capable of defending her right to happiness. That is why the novel is named after her “The Captain's Daughter”. She is a true heroine. Her best features will develop and manifest themselves in the heroines of Tolstoy and Turgenev, Nekrasov and Ostrovsky.

Bibliography.

1. D.D. Good. From Cantemir to the present day. Volume 2 – M.: “ Fiction", 1973

2. A.S. Degozhskaya. Story by A.S. Pushkin's "The Captain's Daughter" school study. – M.: “Enlightenment”, 1971

3. Yu.M. Lotman. At the school of poetic word. Pushkin, Lermontov, Gogol. – M.: “Enlightenment”, 1988

4. N.N. Petrunina. Pushkin's prose (paths of evolution). – Leningrad: “SCIENCE”, 1987


A.S. Degozhskaya. Story by A.S. Pushkin's "The Captain's Daughter" in school study. – M.: “Enlightenment”, 1971

D.D. Good. From Cantemir to the present day. Volume 2 – M.: “Fiction”, 1973

Page 6

Depending on the conditions of the problem, we will connect the points with segments if there is a correspondence between these elements, or with a dotted line if there is no correspondence.

The task boils down to finding three solid triangles on the graph with vertices in different sets (they can be highlighted in different colors on the board and in the notebook).

So, using condition 1), we draw a dotted line connecting the objects Ivan and Moscow, Dmitry and Novgorod.

In accordance with condition 2) we connect the peaks Moscow and Physics with a solid line, and condition 3) we express with a solid line from the point Novgorod to the point Chemistry.

Dmitry and Stepan do not teach biology, let's connect the corresponding vertices with dotted lines. Who teaches biology? If it is not Dmitry or Stepan, then it turns out that Ivan teaches biology. These objects are connected by a solid line.

Where does the biology teacher live? It is known that the chemist lives in Novgorod, and the physicist in Moscow, therefore, the biologist lives in Tula. Let's pay attention to the triangle formed by the vertices Ivan, Tula, biology: it has two solid sides, which means the third side can also be highlighted with a solid line. In fact, if Ivan teaches biology, and the biologist lives in Tula, then Ivan lives in Tula.

What is known about Dmitry? Dmitry does not live in Novgorod (according to the condition) and does not live in Tula (Ivan lives there), which means Dmitry lives in Moscow - let’s carry out the corresponding solid line. But a Muscovite teaches physics - this line is also continuous. In a triangle with vertices at the points Dmitry, Moscow and physics, two sides are solid, therefore, the third side can also be highlighted with a solid line.

What is known about Stepan? Stepan does not live in Tula (Ivan lives there) and does not live in Moscow (Dmitry lives there), therefore, Stepan lives in Novgorod - let's draw a continuous line. But the one who lives in Novgorod teaches chemistry - this line is also continuous. This is how the third triangle of solid lines appears.

The answer is indicated on the graph by triangles. The problem is solved.

4. Modeling techniques using a block diagram.

If in a problem it is necessary to consider various options for a situation, analyze them and draw appropriate conclusions, such a situation can be visually represented by a flowchart, where each step in the reasoning is highlighted by a separate block (rectangle).

Task. On some island, “truth-lovers” and “jokers” live in separate villages. “Truth-tellers” always tell only the truth, and “jokers” constantly joke, and therefore always lie. Residents of one tribe visit the village of another tribe, and vice versa. The traveler ended up in one of the villages, but he doesn’t know which one. To prove that a traveler only needs to ask the first person he meets the question: “Are you local?” in order to determine from the answer which tribe’s village he is in.

Solution. The traveler can end up either in the village of “truth-lovers” or in the village of “jokers” - two appear various options. In the village of “truth lovers,” a traveler can meet both a “truth lover” and a “joker.” Similarly, in the village of “jokers,” a traveler can meet both a “joker” and a “truth-teller.” Possible options there are already four (Fig. 5).

The flowchart allows you to visualize them and notice that a positive answer in any case is possible only in the village of “truth lovers”, and the answer “no” is only in the village of “jokers”.

In these examples of models (half-line with points, table, graph, block diagram) the “main heuristic function” of the models is clearly visible (D.B. Bogoyavlenskaya – generative, i.e. from the model “one or another decision principle is read ( idea, hypothesis, concept)".

Education information:

The concept of pronunciation side of speech
Speech is the basis of every mental activity, means of communication. The ability of students to compare, classify, systematize, generalize is formed in the process of mastery, through speech, and also manifests itself in speech activity. Logically clear, demonstrative, figurative oral and written speech...

What is deaf-muteness
Deafness - complete absence hearing loss or such a degree of hearing loss that speech perception becomes impossible. Absolute deafness is rare. Usually there are remnants of hearing that allow one to perceive very loud sounds, including some speech sounds pronounced in a loud voice above...

The most important aspects of the formation of verbal speech
Within the framework of the theory of education of a deaf child, it is important to highlight the following aspects of education verbal speech. 1. Teaching verbal speech is the basis for personal development, for socialization and inclusion in culture. Without speech ability It is impossible to formulate ideas about...


Ivan does not work in Moscow, and Dmitry does not work in Novgorod;

a Muscovite teaches physics;

the one who works in Novgorod teaches chemistry;

Dmitry and Stepan do not teach biology;

What subject and in what city does each person teach?

Solution. In the problem, three sets can be distinguished: educational subjects, cities, teachers. Each set contains three elements. Let us denote them by points - the vertices of the graph (Fig.).

Depending on the conditions of the problem, we will connect the points with segments if there is a correspondence between these elements, or with a dotted line if there is no correspondence.

The task boils down to finding three solid triangles on the graph with vertices in different sets (they can be highlighted in different colors on the board and in the notebook).

So, using condition 1), we draw a dotted line connecting the objects Ivan and Moscow, Dmitry and Novgorod.

In accordance with condition 2) we connect the peaks Moscow and Physics with a solid line, and condition 3) we express with a solid line from the point Novgorod to the point Chemistry.

Dmitry and Stepan do not teach biology, let's connect the corresponding vertices with dotted lines. Who teaches biology? If it is not Dmitry or Stepan, then it turns out that Ivan teaches biology. These objects are connected by a solid line.

Where does the biology teacher live? It is known that the chemist lives in Novgorod, and the physicist in Moscow, therefore, the biologist lives in Tula. Let's pay attention to the triangle formed by the vertices Ivan, Tula, biology: it has two solid sides, which means the third side can also be highlighted with a solid line. In fact, if Ivan teaches biology, and the biologist lives in Tula, then Ivan lives in Tula.

What is known about Dmitry? Dmitry does not live in Novgorod (by condition) and does not live in Tula (Ivan lives there), which means that Dmitry lives in Moscow - let’s draw the corresponding solid line. But a Muscovite teaches physics - this line is also continuous. In a triangle with vertices at the points Dmitry, Moscow and physics, two sides are solid, therefore, the third side can also be highlighted with a solid line.

What is known about Stepan? Stepan does not live in Tula (Ivan lives there) and does not live in Moscow (Dmitry lives there), therefore, Stepan lives in Novgorod - let's draw a continuous line. But the one who lives in Novgorod teaches chemistry - this line is also continuous. This is how the third triangle of solid lines appears.

The answer is indicated on the graph by triangles. The problem is solved.

3. Method of modeling on a half-line

If the problem has many objects and it is necessary to establish the relationship between the elements of this set, then the problem can be solved on a half-line.

Task. Four friends gathered for the party: Anya, Vika, Misha and Kolya. Kolya arrived earlier than Anya, but was not the first. Determine in what order the friends came to the meeting place if Vika arrived last.

Solution. Let's build a model of the described situation, considering an ordinary beam to be a “time line”. Friends who came to the party will be indicated by dots with corresponding letters. Let’s agree that the one who came to the party earlier will be designated on the half-straight line (by the first letter of his name) to the left, and the one who came later will be designated to the right. In order, we mark each condition on the half-line ( a-d).

In Fig. A it is shown that Kolya arrived before Anya. According to Fig. b we see that one of the friends is ahead of Kolya, and, consequently, Anya. The appearance of another right point in Fig. V conveys the condition “Vika was the last.” Then we will have to conclude that Misha arrived earlier than everyone else. The sequence of friends' appearance at the meeting place is visible in Fig. G.

4. Technique for modeling using a block diagram

Let's consider another modeling method - drawing up a flowchart in which each step in the reasoning is highlighted with a separate image (rectangle).

Task. On some island, truth-tellers and jokers live in separate villages. Truth-tellers always tell only the truth, but jokers constantly joke, and therefore always lie. Residents of one tribe visit the village of another, and vice versa. The traveler ended up in one of the villages, but does not know which one. To prove that a traveler only needs to ask the first person he meets the question: “Are you local?” in order to determine from the answer which tribe’s village he is in.

Solution. The traveler can end up in the village of “truth-lovers” or in the village of “jokers” - two different options appear. In the village of “truth lovers,” a traveler can meet both a “truth lover” and a “joker.” Similarly, in the village of “jokers,” a traveler can meet both a “joker” and a “truth-teller.” There are now four possible options.




The flowchart allows you to visualize them and notice that a positive answer in any case is possible only in the village of “truth lovers”, and the answer “no” is only in the village of “jokers”.

Let's consider task.

Katya, Anya and Lena bought three tickets: to the cinema, to a rock concert and to the theater. Lena is not into loud music. Anya doesn't like rock concerts, and watching TV movies quickly tires her eyes. Where did each of the girls go?

1. Analysis of the problem conditions

- What does the problem say?

The problem talks about three girls who bought tickets.

- What is known about the girls in the problem?

It is known that Lena is not fond of loud music, Anya does not like rock concerts, and her eyes get tired from watching TV movies.

- What do you need to know in the problem?

The task requires you to find out where each of the girls went.

- Can we immediately answer the question of the problem?

No we can not.

- How, in what way (method) will we look for the answer to the question of the problem?

To find out where each of the girls went, we will use graph construction.

- How do we stand in this problem?

In the problem we distinguish two sets: the set of girls and the set of tickets. Each set contains three elements. Let us denote them by points - the vertices of the graph (Fig.). Depending on the conditions of the problem, we will connect the points with segments if there is a correspondence between these elements, or with a dotted line if there is no correspondence.

3.4.2. Finding a way to solve a problem and drawing up a plan for its path solutions

At this stage of solving the problem, the establishment of connections between these desired and desired quantities is completed and the sequence of using these connections is indicated.

After analyzing the conditions of a problem, we cannot always immediately find a way to solve it. Basic techniques used when searching for ways to solve a problem:

1. Analysis of the problem according to the text or its auxiliary model. The search for ways to solve a problem can be carried out from the problem data to the question (synthetic path) or from the problem question to the data (analytical path).

Synthetic route. The solver identifies two interrelated data in the text of the problem and determines which unknown can be found from these data and using what action. Then, considering the resulting number as data, the solver again selects two interrelated data and determines which unknown can be found from them and with the help of which action, etc., until performing the next action leads to finding the desired one.

Analytical path. Based on the analysis of the problem, it is necessary to clarify what needs to be found in the problem and determine what is enough to know to answer this question. To do this, you need to find out which of the necessary data is available in the problem statement. If they are missing, it is necessary to determine what needs to be known in order to find the missing data, etc., until both data are known to determine the next unknown. Finding a solution ends with drawing up a plan to solve the problem.

When solving problems, analysis and synthesis in reasoning are usually intertwined. When searching for a solution to a problem synthetically, analysis is often carried out “to oneself.” At the same time, no matter how we search for a solution to a compound problem, its preliminary analysis is inevitable.

2. Finding a way to solve a problem (p.79)

Finding a way to solve a problem takes place within the framework of working with the model. Consider the graph. Since Lena is not into loud music, we will connect “L” and “R” with a dotted line. Anya doesn’t like rock concerts, and her eyes quickly get tired from watching TV movies, so let’s connect “A” and “R”, “A” and “K” with dotted lines.

So we have to answer the questions:

1) Where did Lena go?

2) Where did Anya go?

3) Where did Katya go?

So, a solution has been found.

2. Another technique that helps to carry out the stage of searching for a solution to a problem is dividing the problem into semantic parts, the sequential solution of which allows one to obtain an answer to the requirement of the problem.

Example: A certain ruler, wanting to test his three sages, told them: “Before you are five caps: three black and two white. You will be given a cap. The one of you who is the first to guess what color the cap he is wearing will receive a reward." Then the wise men were blindfolded and a cap was put on each of their heads. After the bandages were removed from them, the wise men were silent for a long time. Finally one of them said: "I'm wearing a black cap!" How did this sage reason?

Solution. The problem can be divided into three subtasks, into three options for distributing the caps: black, white, white; black, black, white; black, black, black.

1. Two white caps and one black.

In this case, the one of the participants who is wearing a black cap reasons like this: “I see two white caps, but there are only two of them. That means I’m wearing a black cap!”

2. One white and two black caps.

In this case, he reasons as follows: “I see one white cap and one black one, which means that if I had a white cap on, then the one with a black one on his head would say which cap he is wearing (black), but he is silent. So I’m wearing a black cap.”

3. Three black caps.

He reasons like this: “I see two black caps. I can wear white or black. If I am wearing a white cap, then one of the wise men, reasoning (2nd option), will guess that he is wearing a black cap. But they are silent, which means I’m wearing a black cap!”