Mystery in the behavior of three dice. Dice

At some stage of development, dice turned from an attribute of fortune telling into an instrument of gambling. For this purpose, unknown craftsmen began to make dice from wood, stone, elephant ivory, etc. History convincingly shows that gambling with dice appeared long before the construction of the Cheops pyramid, i.e. 3000 years BC they already existed. Various museums around the world store samples of ancient Egyptian, ancient Greek, Roman, and Chinese gambling dice. Most often they had the shape of a cube with notches on the sides indicating numbers from 1 to 6. Although there are examples in the form of other polyhedra: a straight prism with a different number of side surfaces; cuboctahedron with 14 faces; in the form of a prismatic top and others. Dice in the form of a cube have not gone out of use to this day; the rest are kept as museum exhibits. The advantages of the cubic shape of the dice have quite reasonable explanations:

Only a regular polyhedron ensures complete equality of all faces;

Of the five regular polyhedra existing in nature, the cube is the easiest to make;

It rolls easily, but not too much. A tetrahedron rolls more difficult, but a dodecahedron and an icosahedron are so close in shape to a ball that they roll quickly.

The Western standard requires that the sum of the numbers on opposite sides equals seven: 6-1.5-2, 4-3. There are only two different ways of numbering dice, one of them being a mirror image of the other and, moreover, all modern dice are numbered the same.

If you hold the cube so that the three numbers 1, 2 and 3 are visible, the numbers will be arranged in the reverse order of clockwise movement.

Why were these games specifically gambling, that is, they involved some kind of bets in the game, money or things that could be won or lost?

Probably because when throwing a dice you didn’t need to think - you threw it up and left it to chance. If you don’t sweeten this action with the opportunity to hit the jackpot, then there is simply no other point in stupidly throwing dice. Unlike, for example, chess, where the long process of battle of minds itself brings satisfaction, people play with pleasure without additional incentives, and even then not always.

Gambling with dice, as strange as it may sound, benefited science and served as an impetus for the development of combinatorics and the mathematical theory of probability. This theory began with the study of various types of gambling, with the goal of establishing patterns in random events and determining the probability of winning or losing. In the fight against chance, this knowledge does not change anything, but it can warn you, give you the opportunity to realistically assess your chances of winning, and only then decide whether to get involved in the game or wisely refuse. Knowledge of chess openings and chess theory will be useful in the game itself and can lead to victory, but knowledge of probability theory will not affect either the dice or the ball in American roulette; you will be left alone with chance. Although it is still interesting to know that randomness also has its own patterns.

Dice games can be played with different numbers of dice thrown at the same time. Let's start with one bone.

The game is primitive

A primitive game with one die consists of players taking turns throwing it and the one with the most points wins. If the points are equal, the players repeat the throw. It is unlikely that anyone will be interested in such a game, so this procedure is used more often not for the game itself, but when drawing lots in some other games or matters.

But even this simple option allows us to train our logical thinking. In the history of the development of the mathematical apparatus of gambling, there were many cases of incorrect logic that led to incorrect results. Let's look at a similar example.

When throwing one die, the probability of appearing one is 1/6. The same goes for the second toss. This means that if you make two throws, then the probability of one appearing at least once (on the first throw or on the second) is 1/6+1/6=1/3. Reasoning similarly, it turns out that for six throws the probability of getting a 1 at least once out of six is ​​equal to one (1/6-6=1), i.e. is a reliable event. We can apply this reasoning to any of the numbers from 1 to 6, and conclude that each number, when thrown six times, is sure to come up. On the other hand, experience tells us that this is not so. Throw a die six times and it is unlikely that each of the possible numbers will come up exactly once. What is wrong with the reasoning? The statement: “a one came up at least once in two rolls” actually breaks down into several different events:

Dropped out the first time and didn't drop out the second time (1/6-5/6) or

Didn't fall out the first time and dropped out the second time (5/6-1/6) or

It fell out the first time and the second time too (1/6-1/6).

The corresponding probability is calculated as 5/36+5/36+1/36-11/36, which is slightly less than 1/3. For six throws, it is better to start counting differently. The probability that a 1 did not appear with one throw is 5/6, with two throws 5/6-5/6, respectively, the probability that a 1 did not appear with six throws is (5/6)6. This means that the probability that it appears at least once in six throws is 1-(5/6)6 = 0.66510.

Game with expansion

The first player rolls the die and adds the number on the top side to any number on one of the four sides. His opponent adds up all the remaining numbers on the three side faces. The lower edge is not taken into account. The second player then rolls the die and they make similar calculations. The player who, after the throws of both players, has a greater total wins. To the blind chance there was added a small opportunity for the player to choose one of the side numbers, although what to choose there - you need to take the largest one. In addition, you will have to add numbers in your head, it turns out that you have added thinking.

Dice flips

This game again requires one die. The first player calls any number from 1 to 6, and the second throws the die. Then they take turns turning the bone over its edge in either direction a quarter of a full turn. To the number of points named by the first player, the number of points that fell on the top side after throwing the die and after each turn is added. The winner is the player who manages to reach the total of 25 points at the next turn or force the opponent to exceed 25 points at the next turn.

In just the third step, left with only one die, we came to the need to think seriously.

What number should the first player call to have the best chance of winning?

Two-dice games have been so popular for centuries that they have their own historical names and specific terminology.

Hazard

The name of the game comes from the Arabic expression “az-zahr” - “dice”.

The player acting as the banker bets against other participants, whose number is unlimited, that he will be able to roll one of the following numbers using two dice: five, six, seven, eight or nine. Opponents, in turn, are obliged to equal his bet.

The number guessed by the banker is called “main”. If after his throw the “main” appears, then the banker receives all the money at stake. This successful move was called “nick”. If some other number comes up, it is called “chane”, then all is not lost for the banker. He must continue to throw the dice until he rolls “chane” again - then he wins, or the “main” rolls up - then he loses and must pay out the money.

Gambling with throwing three dice and other rules was widespread in casinos; we’ll talk about it later.

Craps

The game Craps is one of the most popular in America. Invented in the 9th century by black slaves from the banks of the Mississippi. The player rolls two dice and calculates the total points. He immediately wins if this sum is 7 or 11, and loses if it is 2, 3, or 12. Any other sum is his “point”. If a “point” is rolled for the first time, the player rolls more dice until he either wins by rolling his “point” or loses by getting a score of 7. Let’s do some thinking about throwing two dice. First, let's calculate the probabilities for the total number of points on two dice. Let's assume that one of them is white, and the second is black. This is an important detail in the reasoning, since we must distinguish between dice, and, consequently, such options for possible outcomes as (3.5) and (5.3). Tossing two dice has 36 equally likely outcomes, which we have summarized in a table.

The cells of the table indicate the amount of points received. Based on the first table, it is possible to calculate the probability distribution of obtaining a certain amount of points when throwing two dice. We will present these values ​​in a table.

Here the bottom line indicates the probability of occurrence of the corresponding score. The table allows you to calculate the probability of winning after the first throw

Р(7)+Р(11)=6/36+2/36=8/36=2/9

The probability of losing after the first throw is

Р(2)+Р(3)+Р(12)= 1/3 6+2/36+1/36=4/3 6= 1/9

Thus, the theory says that the probability of winning on the first throw is 2 times greater than the probability of losing, but even greater (2/3) is the probability that the game will not stop on the first throw, but will continue. Try to conduct your own research into the probability of throwing it again the first time you throw a point in the next game.

Try your luck

This is a game of chance with three dice. It is often played in gambling houses and during public festivities at fairs or carnivals. There are six squares on the counter, marked 1, 2, 3, 4, 5, 6. Players make standard equal bets on one of the numbers, after which three dice are tossed. If the player's number appears on one, two or three dice, then for each appearance of this number the player is paid the original bet, and his own money is also returned. Players whose number is not drawn lose their bet even once. A player can bet on several numbers at the same time, but each bet is considered separately.

The game is simple and exciting. Only lack of education explains the fact that our “scammers” ignored her, because there was no crime.

Let's assume for simplicity that there is a single bet on each number. The game is harmless only if all three numbers drawn are different. Then, having received six bets on six numbers, the gambling house pays with this money to three lucky players, giving them three won bets and returning three bets. In this case, the organizers of the game have nothing, but only redistribute money between the lucky ones and the losers. This will always happen when three different numbers are drawn, but not all different numbers will always be drawn.

Now suppose that after throwing the dice, exactly two identical numbers come up. Of the six bets received, three will be given to the player whose number is drawn twice (taking into account the returned bet) and two will be given to the player whose number is drawn once. It turns out that in this situation, one bet remains with the gambling house.

Finally, let the same number come up on all three dice. Then one player receives four bets, three won and one returned, and the gambling house is left with two player bets.

Let's consider the probability of these cases. Let the dice vary in color, such as red, green and blue. They can appear in 6*6*6 = 216 ways.

It is easy to calculate the last case when three identical numbers are drawn. The number of such options is only 6, since the red die can fall on any of the 6 faces, and the green and blue ones can only fall on the only one that has already landed on a red die. Let's determine how many ways three different numbers can appear. For a red die there are 6 different options, for a green die there are only 5, because the number rolled on a red die should not be repeated, similarly reasoning, a blue die can only land on one of the 4 faces. Total 6*5*4 = 120 options.

It follows that in 90 cases two identical numbers are drawn (216 - 126 = 90). The probability of a gambling house receiving a bet is (120/216)*0+(90/216*1+(6/216)*2 = 102/216.

This means that the number of single player bets remaining in the gambling house is approximately equal to half of the games played and no losses. In this situation, it is profitable to work around the clock.

Now let's look at this game from the player's point of view. Out of 216 equally probable outcomes, he wins only in 91 cases and loses in 125. Where did we get the number 91 from? Let’s say a player bets on “one”. One in 216 outcomes is when all three ones are rolled; out of 90 cases with two identical digits, the third part includes one; out of 120 options with three different numbers, one is included in half. Total: 1+30+60=91.

This probability is significantly different from the probability of winning for a gambling house. Although the numbers 102/216 and 91/216 are not very different, for a gambling house they mean an inevitable profit, and for a player a loss is more likely than a win.

The calculations will be more complicated if players are allowed to make arbitrary rather than fixed bets on different numbers. With these rules, there is a chance that the gambling house will initially put some money into the game when the small bets of the losing players do not cover the large bet of the winning players, but if the game lasts long enough, then the organizer of the game can hope to receive 7.8% from each dollar bet by the players. Try to figure this figure out yourself.

Three dice

First, each player calls a number from 3 to 18. Three dice are thrown. The player whose sum of points is equal to the number named before the game wins. Let's determine the player's chances depending on the number he named. Three dice are tossed over the table and the sum of the points on the top faces is counted. How many different outcomes are possible for one toss of the dice?

Each die can show one of six numbers on its top face: 1, 2, 3, 4, 5, 6. Combining the 6 locations of the first die with the six locations of the second, we get 6*6=36 options for two dice. Each of these 36 arrangements of two dice combined with one of the 6 arrangements of the third die gives 36-6=216 combinations of 3 numbers. Do each amount have the same probability of occurrence from the smallest (1-3) to the largest (6-3)?

Let us compare, for example, the probabilities of receiving sums 9 and 10. At first glance, the probabilities are the same. Three dice form 6 triplets of numbers, giving a total of 9 - (6, 2, 1), (5, 3, 1), (5, 2, 2), (4, 1, 1), (4, 3, 2 ), (3, 3, 3), and the same number form triplets of numbers with a sum of 10 - (6, 3, 1), (6, 2, 2), (5,4, 1), (5, 3,2 ), (4, 4, 2), (4, 3,3). To avoid errors in reasoning, let's assume that our cubes are colored, for example, according to the RGB system, i.e. red, green and blue. Then the first triple of numbers, giving the sum 9, actually breaks down into six objectively different options: (6, 2, 1), (6, 1, 2), (2, 1, 6), (2, 6, 1), ( 1, 2, 6), (1, 6, 2). In this entry, the number that came up on the red die is in first place, the number that came up on the green die is in second place, and the number that came up on the blue die is in third place. If in a trio of numbers that give the required sum, two numbers are the same, then, taking into account the coloring, three different layouts are obtained. For example, - (5, 2, 2), (2, 5, 2), (2, 2, 5).

If three numbers are the same, permutations do not create different cases and only one option is possible. Now let’s count the number of cases that give a sum of 9, taking into account the individuality of the cubes: 6+6+3+3+6+1=25. A similar calculation for the sum of 10 will give the result: 6+3+6+6+3+3=27. Maybe not by much, but when throwing three dice, the probability of a total of 10 appearing is greater than the probability of a total of 9. Thus, you can calculate the probabilities of appearing for each of the possible totals from 3 to 18. As a result, all 216 possible outcomes will be distributed according to their amounts. The first person to correctly carry out such reasoning was the famous scientist Galileo Galilei.

Three dice hazard

This game is common in casinos and is therefore played by the casino, represented by the dealer, against the bettors.

The game table has a special layout so that players can bet on different outcomes when throwing three dice. By placing a chip on any of the 6 combinations in the Raffles field, the player thereby bets that exactly this number of points will be rolled on all three dice at the same time. If he is lucky, he will win at a ratio of 180:1. By betting Any raffle on the field, the player wins if after throwing all three dice there are the same number of points, but it doesn’t matter which one. Winnings are paid out at a ratio of 30:1. On the Low field (little) they win when the sum of the points drawn is no more than 10. On the High field (many) - when the sum of points is not less than 11. Winnings on Even (even) and Odd (odd) are paid out if any even number is rolled or, accordingly, an odd number. But if the resulting number consists of three identical digits, this means the player loses. In addition to these bets, there are bets on a specific amount of points, “on numbers”. The table layout shows the ratio in which winnings are paid out when betting on a particular number. The ratios are different and depend on the probabilities of throwing out each amount.

We will not repeat the probability calculations for rolling three dice; we will only note that for any bet, the ratio paid to the player is less than what it should be based on theory. In the Raffles field, the true ratio is 215:1, which means the casino keeps 16 2/3% of the winnings. Each field has its own percentage, which remains with the casino. We outlined how to calculate this in the discussion of the previous game, and you, if you wish, can complete the calculations. Thus, arm yourself with knowledge, the main thing of which is that the casino always wins.

To play, you must have five standard dice. The dice are thrown from the hands or from any glass onto a flat surface. The game can be played by two or more players. The goal of the game is to complete certain figures with the maximum number of points. The first throw is to draw lots for the turn order between the players. The player with the most points starts, and then in descending order of points.

The set of figures consists of two programs: compulsory and free.

Mandatory program:

ones, twos, threes, fours, fives, sixes. (You need to throw out at least 3 dice of a specific value).

Free program:

One pair (1 p) - 2 dice of the same value;

Two pairs (2p) - 2 dice of one value and 2 dice of another value;

Any three (3) - 3 dice of the same value;

Small Straight (LS) - 5 dice with values ​​of 1, 2, 3, 4, 5;

Big Straight (BS) - 5 dice of 2, 3, 4, 5, 6;

Full (F) - 2 dice of one rank and 3 dice of another rank;

Four of a kind (C) - 4 dice of the same value;

Poker (P) - 5 dice of the same value;

Chance (Sh) - 5 dice of any value.

The execution of figures begins with a mandatory program. Figures of the free program can only be performed after the completion of the compulsory program. The order of execution of figures in programs is arbitrary. With each move, the player has the right to three attempts to complete one of the pieces. After the first throw, he keeps the dice necessary for the intended figure, and in subsequent attempts he throws away the remaining ones to obtain the desired result. With any of the three attempts, you can start performing another figure, depending on the situation.

The results of the moves are recorded in a special, pre-drawn table. After completing each move of a mandatory program, the following options may arise:

1. 3 dice of the same value fell out: then a “+” sign is placed in the corresponding cell of the table, marking the completion of the figure;

2. Less than 3 dice of the same value fell out: a negative result is entered into the table, equal to the number of dice missing up to three, multiplied by their value (for twos 2, for threes 3, etc.);

3. More than 3 dice of the same value are rolled: a positive result equal to the number of dice in excess of three multiplied by their value is recorded in the table.

4. Not a single dice of the desired value fell out: then the table indicates a negative result equal to the value of the desired dice multiplied by 3.

Each participant can perform the combination only once. For example, if one of the participants gets the mandatory “four” combination for the second time, and possibly with a better result, then he cannot enter this result into the table again, but must perform one of the remaining combinations.

After the mandatory program, an interim result is summed up. Each player's points are summed up. If the total is zero or more, a bonus of 50 points is added. When performing a free program figure from the first throw, its total points are doubled, except for chance. If, when making a move, it was not possible to discard the desired piece, then, at the player’s request, points for any already completed piece are crossed out from the table. When performing poker, a bonus of 50 points is given. The game ends by filling in all the cells of the table. The points of each player are summed up and then the calculation is made. The arithmetic average of the sum of all players is subtracted from the points of a particular player. Positive result- this is a gain, negative - a loss. Let's show an example of filling out a table with scoring for one of the players and comments on the game process.

This game is a variation of card poker. Moreover, poker with ordinary dice is described here, and there are special poker dice, on the sides of which there are card symbols: nine, ten, jack, queen, king and ace.

So, we looked at several dice games and showed some methods for calculating the probabilities of individual outcomes. There is also a variant of craps for casinos with its own table layout, the popular game passe di and many others. But poker, it seems to me, is the most intellectual of the dice games, so we’ll finish our conversation about this group of gambling numerical games. Dice gave the main impetus to the development of combinatorics and probability theory. And such great mathematicians as Tartaglia and Galileo, Fermat and Pascal, who left their names in science in connection with other major discoveries and research, were engaged in theoretical studies of dice games.

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AMAZING WORLD
MATHEMATICS
(pedagogical project for mathematics teachers)
Subject week of mathematics “As a means of development
individuality of the student’s personality through involvement in
creative activity by subject"
Author of the project: mathematics teacher Olga Viktorovna Gladkova,
Tyumen city
Justification for the need for the project:
Low level of mathematical literacy of school graduates.
A graduate of a modern school must think creatively and be able to
find non-standard solutions, be competitive (for
This requires the ability to take initiative).
Relevance of the selected topic
significant increase in motivation and interest of students in
teaching mathematics;
deeper and more lasting assimilation of knowledge by students, the opportunity
their independent movement in the study area;
providing conditions for general cultural and personal development
Hypothesis
Subject week communication system that allows
to express oneself, to assert oneself, to realize oneself with all its
participants
Target

Creating optimal conditions for the development of individual
intellectual, creative, social abilities of children in
educational institution.
Project objectives
1) Ensuring the possibility of creative self-realization of the individual in
various types activities.
2) Formation of key competencies among students: subject,
social, informational, communicative.
3) Improving methodological support for educational
and educational process in exact cycle subjects.
4) Development of mass, group and individual forms
extracurricular activities
Participants and their role in the implementation of the project
 Students – actively participate in the project;
 Parents receive information, interact with
teacher;
 Teachers interact “parents + children +
supervisor";
 The administration provides regulatory conditions
for the implementation of the project (provision on the subject week),
rewards project participants
Expected results
For the teacher
creating conditions for the formation of information,

communicative, social, cognitive and subject
competencies of their students;

subject;
mastery creative approaches to teaching your

improvement of professional skills through

preparation, organization and conduct of subject-related events
weeks.
For students
 importance of mathematics in Everyday life, level up
mathematical literacy
 ability to understand the task at hand, the nature of interaction
with peers and the teacher, the ability to plan the final
the result of work, searching and finding the necessary information,
 confirmation of existing basic knowledge in accordance with
theme of the subject week,
 expansion of historical and scientific horizons in subject area.
At the administration level
 Monitoring the level of teacher professionalism.
 Submission of materials about the teacher’s experience for certification,
awards, competitions.
 Preparation of materials for publication.
At the parent level
 Formation of motivation to cooperate with the school.
 Increasing the degree of parental involvement in activities
schools.
 Improving communication culture.
Project implementation stages
1. Methodological and motivational
2. Preparatory
3. Organizational

4. Implementation
5. Reflective
1. Methodological and motivational
Stage objectives:
Studying the work experience of school teachers and other educational institutions, methodological
literature on conducting subject weeks.
Formulation of the main goals and objectives of the subject week.
The purpose of the subject week is to develop personal qualities
students and activation of their mental activity, support and
development creativity and interest in the subject, the formation
conscious understanding of the significance of mathematical knowledge in everyday life
life.
Objectives of holding Mathematics Week at school:
1. To develop students’ interest in mathematics.
2. Identify students who have creative abilities and strive
to deepen your knowledge in mathematics.
3. Develop speech, memory, imagination and interest through the use of creative
tasks and assignments of a creative nature.
4. Foster independent thinking, will, and perseverance in achieving
goals, a sense of responsibility for one’s work to the team.
5.Developing the ability to apply existing knowledge in practical situations.
Principles for organizing Mathematics Week:
1. The principle of mass participation (the work is organized in such a way that the creative
activity involves as many students as possible).
2. The principle of accessibility (multi-level tasks are selected).
3. The principle of interest (tasks should be interestingly designed,
to attract attention visually and in content).
4. The principle of competition (students are given the opportunity
compare your achievements with the results of students in different classes).
Determination of the main activities, their forms, content and
participants.
Activity:
1. Competition of mathematical fairy tales and puzzles.
2. Presentation competition in nominations.

3. Game “What? Where? When?” (Grade 711).
4. Virtual excursion (history of mathematics).
5. “Own game” (grade 56)
Motivating and attracting active children and parents to conduct
subject week.

Duration:2 months
2. Preparatory
Stage objectives:
Approval of the subject week plan. Approval of provisions,
chairpersons and jury members of competitions.
Distribution of responsibilities between MO teachers for conducting
subject week.
1. Dudina A.A., Sadykova Z.G. – “Own game” 56th grade
2. Grekova N.V., Timofeeva V.M. - game “What? Where? When?"
3. Safronova E.S. virtual tour.
4. Shirshova E.V. – competition of mathematical fairy tales and puzzles.
5. Gladkova O.V. – presentation competition, preparation for project defense
students.
Release of an extended announcement on the subject
weeks.
Identification of creative groups of schoolchildren, teachers, parents
for conducting a subject week (distribution of roles,
preparation of registration).
Main participants: teachers of mathematics and computer science, MO
Duration: 1 week

3. Organizational
Stage objectives:
Self-determination of children to participate in competitions.
Creation of creative groups of students for final events
subject week.
Groups are formed by sections:
 Fun math
 History of mathematics

 Mathematics in everyday life
 Hard math problems
 To help the teacher
Work of creative groups.
Main participants: students, teachers, parents.
Duration: 1 week
4. Implementation
Stage task:
Work according to the approved subject week plan.
Main participants: school students, teachers
Duration: 1 week
5. Reflective
Stage objectives:

Summing up the results of the subject week, awarding the winners
and active participants.
Analysis of the work performed.
Development of recommendations for conducting a subject week.
Main participants: teachers of mathematics and computer science, MO,
school administration
Duration: 1 week
Types and forms of events
● Training activities:
poster subject assignments
project activities
non-traditional lessons on the subject
● Collective creative activities
 creative competitions for wall newspapers, crosswords, puzzles,
poems, fairy tales, etc.
 Virtual tour
 “Own game”
 Quiz
 What? Where? When?
The role of the teacher in organizing and conducting a subject week
Leading
determining the content of the work;

setting tasks;
indication of the main sources of knowledge.
Tutoring
assistance in choosing forms of work;
consulting students in the process of completing assignments and
coordinating their activities;
studying together with students the information they have identified;
participation in the design of material collected by students
Forms of encouragement for subject week participants
Awarding diplomas from educational institutions:
1) individual winners of a creative work competition.
2) classes for the best newspapers;
3) teams – winners of various competitions.
Presentation letters of thanks the most active participants
subject week from among schoolchildren and their parents.
The success of the project and its significance for the educational institution
1) Mass scale of the project (involvement of students in the project,
involving parents in joint activities with children)
2) Satisfaction of project participants with their activities
What does the project benefit the school?
For students
 Self-affirmation
 Opportunity for self-realization

 Test your strength in the subject
 Interesting
 The result is visible immediately
For teachers
 Involving students in independent creative
activity
 Feeling of professional satisfaction
 Opportunity to exchange experiences
 Opportunity for creative self-expression
 Increasing pedagogical authority.
Parents
 Disclosure of interests and inclinations of students
 Increasing interest in the subject.
 Promoting vocational guidance for high school students
 Instilling students’ interests in studying mathematics
 Improving the image of the educational institution
Development of the individuality of the student's personality
1) manifestation of individual abilities, creativity
self-expression, leadership qualities in a child
2) ability to work in a group
Further development of the project
A special feature of the project is its complementarity.
Based on this project it is assumed:
participation in various methodological competitions;
publications dissemination of experience,

development of the virtual component of the project in order to attract
more participants.
Mathematics week plan
1. Game “What? Where? When?" (grades 5-11)
2. Results of the competition of mathematical fairy tales and puzzles.
3. Results of the presentation competition in nominations:
 History of mathematics;
 Mathematics – orientation towards life in
in today's changing world;
 To help the teacher (summarizing the topics studied in
lessons);
 Connection of mathematics with other subjects.
4. Defense of projects in sections:
 Fun math
 Benefit of one task
 Mathematics in the knowledge system of other subjects
 Mathematics exam (different ways
solving difficult problems of the second part)
Subject
ika
project
comrade
And I fell in love with the circle and on it
has stopped.
What is your area?
Axiomatic method
Axioms of planimetry.

Euclid's algorithm
Arithmetic of figures
Bimedians of a quadrilateral
Bisector - familiar and not so familiar
In the world of triangles.
In the world of figures
In the world of quadrangles
Geometry is in fashion!
The most important theorem of geometry
The Great and Mighty Theorem of Pythagoras
Great problems of mathematics. Squaring the circle.
The Great Mysteries of the Pythagorean Theorem
The whole world as visual geometry
A look at elementary geometry.
Excircle
Inscribed and circumscribed polygons.
All about the right triangle
All about the triangle.
All about the compass
Second midline of trapezoid
Derivation of formulas for the areas of a rectangle, triangle and
parallelogram according to the coordinates of their vertices.
Calculating the circumference
Calculation of the area of ​​a maple leaf.
Harmony of the golden ratio
Geometric illusion and optical illusion
Geometric illustration of averages
Geometric mosaic.
Geometric cheat sheet
Geometric analogies
Geometric puzzles.
Geometric problems of the ancients in the modern world
Geometric problems with practical content
Geometric problems through centuries and countries.
Geometric toys - flexagons and flexors
Geometric lace.

Geometric methods for solving algebraic problems.
Geometric impossibilities
Geometric surprises
Geometric paradoxes
Geometric parquets
Geometric scissors in problems.
Geometric constructions and their practical application
Geometric tales
Geometric tales on the topic "Length"
Geometric figures
Geometric shapes in the design of paving slabs.
Geometric shapes in the modern world
Geometric figures in the Pythagorean theorem.
Geometric shapes around us
Geometric ornament on dishes.
Geometric dictionary.
Geometric constellation
9th grade geometry in puzzles
Geometry of Lobachevsky. Definition of a straight line
Geometric ornament of the ancient Arabs and its modern
reading
Geometry in the architecture of buildings and structures
Geometry in geodesy
Geometry in painting, sculpture and architecture
Geometry in Winter Olympic Sports
Geometry in the beauty of ornaments
Geometry is in fashion
Geometry in folk art
Geometry and art
Geometry and cryptography
Geometry and character
Geometry of measurements
Geometry of measuring instruments
Geometry of beauty
Geometry on paper

Geometry on checkered paper
Geometry on a plane
Circle geometry
Parallelogram geometry
Triangle geometry
Geometry. Remarkable theorems
"Double bisector" of a triangle
Two remarkable theorems of planimetry
Movement geometric shapes on surface
Cartesian sheet
Cartesian coordinate system
Cartesian coordinate system on a plane
Dividing a circle into equal parts
Dividing a segment into equal parts
Dividing the side of a square in a given ratio by
folding
Length and its measurement
Circumference and area of ​​a circle.
Proofs of the Pythagorean theorem
Proof of Napoleon's theorem
Additional properties of a parallelogram
Euclidean and non-Euclidean geometry. Euclid's fifth postulate
Another property of trisectors of a triangle
Dependence of the number of segments on the number of points marked on
straight
Dependence of the number of diagonals of a polygon on the number of its
peaks
Riddles of the circle
Triangle Riddles
Mysterious and unique geometry
Mysterious ellipse
Entertaining geometry
An entertaining and educational journey to the country of "Geometry"
Entertaining problems in geometry and drawing
Fun problems (geometric problems, match puzzles)
Geometric probability

Famous problems of antiquity. Trisection of an angle
Golden ratio in geometry
Golden triangle in problems
From the history of the emergence of squares
From the history of the emergence of trigonometric terms
From the history of the Pythagorean theorem
Isoperimetric theorem
Studying the method of tiling a plane with equilateral
pentagons
Inversion as symmetry about a circle
Using geometry to solve some types
trigonometric problems
Using flat models when studying the topic "Area"
Study of the influence of the radius of a circle on the circumference and
area of ​​a circle
Study of properties of polygons
Measuring the height of a building in an unusual way
Measuring the height of an object
Length measurement
Measuring long distances. Triangulation
Measurements on the ground in the history of our region
Measuring instruments are our assistants
On-site measurement work
Image of points on the coordinate plane
Study of symmetry in nature
How to find the area of ​​a hole?
Square
Pearson square
"Pythagorean Square" in my life

Squaring a circle
Key tasks in teaching 7th grade geometry
Geometry wheel
Complex numbers in geometry problems
Square wheel - truth or myth?

Magic squares
Median and bisector
Medians of a triangle and areas of figures
Metric system
Metric theorems of planimetry
Mysticism of the triangle
The many faces of symmetry in the world around us
The variety of the circle
Polygons
Polygons. Types of polygons
A set of problems on calculating the areas of figures for 5th and 6th grade students
classes
Names of geometric shapes in surnames
Finding the area of ​​plane figures using the area of ​​a rectangle
Initial geometric information
Celestial geometry. Geometry of snowflakes
Impossible figures
Non-Euclidean geometry
The unknown about the known triangle
Unknown pages of Pythagorean theorem
Some problems for constructing a parallelogram
Several proofs of the Pythagorean theorem
Several approaches to solving geometric problems
Several ways to solve one geometric problem
Several ways to solve a planimetric problem
New criteria for the equality of triangles.
Triangles
About coordinates with a smile
About some remarkable theorems of geometry
About the midline of the trapezoid
About the Pythagorean theorem
triangle of a circle for the multidimensional case
Generalization of the radius formula described around a rectangular
triangle of a circle for the three-dimensional case

Generalizations of the problem of the smallest sum of distances from two points to
straight
Circle in Cartesian coordinate system
Circle of nine points
Circle and circle around us.
Determining the distance to an object. Rangefinder
Determining the center of gravity using mathematical means
Origami and geometry
Orthotriangle and its properties

From segment to vector
From parallelogram to golden ratio
Discovering non-Euclidean geometry
Segments
Parallelogram and trapezoid

Parallel lines
Parallel translation and rotation.
Parquets and ornaments
Parquets on a plane
Parquets, mosaics and the mathematical world of Marius Escher.
Parquets: regular, semi-regular. Paradox M.K. Escher.
Perimeter and area of ​​polygons
Pythagorean pants. Are all sides equal?
Areas of "composed" figures
Areas of geometric angles
Areas of polygons
Area of ​​orthogonal projection of a polygon
Area of ​​a rectangle, units of area measurement.
Area of ​​trapezoid
Following the Pythagorean Theorem
We repeat the chapter "Triangles"
Similar triangles
Similarity in life
Similarity of triangles
Similarity of triangles in solving problems and proving theorems.

Let's talk about a rhombus
Finding an angle in geometric problems
Useful geometry
Constructing acute angles on checkered paper
Drawing lines in the polar coordinate system
Construction of regular polygons
Constructing regular polygons using a ruler and
compass.
Construction of regular triangles with compass and ruler.
Regular polygons
Practical geometry
Practical orientation in the study of geometry
Practical applications of the parallelogram and its types
Practical application of geometry
Practical application of tests for the equality of triangles.
Practical application of the Pythagorean theorem
Transforming a square
Napoleon Transformation of Polygons
Napoleon transformation of quadrangles
Approximate construction of regular polygons.
Signs of a parallelogram
Signs of similarity of polygons
Signs of similarity of triangles
Signs of equality of triangles
Tests for the equality of quadrilaterals
Application of the theorems of Ceva and Menelaus
Application of the theorems of Cheva and Menelaus to solve advanced problems
difficulties
Application of trigonometry in planimetry
Proportional segments in a triangle
Proportional segments. Ways to solve problems
The simplest construction problems
Simple and inexhaustible triangle
Euler's line and circle
Rectangle in visual geometry problems

Right Triangles
Journey through the land of geometry
Fifth postulate of Euclid. Non-Euclidean geometry
Isosceles trapezoid, its properties
Equal and equal plane figures
Equal area polygons
Equally self-intersecting broken lines
Various proofs of theorems of elementary geometry, not
studied at school.
Cutting and folding polygons.
Cutting a square into equal parts
Cutting shapes into equal parts
Distance between remarkable points in a triangle
Solving geometric problems using meshes
Solving geometric problems with practical content
Solving geometric problems using algebra and trigonometry
Solving inscribed and circumscribed circle problems
Solution of the problem of squaring a circle in its medieval formulation
Solving complex geometric problems using the construction method
straightening.
Rhombus and its properties. Problem solving.
Diamond and square
Properties and signs of an isosceles triangle
Properties of the median of a right triangle drawn to
hypotenuse.
Properties of Quadrilaterals
Symmetry in geometry
Symmetry on the plane
Geometry snowflakes
Relationships between sides and angles of a triangle
Sophisms and paradoxes
Treasures of Geometry
Methods for measuring the height of an object in a real environment.
Sum of triangle angles
Bisector surprises

The Mystery of the Four Corners
Secrets of the star pentagon
Morley's theorem
Pythagorean theorem
Pythagorean theorem outside the school curriculum
Pythagorean theorem and its relevance
Pythagorean theorem and various ways her evidence.
Ptolemy's theorem
Thales's theorem
Ceva's theorem
Theorem of Ceva and Menelaus
Cosine theorem
Theorems of Menelaus, Cheva, Ptolemy
Relativity and geometry
Point FarmTorricelli
A point, a straight line... what is it?
Trapezoid
Triangle
Triangles
Reuleaux triangle
Triangle and circle
Triangle is the youngest of the polygons.
Three signs that triangles are equal
Trisection of an angle
Angles and segments associated with a circle.
Amazing square
Polygon patterns
Shapes of constant width. Reuleaux triangle.
Figures drawn with one stroke.
Flag geometry
Flexagons
Formulas of Heron and Brahmagupta
Formulas for finding the area of ​​a triangle
Floral geometry
Center of mass and its application in solving problems
Central symmetry

Central symmetry as a type of movement
Four wonderful points of the triangle
Quadrilaterals
Quadrangles in our lives
Quadrilaterals: their types, properties and characteristics
Numerical methods for calculating the areas of figures of complex shapes.
Extreme problems in geometry.
Ellipse.
Topics of work on mathematical games and puzzles:
Games and tricks with matches
Games with numbers and digits that make up their notation
World games
Games that are played without stopping
Puzzle games of the peoples of the North
Intellectual games on the table of prime numbers up to 1000
Rubik's cube mental gymnastics!
Rubik's cube and its relatives
Rubik's cube is not just fun
Labyrinths are interesting!
Labyrinths: finding a way out
Math in games
Math Quiz
Mathematical game "Tic-Tac-Fac"
Mathematical game "The Adventures of the Three Little Pigs"
Mathematical game "Tangram"
Math games and puzzles
Math Lotto
The imaginary mystery in the behavior of dice
My favorite pastime is checkers
Is mosaic just a game?
Math board game
The role of games and drawings in mathematics
Mathematics in chess
Mathematics in chess
Math on a chessboard

Unusual chess
Chess mathematics
Chess pieces on the coordinate plane
Chess teaches you to think
From play to knowledge
Solving chess problems. World of chess.
Tangram is an invention of ancient times
Tangram is not just a game, but mathematical entertainment.
Flexagons and flexors
Flexagons, flexmans, flexors
Amazing puzzles - flexagons.
Mathematics in crosswords and puzzles
Math Crosswords
Crosswords on cubes
Mathematics in puzzles
Math crosswords
Mathematical crosswords for primary schoolchildren.
Mathematical puzzles
Mathematical puzzles and crosswords.
Mathematical terms in puzzles
Mathematical crossword puzzle on the topic "Actions with natural
numbers."
Sudoku
Stereometry in crosswords
Math puzzles
Puzzles on famous mathematicians
Solving math crosswords
Solving digital puzzles.
Mathematical riddles and puzzles
Research paper topics on Mathematical riddles and
puzzles

Math riddles
Mathematical riddles "Around the world"
Mathematical riddles in the works of Lewis Carroll
Mathematical riddles, charades, puzzles
Math puzzles
Puzzle examples.

Paradoxes and sophisms in mathematics
Mathematical paradoxes
Mathematical sophisms
Math Tricks
Paradox... Trick... Focus
Paradoxes in mathematics
Paradoxes and sophisms in mathematics
Optical illusions and their applications
Origametry
Origami + geometry = origami
Origami helps math
Origami - paper sheet geometry
Ornament
Features of construction on checkered paper
Mathematical tales
Mathematics in fairy tales
Mathematical fairy tale "In the land of unlearned lessons"
Mathematical tale "How Division learned to divide"
Mathematical fairy tale "Kolobok"
Mathematical tale "The Legend of the Chessboard"
Mathematical fairy tale "The Adventures of Fedya Plyushkin visiting
queens of mathematics"

Mathematical tale "Ice Box"
Mathematical tales
Mathematical tales on the topic "Time"
Mathematical tales on the topic "Addition. Subtraction"
Mathematical tales, poems, riddles, jokes, songs, puzzles. Numbers
and the bill
Math tricks
Games and tricks with matches
Exploring the essence of mathematical tricks
Math tricks
Unusual in the ordinary, or Mathematics tricks
Tricks in mathematics
Tricks and curiosities of mathematics
Tricks. What is their secret?
Magic in mathematics
Magic square - magic or science?
Magic of squares
The magic of prime numbers.
The magic of numbers
The magic of numbers 3, 11, 13
Scheherazade's magic number.
Mathematical wonders and mysteries.
The relationship between mathematics and literature
In the world of numbers. Poems
Entertaining literary mathematics
Mathematics in verse
Cryptography in literature
Literature in geometry.
Literary and mathematical interpretation of the tragedy of A.S. Pushkin
"Mozart and Salieri"
Literary and artistic problems in mathematics

Mathematics in legends and fairy tales
Mathematics in proverbs
Mathematics in proverbs and sayings
Mathematics and literature - two wings of one culture
Mathematics and literature - two intersecting planes
Mathematics and literature. Non-Euclidean parallels
Mathematics and poetry
Mathematics or philology
Mathematical poem "Ray, segment and line"
Mathematics in fiction
Mathematics and poetry
"Mathematics and poetry are expressions of the same power
imagination, only in the first case the imagination is directed to
head, and in the second - to the heart" (T. Hill)
Folklore tasks
Mathematics is one of the topics of literature
Mathematical problems in literary works.
Math problems in verse
Mathematical problems from Baba Yaga
Mathematical problems based on the fairy tale by A. Lindgren "Carlson,
who lives on the roof."
Mathematical and physical concepts in proverbs.
Mathematical motives in fiction.
Mathematics in verse
Proverbs and sayings containing numbers
The use of numbers and the range of colors in the poems of Gabdulla Tukay.
A tale of geometry in verse
Numbers in the magical world of riddles.
Mathematics in history
The use of historical and local history material in
creating math problems
Mathematics during the Great Patriotic War

Mathematics to the front, or How plywood defeated duralumin
Mathematical problems with local history content
Mathematics in biology
Study of the species composition and size of trees on
school mathematical methods.
Study of the main types of symmetry in plants and animals
world.
Medicinal plants in mathematical problems.
Mathematics and nature are one
Mathematical harmony in the surrounding world
The mathematical beauty of plants
Mathematical walk in an unusual garden
Mathematical patterns in biology: group inheritance
blood.
Mathematical portraits in nature
Math Zoo
Mathematical Reserve
Mathematical modeling of the environment
Mathematics in nature
Records in the world of birds
Can animals count?
Mathematics in Russian
Grammatical norms of the modern Russian language in the classroom
mathematicians
Study of the frequency of use of Russian letters in texts
Which letter of the alphabet is the most necessary?
Mathematical models in language and science
Mathematical shoots on the Russian language tree
Mathematics in ecology
Environmental Pollution: Geographical and Mathematical
aspect.
Introduction to ecology using quadratic equations.

Using mathematical methods to assess environmental
environmental conditions.
Quadratic function for environmental friendliness and efficiency under
hood.
Mathematics at the service of ecology
Mathematical methods in ecology
Mathematical analysis of the environmental situation.
Environmental problems in 2nd grade
Ecology and mathematics
Ecology in numbers and tasks.
Interdisciplinary connections between ecology and mathematics. Mathematical
tasks of environmental content.
Mathematics in Physics
Vectors and their applied orientation in geometry and physics
Mathematical calculations in physics
The place of mathematics in the study of the acoustic characteristics of hearing
devices
Application of graphs in physics
Application of trigonometry in physics and technology
Application of trigonometry in solving physical problems
Application of mathematical apparatus for solving problems in
physics
Proportional quantities in physics problems.
Mathematics in Astronomy and Astrology
Starry sky and mathematics
Coordinate plane and zodiac signs
Legend of the starry sky and mathematics
Mathematical problems of spaceships
Using space images in a math lesson
Mathematics in Chemistry

Mathematics and music - the unity of opposites
Mathematics and music: do they have a connection?
Mathematical analysis of music of the XVIIX-VIII centuries.
Folklore tasks
The mathematical nature of music
Mathematical Rhapsody
Mathematical component of musical language
Musical harmony of proportions
Rhythm in music and mathematics
Mathematics in art
The relationship between geometry and fine arts
Coded drawings
The golden ratio in the paintings of the Estonian artist Johann
Köhler
Golden ratio in art
Exploring the possibility of using drawing in mathematics lessons
Paintings by famous artists and coordinate system
The coordinate plane through the eyes of a mathematician and artist
Mathematics in a female form
Mathematics in painting
Mathematics in art
Mathematics in pictures
Mathematics and the laws of beauty
Mathematics and art
Math coloring book
The mathematical component in the construction of the ornament (for example
arts and crafts products)
Mathematical foundations of the laws of beauty
Between mathematics and art
Perspective in painting and architecture
Regular polyhedra: mathematics, art, origami
Transforming space using the Origami technique
Proportions and their application in art
Perspective in Geometry and Art

Parallelogram and clothing design
Mathematics in physical education, sports and basic health
Basketball shot through the lens of mathematics
The influence of study load on the health of students
Human health, psychology, mathematics
Math for a healthy lifestyle!
Mathematics of health
Math and bicycle
Math and smoking
Mathematics and tourism
Mathematics and sports
Mathematics and sports for a healthy future
Mathematics to protect your health, or Everything about the school bag
Mathematics for health
Mathematics against smoking
Mathematics through the prism of gymnastics
Math on a chessboard
Mathematical model of throwing a ball into a basket
Mathematical problems about the dangers of smoking
Mathematical methods for studying compliance
anthropometric data of a teenager to the standards of his physical
development
Mathematical methods for studying the physical process
student development
Proportions of height and weight of schoolchildren
Mathematics in sports
Mathematical calculations and water polo
Sports and mathematics.
Mathematics in Defense of the Fatherland
Mathematics and military science
Mathematics and national defense
Mathematics in the service of peace and creation
Mathematical models in military affairs

Mathematics in construction
Mathematics and apartment renovation
Platonic solids and large-scale construction
Application of the Pythagorean theorem in construction
Practical application of similarities and trigonometry formulas to
measuring work
Help of mathematics in repairs
Mathematics in Architecture
Architecture and mathematics
Types of domes and some of their mathematical characteristics
Golden ratio in architecture
Golden ratio in city architecture
Irrationality in architecture.
Irrationality in the construction of arches and domes
Circular patterns in architecture
Mathematics in Architecture
Mathematics in architecture and painting
Mathematics and Architecture
Polyhedra in architecture
Geometry - the servant of architecture
Proportional relationship between music and mathematics in architecture
using the example of churches and temples
Proportion is the mathematics of architectural harmony.
Mathematics in culture
Mathematics and tolerance
Platonic solids in world culture
Mathematics and culture are two wings of one culture

Municipal educational institution

secondary school no. 105

Voroshilovsky district of Volgograd

Research project

"The Mystery of the Dice"

Collective of students of 1st "A" class

under the direction of

Ternova E.V. and Karnova T.I.

Volgograd

2016

1. Preparatory

Relevance and statement of the problem.

World of mathematics at all not boring, as many people think.With the right approachifras can become magician's tools. Such f Ocuses 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. It is well known thatTricks are best suited for children aged 8 years, since it is at this age that the child is able to appreciate them. Most likely he will want to knowand myselfsecret of focus.It is especially useful for shy, insecure kids to learn magic tricks. After all, in order to show a prepared trick, you need to go, if not on stage, then at least to the center of the room where people have gathered for the performance spectators . And thunderous applause and surprise from friends will be the best cure for low self-esteem. Unfortunately, f ocuses, as teaching aids, are rarely used in the educational process, although theyapplicationin mathematics lessons and in extracurricular activitiescontributedevelopYu logical thinking, spatial imagination, the ability to think outside the box, and also increase interest in the subject. It is clear that m athematic tricks are a kind of demonstration of mathematical laws. If during educational presentation they strive to reveal the idea as much as possible, here, in order to achieve efficiency and entertainment, on the contrary, they disguise the essence of the matter as cunningly as possible. That is why, instead of abstract numbers, various objects or sets of objects associated with numbers are so often used.M We decided to look at this topic and created a project in which we highlighted:

Hypothesis: Tricks with dice are based on mathematical principles.

Name: The mystery of the dice.

2. Main stage

A trick is a skillful trick based on deceiving the eye with the help of deft and quick techniques.However, mathematic tricks are observable experiments based on mathematics, on the properties of figures and numbers, presented in a somewhat extravagant form. They combine the elegance of mathematical constructions with entertainment.The focus is always half hidden from the audience: they know about the existence of that secret half, but imagine it as something unreal, incomprehensible. This reverse side of the trick is based either on sleight of hand or on a variety of auxiliary devices. The amazing is not born in a vacuum. It, driven by a person’s fantasy, always grows out of what is already known.That's why we decided that our

Target: Study the mathematical principles of tricks with dice.

Tasks: Learn to perform tricks with dice.

Analyze the mathematical properties of dice, which make it possible to demonstrate tricks with them.

Get viewers interested in mathematical tricks.

At the beginning, we looked at all the possible tricks with dice in books and on the Internet. It turned out that there are not very many of them (Appendix No. 1). Some of them were based on the obvious “deception” of the audience, that is, the use of sleight of hand, rather than the mathematical properties of the dice. Therefore, we selected only those tricks where it was necessary to make calculations. Then we abandoned those tricks that required multiplication or division, since first-graders do not yet know how to do this. As a result, we had only two focuses at our disposal:"Arrangement of cubes" And "Tower of cubes" (Appendix No. 1).

The project participants (1st grade students) tried to perform these tricks with ordinary board game dice. They managed to perform the second trick (“Tower of Cubes”) without any problems, but they had difficulties with the first, because due to their age, they could not remember the order of the mathematical operations of the trick. That's why we settled on demonstrating the "Tower of Cubes" trick. However, to demonstrate tricks in public, large cubes were required, that is, there was a need forproduction of props.EThatwas fascinatingcreative activity.Tum, whereGuysNotcouldwill copebmyselfAnd, them parents and teachers helped. While assembling the cubes, the guys did not pay attention to the location of the values ​​​​on the faces, and the attempt to demonstrate the trick failed. This made the participants think that the cubes must follow certain mathematical laws. Having carefully examined the factory-made dice, we came to the conclusion that the sum of the opposite faces of the dice is 7 (1 and 6, 3 and 4, 2 and 5). And that is why, in the above tricks, the magician could predict the result. Having arranged the values ​​of the faces on the cubes in accordance with the assumption we received, we tried to demonstrate tricks and... we succeeded (Appendix No. 2).

Having understood the pattern underlying these tricks, we assumed that these tricks can be demonstrated with other cubes in which the sum of opposite faces will have different but equal values. We made cubes in which the sum of opposite faces was equal to 33 (these cubes contained two-digit numbers) (Appendix No. 3). In addition, we came up with another trick of our own - we covered three adjacent faces of the cube with paper and could write the meanings of the faces hidden under them.

We understood well thatThe success of each trick depends on good preparation and training, on the ease of performing the act, accurate calculation, and skillful use of the techniques necessary to perform the trick. Such tricks make a great impression on the audience and captivate them.Even the most amazing “magic” will be boring if the “wizard” silently waves his wand. It’s a completely different matter when an artist smiles and jokes with the audience.The project participants triedwill teachbnot only to talk casually during the performance,but also to react correctly to difficult situations (Thisshould havepromote the development of a sense of humor), which were created for them by adult viewers. As a result, we found out thatfocuswith dicewill be successful only if the audience does not make a mistake in their calculations. Therefore, if there are several spectators, then it is best to use not one, but several or all of them in the focus.Xspectators. Let only one person roll the dice, but each spectator calculates the sum in his heador do it in unison.

We devoted a lot of time to practicing tricks. We drew up a performance script based on a pirate theme (pirates often played dice) (Appendix No. 4), developed words, carefully rehearsed performing tricks in front of a mirror (this helpedunderstand what viewers will see and correct possible errors) (Appendix No. 5).

In addition, to demonstrate the tricks, it was necessary to hone the skills of adding single and double digit numbers, as well as high-speed subtraction of numbers from 8 and 9:

four regular dice give a sum of hidden faces equal to 28 minus the top face (1,2,3,4,5 or 6);

three dice with a sum of opposite faces equal to 33 give the sum 99 minus any number up to 32 (32+1=33);

finding the sum of the faces is a demonstration of the magician's “superpowers.”

Results The implementation of the project “The Mystery of the Dice” included:

The mathematical laws of dice have been determined - the sum of the opposite faces of the dice must be equal.

Props have been created to demonstrate magic tricks.

We developed our own tricks based on the obtained patterns.

A script for the magicians' performance has been developed.

Skills in quickly adding numbers up to 99 and subtracting numbers 1,2,3,4,5,6,7, 8 from 8 and 9 have been developed.

Sources of information used

Wilson M. Complete pocket encyclopedia. Tricks and tricks. - M: Eksmo Publishing House, 2003

Postolaty V.K. Tricks at school and at home. - M.: Sphere shopping center, 2000

Postolaty V.K. Vacation tricks. - M.: Sphere shopping center, 2000

Kordemsky B.A. Mathematical savvy. - M.: “Science”, 1965

Minskin E.M. Games and entertainment in an after-school group: A manual for teachers. - 3rd ed. - M.: Education, 1985

Nikitin B.P. Steps to creativity, or educational games. - 3rd ed., add. - M.: Education, 1990

Video recordings of the School of Tricks programs (Carousel channel) on the Internet.

Appendix No. 1

1. Focus “Guessing the amount”

Focus: The person demonstrating turns his back to the audience, and at this time one of them throws three dice onto the table. The spectator is then asked to add up the three numbers drawn, take any die and add the number on the bottom side to the total just obtained. Then roll the same die again and add the number that comes out to the total again. The demonstrator draws the audience's attention to the fact that he can in no way know which of the three dice was thrown twice, then collects the dice, shakes them in his hand and immediately correctly names the final amount.

Explanation. Before collecting the dice, the person showing adds up the numbers facing up. By adding seven to the resulting sum, he finds the final sum.

2. “Cube and scarf” trick

Focus: The performer brings out in his hands a cube measuring 10x10x10 cm, glued together from cardboard, and shows it to the audience from all sides. And they see that on one side of it five points are drawn in black ink, and the rest of the sides are clean. The magician covers this cube with an opaque scarf, pulls off the scarf and shows the cube again. Now six points are drawn on one of its faces in black ink, and the remaining five faces are blank.

Explanation: The secret to performing this trick from a drawing is that a five and a six are drawn on two adjacent faces of this cube in black ink, and a cardboard flap made of the same material as the cube is glued to the edge of the cube located between these two faces. It certainly closes one or the other facet. Of course, if the performer masters the technique of turning the cube well enough, then the trick can be performed without a scarf. Then the trick looks more effective, but it is more difficult to perform.

3. Focus "Arrangement of cubes"

Focus: The magician gives three cubes, paper, a pen and offers, by randomly arranging the cubes in a row, to create a three-digit number from the number of points on the top edge of each cube. Then three numbers must be added to this number, indicating the number of points on the corresponding lower faces of the cubes. The resulting six-digit number must be divided by 111 and the result reported to the “magician”.

It tells you very quickly what order the cubes were placed in.

Explanation : You need to subtract 7 from the declared quotient and divide the difference by 9. The numbers of the resulting quotient will show the initial arrangement of the cubes.

4. “Tower of cubes” trick

Focus : The magician asks any of the spectators to place several cubes on top of each other. Then asks them if he can see the hidden faces of the cubes. Having received a negative answer, he declares that he can name the sum of these hidden faces and... successfully does so.

Explanation: The sum of the opposite faces of the cubes is 7. This means that the sum of the hidden faces of the cubes is 7 times the number of cubes minus the value of the top face.

5. Trick “Turning a black cube into a white one”

Focus: At the bottom of a plastic container with a black wide lid there is a black cube. The magician shakes the jar sharply and a white cube appears in place of the black cube.

Explanation: The black cube has no bottom edge and a white cube is inserted into it. There is a magnet attached to the top edge of the case cube, and metal to the lid. When shaken sharply, the black cube sticks to the lid, and the white cube falls into the container.

6. Focus “Identical values ​​on the dice - easy!”

Focus: A magician demonstrates a box of dice. All dice have different values. He then closes the box, shakes it, and displays all the cubes with the same values ​​on their faces.

Explanation: The magician arranges the cubes in advance so that one side has the same value of the faces. Then he pushes them with this side towards the wall of the box. After shaking, he turns the box over and the cubes turn out to be the “prepared” side up.

7. Focus “Different Facets”

Focus: The magician demonstrates two cubes held between his fingers. The values ​​of their faces are the same. He turns the cubes and the audience sees different values, then equal again, and then different again.

Explanation: When turning, the magician rotates the cubes unequally, but the spectator does not notice this.

Appendix No. 2

Rehearsing a magic trick with homemade dice

Appendix No. 3

Is it possible to do a trick with these cubes?

The focus works. The law is in effect.

Appendix No. 4

Scenario for magicians performing with dice

"Pirates"

Materials and equipment:

table and tablecloth,

phonogram of music by D. Bodelt for the film “Pirates” Caribbean Sea»,

opaque glass, 4 regular dice,

4 large (simulating regular) dice,

3 cubes, the sum of opposite sides of which is 33, 2 markers, a folder, sheets of paper or a board and chalk,

a paper funnel covering three adjacent faces of the cube, a marker,

3 pirate costumes.

Progress of the event:

On the stage there is an improvised barrel (a disguised stool) or a table covered with a tablecloth. Two pirates come out to the music of D. Bodelt for the movie “Pirates of the Caribbean”. They take out dice and a glass and begin to “play.” When the musical rhythm changes, the Captain's wife comes out.

Captain's lady (menacingly): What are you doing here?

Pirates (in unison): We play dice.

Captain's lady: Are these bones? These are bones!

Snapping his fingers, the pirates take out 4 large dice from under the table and place them on the table.

Captain: Play this!

1st pirate: Easily!

The “Tower of Cubes” trick is demonstrated. The second pirate goes backstage.

Captain: It's really easy. Come on, bring in my special cubes.

To the music, the 2nd pirate brings in 3 cubes with the sum of opposite sides equal to 33. The captain demonstrates a complicated trick “Tower of cubes”.

2nd pirate: Ah, I think I understand everything. And now I personally can predict the number of points on three hidden faces of one cube at once.

Take out a paper corner funnel that covers three adjacent faces of the cube. A trick involving guessing hidden edges is demonstrated.

Captain's lady: Well done!

1st pirate: Talent!

2nd pirate: No, I just love math!

Captain and 1st pirate (in unison): And us too!

They bow to the music and leave the stage.

Appendix No. 5

What will the audience see? Rehearsal in costumes.

“Trembling nuts from a huge tree intoxicate me.
Born of a hurricane, they roll along the groove.
Like soma drink from Mujavat Mountain,
A waking dice appeared to me."

Rig Veda "Hymn of the Player"

If a person tells you that he has never held dice in his hands, this is most likely not true. It all begins... since childhood. Each of us has had board games where, in addition to multi-colored chips, a “special die” was included, but few people think that these are also dice.

The history of the appearance of dice.

Their history is one of the richest and most interesting among games, and its origins lie in more than ancient times, because, according to archaeologists, it was dice that began the path of gambling in the world. Dice are the basis of the Game and its philosophy; it is no coincidence that the word “gambling” itself comes from the Arabic name for this game. When man's task was to survive in the harsh conditions of the cave and the lack of mammoths, Pithecanthropus and others like them used prototypes of dice for magic and fortune telling. So, when you throw the dice during the game, remember that this is an echo of those ancient rituals about calling on the gods to help.

Later, when dice became a “pleasant pastime,” the Greeks, at the suggestion of Sophocles, tried to “appropriate” their invention: while talking about the legendary Troy, he mentioned a certain Palamedes, who invented the game during the siege. But even the Greeks could not agree on the discoverer of the “cubes” and Herodotus, in his chronicles about King Atis, told about the Lydians who played this game. During the Crusades, a popular version was about her Palestinian origin. Thanks to the archaeologists who proved that zara (and this is another name for them) are, perhaps, one of the oldest gaming “artifacts”, known long before the Greeks and even more so the Romans.

Many scientists have repeatedly tried to prove that our ancestors, living on different continents, communicated with each other, and they usually show photographs of the pyramids of Cambodia, Peru and Tenerife, Indian and Indian creativity, household utensils of the tribes of the Dark Continent and Australia. But few people compare bones. But the Aztecs, and the Mayans, and the Papuans of New Guinea, and the cannibals who lived in Central Africa, and the peoples of the North who lived thousands of years ago were no strangers to excitement, and the zaryas helped them a lot in this, and they were made from materials characteristic of a particular area, the “dots” (more correctly, markings) were very different, but the principle was the same - Game and rituals (which is also a kind of game, only for the elite). All over the world, modern Indiana Joneses find bones made from fruit seeds and nut shells, from bones, teeth and animal horns, from stones, and sometimes they are real works of art - the further human civilization developed, the more sophisticated they seemed to become would be banal cubes that can tell a lot about the culture of the people who made them: ivory, bronze, precious and semi-precious stones, crystal and amber and even porcelain were used. It is assumed that they initially became widespread due to their low cost and ease of manufacture, as well as the fact that from one to six it is quite convenient to learn to count.

Methods of playing dice were carved on stones by the Egyptians and written by the Hindus in the Mahabharata 2000 years ago: the legends of Prince Nala and the Pandava brothers tell about the game of zara, its secrets, loss and winning - this is the most quoted of the ancient monuments dedicated to dice.

But much more interesting are several works about a player from the Rigveda, dedicated specifically to the zarams. In “The Gambler's Complaints” where God Savitri gives the instruction: “Do not play dice, but plow your harrow! Find pleasure in your property and its prices are high! Look after your cattle and your wife, you worthless gambler.” In ancient India, the game vibhidaka was widespread, which is described in the “Gambler’s Hymn”: a lot of bones “a flock of them are frolicking, three times fifty” were thrown out of the vessel, and sometimes simply snatched from the heap, and if they could be divided into four, then the player won; if there were extra dice, he lost. But at the same time, the Rig Vedas were very disapproving of this game:

“After all, the bones are strewn with thorns and hooks,
They enslave, they torture, they incinerate,
They give gifts like a child, they again deprive the winner of victory.”
(lane T. Elizarenkova)

Playing dice deprived not only money, but also personal freedom; in particular, the ancient Germans, after making material bets, could put themselves on the line, and in case of loss, becoming the slave of the winner.

And what is characteristic is that for some reason it was the Zariks who were disliked by those in power. Although Julius Caesar was their biggest fan: his phrase “The die is cast” when crossing the Rubicon is directly related to this game, since he was a great admirer of dice and believed in their mystical ability to predict the future, the palm here belongs to the Romans. It was they who issued the first known law on gambling, Lex aleatoria (alea (Latin) - dice). And this despite the fact that in Rome dice were one of the most popular games: Pompey played them at his triumphs, Juvenal, at whose suggestion the law was passed, complained about the too great popularity of dice as an excessively gambling game; It was especially fashionable to play them during the Saturnalia. They played even and odd, throwing dice into a hole in the board or a drawn circle. Various combinations of points on the rolled dice bore the names of gods, heroes, hetaera (the minimum roll of 4 points was called “dog”, the maximum - “Aphrodite”), they were happy and unlucky. This law regulated gladiator fights, sports competitions, social events and games. Alea was banned not only as a game, but also for storage.

Because Roman law was taken as a basis in medieval Europe, it is not surprising that dice were prohibited until the end of the 14th century: laws 1291, 1319 prohibited this game. According to historians, here, again, the Holy Inquisition could not have happened: according to the New Testament, Roman soldiers at the foot of the Holy Cross (the place of execution of Jesus Christ on Calvary) played precisely in them. Although here one can trace the illogicality of the ban: bones are prohibited by Rome for storage, but Roman soldiers play in front of people.

In 1396, an amnesty was declared for the Zars - only the distribution and production of fake bones was prohibited. This game was very popular in wealthy houses. Three dice, denoting the present, past and future, were thrown onto the board, or the dice were used as a fortune-telling game, for example, in France the Christmas game “Goose” was very popular - the dice were thrown on a board with the image of a palm-fingered bird.

In the Middle Ages, the Church, an ardent opponent of games, suddenly discovered that not only nobles played them, but the clergy were no strangers to gambling. Urgent measures were required and Bishop Witold of Cambresia popularized the game “Virtues”. Instead of numbers, virtues were symbolically designated on the sides of the cubes: 1.1.1 - love, 1.1.2 - faith, 1.2.4 - chastity, etc. The victorious clergyman had the right to instruct other monks in virtues. And Pope Sylvester P invented rhythmomachy - a game based on chess, only instead of pieces there were dice with numerical designations on the edges. But nevertheless, in church and near-religious books of that time, dice were described as nothing other than the creation of the devil, in order to win the souls of mortals. The designations on the edges of the zariks are the main enemies of the devil in the Christian religion, against whom Satan acts: one - the devil acts against God, two - against God and the Mother of God, three - against the Trinity. But again, the Apostle Peter, having come to Hell, must beat the juggler at dice, who guards sinners, beats and saves suffering souls. And even despite the new games, and the “history” of the origin of the game, the popularity of dice grew among both secular people and clergy. Even schools have appeared to teach the intricacies of the game. Usually they played with two or three dice, which were thrown onto the table from a barrel, a hand, and even a knight's glove. The most popular game was the game for a large sum of points.

But the Slavs played kostigi and roe, and, unlike the Europeans, most of them were played by the poor. The most popular game was “grain”: before the start of the game, the opponents agreed on which sides of the cubes would be considered winning. After that, small white and black zariks were thrown onto the table, the one who guessed the color won. Like cards, dice games were condemned and severely punished. But Tsar Alexei Mikhailovich allowed playing cards and grains in Siberia, however, the permission lasted exactly a year and was canceled. As usual, the most popular places for games were taverns, taverns and secret tavern baths. The game of grain was more than popular; it had its fans and professional players and sharpers. And in the north of Russia at the end of the 19th century, dice, or in the local dialect " ankles ", were played on Christmastide; the cubes were painted red, black and yellow and were stored for decades, as they were used as payment for forfeits or in card games on Christmastide.

Types of dice

And in Russian prisons and prisons for the game they used a pair of points with “bulls” - that’s what the points on the edges were called, and each combination of points had its own name: 1-1 - goal, 1-2 - three, 2-2 - chikva, 2 -3 - rooster, 5-6 - with a pound, 6-6 - full. And by the way, Russian peasants used bones to divide land plots and agricultural work, and also litigate - in all these matters, exclusively lot played a role.

And the most ancient bones were found in the southern part of modern Iraq: tetrahedral pyramids made of lapis lazuli and ivory in two corners, decorated with semi-precious stones, date back to about 3 thousand years BC. By the way, we owe our usual “cube-shaped cubes” with dot markings, or, to be precise, six-sided cubes with slightly rounded corners, on which the sum of opposite faces always equals seven, as archaeologists say to the Chinese - they used these in 600. BC. The ancient Egyptians, instead of dots, depicted a “bird’s eye” - one of the most famous symbols of Egypt. The Greeks used both cubes and astragals. Astragals are dice with four sides and markings in the form of indentations 1, 3, 4 and 6; four astragals were taken for the game. In Ancient Greece, there were two types of dice: cubes, identical to modern dice (called “barrels”, played with three, later with two) and astragals.

By the way, even now in the game they use not only the cubes with dot markings that are familiar to us. For poker, dice are taken with card symbols from Ace to Nine, and for the game “Crown and Anchor”, dice with a crown, anchor and symbols of four card suits on six sides are taken.

In Europe and the Americas, machine-made dice, or “imperfect” dice with rounded corners at the edges, are purchased for playing at home. And in gambling houses and casinos you will see only perfect dice on the tables: they are made by hand, according to very strict standards, with an error of no more than 0.013 mm. And this clarity is explained quite simply: the ancients proved that if the bone does not have an ideal cubic shape, then the laws of probability will be violated - after all, the loss of different faces will not equally probable. It is no coincidence that the most famous cheating technique is the use of irregularly shaped dice, of which there are only three types: dice with a displaced center of gravity, dice with beveled planes and dice with broken markings. The latter will not allow you to roll certain amounts of points, for example, 2 dice marked 3-3-4-4-5-5 and 1-1-5-5-6-6 will never throw 2, 3, 7 or 12.

And some RPG games use dice with 4, 6, 8, 12, 20, etc. sides. There are even dice with 100 sides - zocchiedrons, invented by Low Zocchi. In role-playing games, the type of die is indicated by the letter "d" (dices) or "k", (dice), followed by the number of sides: for example, d4, d8, d20 dice. There is also d% - a percentage cube in the form of two decahedrons, one of which defines tens, and the other defines units.

In the 21st century, when we talk about dice, we either mean the dice used in dice and board games, or we mean games involving dice.

The most famous games that use dice

There are different types of dice games and they differ in equipment (the number of points, the ability to use chips, different ways recording the results), the goals of the game (the one who scores the maximum or minimum number of points wins, or throws out certain combinations of numbers together or in order, or, alternatively, collects all the cubes or, conversely, is left without them), there are games with strict number of players - in general, there are a lot of options and they all have one or another historical roots.

The earliest sign of victory in the history of the game is the highest number of points rolled. Now you can feel like a distant descendant of the Roman patricians by playing “Pig”, “Chicago”, “Lay Down Dead”. And if you believe in the absolute favor of Fortune, then you can take a chance in “Indian Dice”, “Baiburt” or “General” - here your winnings will depend only on the successful combination of the dropped faces. Do you like roulette? You can play “Crown and Anchor”, “Gran Hazard” or “Under and Over the Family” - these games are based on the principle of betting. Are you going to a large group of gambling friends for the weekend? Offer them “Hazard” or “Craps” - time is important here, since the sequence of dropped combinations matters for victory. And for fans of accurate counting, lotto and Sudoku, “Martinetti” is suitable - the drawn numbers will need to be checked against the table and “Help your neighbor” - here you will need to check the numbers assigned to the players.

Games that use not only dice, but also special chips, checkers, which move along the board in accordance with the fallen sides, are now gaining increasing popularity. This is the well-known backgammon with its varieties: short and long backgammon, khachapuri and gulbar, and, of course, children's board games and lotto with dice, where the advancement of chips depends on the number of points on the edge. And the game “Aces” is notable for the fact that the treasures in it are both dice and chips at the same time.

Craps

In any case, all games have the same principle: the roll of the dice determines the winner or loser.
In the world's casinos, the most popular game is craps, which is played with six-sided dice. This game has been known since approximately the 18th century and, according to one version, was invented in New Orleans. African Americans.
The number of craps players, as well as their entry into and exit from the game, is not limited by the rules. At the same time, the throwing order is clearly regulated: two dice must be thrown so that, having hit the opposite edge of the table, they stop on the table. At the first stage of the game (there are two in total), the player must make one throw, and according to the results of the “crepe” (points): if he threw 2, 3 or 12, he is considered a loser, with 7 or 11 points he is considered a winner, and all other combinations ( 4 – 6 and 8 – 10) indicate that the player must repeat the dropped points on the second round. In the next stage, the player rolls until he repeats his points, which means a win, or until he rolls a 7, which means a loss.

In craps, players can bet on any combination of dice, and there are many betting options

Dice Poker

Classic poker served as the ancestor of a number of games with dice, and some games require standard dice, others require special poker dice, where the six sides of the dice have images of nine, ten, figures and ace, and others use a combination of both . Poker with dice is closest to card poker; it requires not only luck, but also the ability to quickly calculate the situation and combine decisions.

Bets are placed before the game, the bank belongs to the winner. Players throw five zariks and, according to the rules of poker, count the combination that comes out: four of a kind, straight, full, etc. The rules allow for an additional throw by prior agreement between the players (by analogy with the ability to discard unnecessary cards in poker and buy new ones in return): the player can, leaving the dice he needs in the same position, re-roll the rest. After the throw, each player can either be satisfied with the results or reroll from one to five dice. After the second throw, it is possible to reroll all the dice except those that remained on the table during the first reroll. The final third throw does not grant the right to re-roll. The winner will be the owner of the highest combination (as in poker): poker, four of a kind, full house, three of a kind, two pair, pair, or, if none have been collected, the player with the most points. The points scored are also taken into account when the opponents' combinations coincide (points are counted on the winnings included in it), and the combinations can be complex: a full house of 3 fives and 2 twos (3x5+2x2-19) is higher than a full house of 3 threes and 2 sixes (3x3+2x6=21). If the combinations and points are absolutely identical, an additional batch of players whose results match is announced.

The player who threw second in the previous game, or sitting to the left of the starter, starts the next game. It is prohibited to interrupt the game in the middle of the circle, when the right of the first move returns to the person who started the entire game.

Game at dawn - Sic-bo (Sic Wo)

The ancient Chinese game Sic Bo, also known as Grand Hazard, is also popular in casinos.
They play with three dice, bets are placed on the numbers of the sides that will appear in the game. The number of players is limited by the size of the gaming table and the space around it. Like other casino games, Sic-bo is played with perfect rounds: a perfectly regular cubic shape with dotted markings. The principle of placing bets is reminiscent of roulette: chips are placed by players on sectors of the playing field according to the types of bets. The dealer launches the popper (from the English pop - clap), a special device that throws dice. The name arose due to the fact that, due to electrical impulses, the bones are thrown upward on a round membrane, and when they hit the dome, a characteristic pop is heard. The device turns off after the announcement of the end of accepting bets, the dome is removed and players see the drawn numbers. Additionally, the dealer calls them out loud. Then the winnings are paid, the chips are removed and bets are accepted on a new game.

As a rule, the casino administration sets the bet sizes independently, which can be seen on the table where they play Sic Bo: a special sign indicates the minimum and maximum bets for all types of bets.

There are 7 types of bets in Sic Wo (Sic Bo). A bet on one number, with payment in a 1:1 ratio. Moreover, if the number you bet on appears on two dice at once, then your bet will be paid twice, and if on all three dice, it will be paid twelve times. Domino bet - involves 15 variants of number combinations, two selected different numbers will be winning. Payment bet 6:1. A bet on a combination of two numbers or a bet on a specific doublet. If your bet wins, you will receive payment at a ratio of 11:1; if your number appears on 3 dice, your bet will already be paid thirty times. A bet on a combination of three identical numbers or on a specific triple will pay out at a ratio of 180:1 if the same number is shown on all three dice. A bet on an arbitrary triplet implies that any triplet that lands will be a winner, but the player does not choose the number; the payment will be in the ratio 31:1. The next bet, over or under, is divided into two subtypes: either the player bets on a “large amount” from 11 to 17 or on a “small amount” from 4 to 10. If the sum of the points of the three dice falls within the player’s range, then his winnings will be calculated at ratio 1:1, the main thing is that the triplet does not fall out, in which the bet loses. And finally, a bet on a certain amount of numbers. There are 14 of them for all amounts from 4 to 17. The amount you specify must match the sum of the numbers on all dice, the winnings are determined by the selected amount.

Backgammon is the most famous and respected game using dice.

One of the most popular dice games is backgammon. It was from them that another name for the cubes came - “zary”. It is approximately known that backgammon has been played for more than 5,000 years; an analogue of this game was found in the tomb of Tutankhamun, and the oldest backgammon board dates back to about 3,000 BC. The Persians considered this game mystical, predicted destinies from it, correlated the game board with the sky, and the movement of checkers with the movement of the stars. Everything on the board is a multiple of six and is related to the passage of time: 12 months - 12 board points, 24 hours in a day - 23 points, 4 seasons - 4 parts of the board, 30 checkers - the number of lunar and moonless nights in a month. The sum of points on opposite sides of the dice is seven - the number of planets known at that time that influenced everything good and bad in the world.

Historians argue about the ancestor country of this game. According to one legend, the Indian ruler sent chess to the Persian ruler, believing that no one would understand how to play this complex game. In response, the Persian sage Büzürkmehr, who immediately unraveled the secret of chess, sent them Nard Takhe “Battle on a wooden board,” the principle of which the Indians had been unraveling for 12 years. Another possible origin of the name is from the Indian "nard" - a plant from which incense and aromatic oils were made. Backgammon is also a name for a special board that serves as a playing field.

Backgammon is a game with many names: in Spain - tablero, in Italy - tavola reale, in Ottoman Empire– tavla – all these words mean “ board game" But the Greeks, French and English gave backgammon their own names, διαγραμισμος, trick-track and backgammon, respectively.

The spread of backgammon, then called backgammon (presumably because of the sound of bones hitting a wooden board), in Western Europe begins with the end crusades HP of the century. In the Middle Ages, only the game of kings was called backgammon - it was the privilege of the highest aristocracy.

The original rules of this game have almost been lost in history, mainly now we play backgammon, the rules of which were established in the mid-18th century by Edmond Hoyle in Great Britain, known as “Short Backgammon”. This name arose as a contrast to the eastern “Long Backgammon”. Another name for short backgammon is Backgammon, which again does not have an exact explanation, but the most popular version is that this name comes from the English “back” and “game”, and contained the basic principle of the game: the opponent’s beaten checker is returned back. Another possible origin of this name is related to the Gaulish language: “Baec” (small) and “Gammit” (battle).

Backgammon is played on a special board - a playing field - of a rectangular shape. The board consists of 24 points, 12 on each of the two opposite sides. Externally, they are usually narrow isosceles triangles, the base of which lies on the side, and the height reaches the middle of the board. The points are numbered from 1 to 24 for each player, most often the even points are colored one color, and the odd points another. The player's house consists of six points located in a row in one of the corners of the board; its location is determined by the rules. Some boards have special areas on the sides intended for placing checkers behind the board. On the sides of the board, areas can be allocated for placing checkers behind the board. In the middle of the board there is a bar - a vertical strip that divides the board. If the game follows the rules where you can hit the opponent’s checkers, then they are placed on the bar.
Each player has his own set of checkers of the same color - usually there are 15 of them (possibly less, depending on the rules). And the dawn itself. At least one pair, but maybe two, for each player, as well as barrels for mixing the dice. If the game is played on a bet, then on the playing field there may also be a “doubling cube”, on the sides of which the numbers 2, 4, 8, 16, 32, 64 are printed - it is convenient to take into account the increase in bets.

Regardless of the many options for playing backgammon, which differ from each other in the rules of moves, bets, and the initial position of the chips, backgammon is united by the general rules of the game. Players take turns, checkers move in a circle, the direction of their movement is fixed in a particular game, but may vary in other versions. The first move is determined by lot: each player throws one die, the winner starts the game.
Before each turn, the player rolls two zara. The dice are thrown onto a free space on the board on one side of the bar - this way the possible moves are determined. Throws are strictly limited by the rules: if at least one of the dice flies off the board, the dice end up on opposite sides of the bar, the dice falls on a checker or stands on an edge (on the edge of the board or on a checker), then the throw does not count and is repeated. In one throw, from 1 to 4 movements of the checker are possible. In each of them, the player moves the checker by the number of points that fell on one of the dice. If a double is rolled, the points are doubled and the player makes 4 moves, while he must use the maximum possible number of points. Each movement of a checker is made for the full number of points rolled on the dice. Moreover, if there are no available movements for the dropped number of points, then the player skips a move, but if it is possible to move a checker, then the player is obliged to do so, even if this worsens his playing position. If there are two options for a move, where one involves using the points of only one of the dice, and the other - both, the player must choose the last option. In the event that it is possible to move one of two checkers, when the move of one checker excludes the possibility of moving the other, the player must make a move by a greater number of points.
After all the player's checkers have reached their home, making a circle around the board, the player begins to place them behind the board. A checker is placed on the board when the number of the point on which it stands coincides with the number of points that fell on one of the coins. If all the placed checkers are closer than the rolled number, then the checker from the point with the highest number is placed on the board.

In backgammon there is always a winner - the one who is the first to remove his checkers from the board. He gets one point. In the case of Mars, when the winner has put all his checkers overboard and the loser has none, then the first one gets two points. Three points are awarded to the winner who has removed all the checkers from the board, while his opponent has not removed any and one of his checkers is in the winner’s house or on board - this is called coke. If the game is played on a bet, then for a regular victory one bet is paid, for Mars - doubled, for coke - tripled. Bets in backgammon can be increased at the request of the player before his move. Before the first move, each player has this right. Refusal to raise bets entails an admission of loss. When a player raises a bet, he takes the doubling cube for himself and sets it up with the side that shows the coefficient of the bet increase. Today, backgammon is so popular that international tournaments are held in it.

Less Popular Dice Games

Another dice game called Under and Over Seven is a variation of Sic Bo and is played with six-sided dice. The gaming table has three fields on which bets are placed. The game is against the bank. The banker throws two dice and the winner is immediately determined. The winner gets paid 1:1 for winning bets in the “Under 7” and “Over 7” fields, and 5:1 for winning in the “7” field.
Under 7 7 Over 7
2-3-4-5-6 7 8-9-10-11-12
1 to 1 5 to 1 1 to 1

Types of Fraud and Illegal Dice Manipulations

Naturally, such an ancient game could not help but attract the attention of scammers: in the tombs of Ancient Egypt, zars were found, on which cheaters clearly worked, archaeologists found fraudulent bones in the burials of the Middle East and the American continents.

If the edges are deviated from the correct shape, the nature of the game will change, and the probability of equal numbers will disappear. Unscrupulous players use dice with beveled surfaces, a displaced center of gravity, incorrect markings, magnets, and mercury in the game. If you hold the cube in the desired position for a few moments, the mercury will move and the cube will fall on the side with which it was held.

The numbers rolled on the marked dice do not follow the correct probability distribution. The most common type used by scammers is sawn bones. Typically, one or more sides of such bones are sawed, which means that the cube will more often fall out on the wide sides. Equipped bones are zara, regular in shape, but on one side, near the surface, a hole is drilled into which a lead sinker is placed. The hole is sealed and the die is more likely to fall out on the side opposite the weighted one.

It happens that the shape of bones is changed: two sides are made slightly concave, and two are made convex. When thrown, such a cube will fall on even sides. You can make the bone slightly elongated, then it will fall on the longer side. Another change to the zar is to round the edges of some of the faces, which will prevent it from falling on them, and making the edges of the face protruding will prevent the bone from rolling.

Another option for cheating is repeating the numbers on the opposite side; professional gamblers and scammers introduce them into the game during the game, and since it is impossible to see all the sides of the dice at the same time, novice players may not notice this.

Magnetic dice can also be used in unfair games. They contain a grid of thin steel wire or steel discs that are inserted into holes that represent the glasses. Usually, 4 edges are filled with metal, which are opposite to those that should fall out according to the scammers’ plan. An electromagnet is inserted into the table, and when it is turned on, the metal edges are attracted.

There are many stories about the “lucky ones of Fortune” who can throw out any combination, but in reality, professional dice players, with long-term training, can perfect their throwing technique, which can significantly increase the probability of a given combination appearing.

If, when throwing, a rotational impulse is given to the dice parallel to the table, at the moment of throwing the dice is with the desired side facing up, and having fallen, it will continue to rotate, preventing it from turning over. You can “roll” the bone in a given plane - the two sides located on the side will then have less chance of falling out. If the game is played on a sufficiently slippery surface, then you can force the dice to slide in the desired direction: one of the dice is lightly held with your little finger, as a result, it will slide rather than roll and will retain a pre-selected number on the top face.

It is very difficult to expose scammers who have the ABILITY to throw dice. Thus, the “Greek” throw, when the lower dice is pressed in the desired direction by the upper one, is practically unnoticeable, and the most talented sharpers can change dice during a throw in less than a second, hiding the false dice inside their palms.

Even a super-professional cannot feel absolute confidence that the game is being played fairly. If a player doubts the integrity of his opponents, then he needs to pay attention to: the numbering of the faces of the cube; that the sum of points on opposite sides is always equal to 7; all faces are equal in area and identical in shape, texture, plane, the tops and edges of the edges have the correct shape, if there are roundnesses, then they are the same at all angles; the gaps between two cubes pressed against each other should be the same; The markings on the cubes are made at the same distance from each other and to the same depth. Bones with a displaced center of gravity can be identified by a rotation test between the fingers (or, if conditions permit, when immersed in liquid).

The most reliable way to avoid ending up at the same table with scammers is to be smart about choosing a company and place to play. The integrity of your partners and the reliable reputation of the gambling establishment guarantees you higher security than if you examine the dice with a magnifying glass after each throw.

Dice in astrology

And zar lovers will also be interested to know that astrologers advise choosing dice in accordance with your zodiac sign. Aries are recommended classic colors - black and white; for variety you can take bright red, orange, blue, lilac, crimson and anything shiny. For Taurus, cubes of nature flowers are suitable: green grass, pink sunset, blue sky, brown bulls. And, of course, no red! Gemini will have luck with purple dice, but it is not possible to use light yellow and gray dice. Cancers will be lucky with pale gold and silver, light green and purple, lilac. Luxury-loving Leos will appreciate purple, gold, orange, scarlet and black bones. And unassuming Virgos will be enriched by grey, beige, dark blue shades, as well as any shades of green. Balanced Libra needs dark blue, sea green and pastel colors, while bright Scorpios are promised victory by bright cubes: rich yellow, dark red, scarlet, crimson. Sagittarius will be lucky with blue, light blue, violet, crimson bones, and Capricorns should never choose light bones, for them the best are dark green, black, ash gray, blue, pale yellow, dark brown and all dark tones . Aquarius will enrich himself when playing with dark blue, sapphire, purple, blue-green and purple cubes, unless, of course, he is opposed by Pisces with white, emerald, light lilac, purple, violet, blue, purple or steel zariks.

If you like tattoos, then dice are a symbol of good luck and success in all matters, because the number of union and balance – 6 – is firmly associated with them.

Buying dice and the criteria you need to pay attention to

The main part of dice games is based on calculating the mathematical probability of the appearance of any sum of numbers on the sides of the dice when throwing dice, while the theory of probability always leaves a chance for a huge jackpot. Total probability is subject to the law of combinations and permutations, but it is now determined by simple mathematics.
They threw dice and threw them into a circle, played and told fortunes with them. They call reverent attitude, as connectors with higher powers - and no wonder, with such a story! It is in the bones that the inconstancy of Fortune is visible, which instantly denies its favor, and then elevates and enriches. Despite numerous prohibitions, dice games have survived to this day and are popular both in ordinary homes and in casinos.

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1 Final test for Foxford courses: Project and research activities. GEF 2. Mark the correct judgments. 1. The research work must include an introduction that presents basic information from the author’s chosen field of knowledge; the introduction can be an independent abstract work. 2. In the abstract work, the student is required to provide a comparative analysis of selected literary sources, their origin and reliability. 3. The purpose of the project work should be aimed at obtaining new information (quantitative, qualitative) about the selected object. 4. The objectives of the research work should include the development of criteria for the practical significance of the results expected to be obtained in the work. 5. The object of research is actually existing in reality, the subject of research is a property (sign, feature) of the object. 3. In what sections of the Federal state standard Basic general education mentions teaching and research activities? 1. The program for the development of universal educational activities and the program of education and socialization. 2. Subject results studying the subject area “Natural Science Subjects” and the conditions for the implementation of the main educational program. 3. Subject results of studying the subject area “Technology” and the program for the development of universal educational activities. 4. Conditions for the implementation of the main educational program and program correctional work. 5. Description of personal educational results mastering the main educational program and the target section of the main educational program. 4. Universal educational activities include the following types: regulatory, reflective, activity-based 2. operational, motivational, personal 3. regulatory, communicative, cognitive, personal 4. communicative, motivational, regulatory 5. abrasive, gender, cognitive

2 5. The concept for the development of additional education involves: 1. Expanding the range of additional general education programs 2. Increasing funding for additional education organizations 3. Compliance with fire and electrical safety requirements 4. Development of partnerships with organizations in science, business, sports, etc. 5. Development standard of additional education 6. The main objective of the program for the development of universal educational activities is: 1. Achievement by students of high meta-subject and personal educational results 2. Improving the quality of educational work; the effectiveness of socialization and development of communication skills of students 3. Professional guidance of students in the field of professions in demand on the labor market 4. Ensuring the dynamics of individual achievements of students in the process of mastering the basic general education program of basic general education 7. Criteria for assessing the research work of senior students should include: 1. Scientific novelty work 2. Practical significance of the work 3. Relevance (interest) of the work for the author 4. Relevance of the work for the development of the chosen field of scientific knowledge 5. The author’s knowledge of the terminological apparatus of the chosen field 8. Extracurricular activities are organized: 1. In areas of personal development (spiritual, moral, physical education, sports and health, social, general intellectual, general cultural) 2. Only for additional general developmental programs 3. Only for the purpose of improving student performance in subjects and working on mistakes made during tests

3 4. In the following forms: clubs, art studios, sports clubs and sections, youth organizations, local history work, scientific and practical conferences, school scientific societies, Olympiads 5. In administrative and other premises equipped with the necessary equipment, including for organizing the educational process with disabled children and children with disabilities health 9. Select the correct pairs of object - subject of research. 1. Object: Spruce growing in Bitsevsky Park. Subject: The amount of annual growth of spruce depending on the year. 2. Object: Baroque architecture. Subject: Winter Palace in St. Petersburg. 3. Object: Volga River basin. Subject: Rybinsk Reservoir. 4. Object: Islamic State, banned in Russia. Subject: Methods of recruiting supporters of the Islamic State. 5. Object: Creating a model of the T-70 tank Subject: Methods for gluing together parts of the model. 6. Object: Environmental situation in Sokolniki. Subject: Creation of environmental teams to clean up the area. 10. Mark correctly formulated (from a methodological point of view) research hypotheses that are not obvious and can be confirmed or refuted during independent student research. 1. The air temperature in the surface layer of the atmosphere decreases at night and rises during the day. 2. An increase in the number of motor vehicles leads to increased air pollution from exhaust gases. 3. An increase in the number of tests in physics in the 10th grade leads to an increase in academic performance. 4. If you turn on classical music when pea seeds germinate, then their germination occurs faster than if you turn on rock music. 5. A manned flight to Saturn is possible subject to the invention of a photon engine. 6. Sociological surveys of 7th grade students do not provide objective information about their level of knowledge.

4 11. The work determines the influence of the talk show “Evening Urgant” on the political views and value preferences of schoolchildren in the city of Kolifeevka using the method of questioning and participant pedagogical observation. 1. Object: LG 42LB677V TV. Subject: features of the color scheme of the display of Ivan Andreevich Urgant on a television of this type. Goal: identifying mechanisms psychological impact Ivan Andreevich Urgant to the audience. Hypothesis: if you don’t watch TV and do your homework, your Unified State Exam results will be better. Methodology: TV screen photometry. 2. Object: Ivan Andreevich Urgant. Subject: students from classes in the Zyablikovo district. Purpose: to identify preferences in spending evening time in families in the Zyablikovo district. Hypothesis: The talk show “Evening Urgant” will be closed within one year. Methodology: sociological survey 7th grade students. 3. Object: students living in the Zyablikovo area. Subject: worldview of class students. Goal: to identify the impact of the “Evening Urgant” program on students’ value attitudes. Hypothesis: watching the program leads to dispersion of motivational attitudes towards continuing education and obtaining a profession in the field of intellectual professions. Methodology: survey of class students. 4. Object: value attitudes of students in classes in the Zyablikovo district. Subject: dynamics of preferences of class students as a result of regular viewing of the “Evening Urgant” program for 3 months. Hypothesis: As a result of watching the program, students' sleep is disturbed.

5 Methodology: longitudinal test studies of students. 12. Find a Soldier Read the text of work 1 at the link. Mark the correct answers 1. Project work, with elements of research 2. Research work 3. Abstract work 4. In conclusion, conclusions are presented that do not fully correspond to the assigned tasks 5. References to literary sources 1-2 are formatted correctly, and 7 and 12 are incorrect 6. The content of the work does not fully meet the stated goals and objectives 13. Read the text of work 2 at the link. Also check out the seven reviews for this work: 1, 2, 3, 4, 5, 6, 7. Rate the quality of the reviews for the work “Mysteriousness in the behavior of three dice” and note the presence of the following characteristics in the 7 reviews presented: The presence of a common characteristics of the work Review 1 Review 2 Review 3 Review 4 Review 5 Review 6 Review 7 Availability of meaningful analysis of the main sections of the work Review 1 Review 2 Review 3

6 Review 4 Review 5 Review 6 Review 7 Availability of a personal appeal to the author, his motivation to continue the work Review 1 Review 2 Review 3 Review 4 Review 5 Review 6 Review 7 Availability of meaningful recommendations for continuing the work Review 1 Review 3 Review 4 Review 5 Review 6 Review 7 Presence of speech and stylistic errors, violation of the logic of sentence construction Review 1 Review 2 Review 3

7 Review 4 Review 5 Review 6 Review 7 Excessive attention to the formal parameters of the work Review 1 Review 2 Review 3 Review 4 Review 5 Review 6 Review 7 The work is not a review, but an annotation of the work Review 1 Review 2 Review 3 Review 4 Review 5 Review 6 Review Read the texts of eight works: 1, 2, 3, 4, 5, 6, 7, 8. Evaluate the quality of the works and note the presence of the following characteristics in the 8 submitted works: Research

8 Work 2 Work 5 Abstract Work 2 Work 5 Project Work 2 Work 5

9 Availability of justification of the topic, introduction to the research problems Work 2 Work 5 Availability of a set structure of the work (introduction, purpose and objectives, methods, obtaining your own data, their analysis, conclusion (conclusions) Work 2 Work 5 Compliance with the goal, objectives, work plan, results Job 2

10 Work 5 Availability of methodology independent work Work 2 Work 5 Availability of independently obtained data Work 2 Work Match the organizers and goals of the conference. Scientific institution - Popularization of the scientific field among young people

11 Company producing intellectual products - Training of qualified users who in the future will provide the necessary demand for the university’s products University - Attracting applicants, popularizing activities General education institution - Inclusion of its students in the system of interregional and interdepartmental relations Educational authorities - Fact of participation in the system of higher education events level 16. Imagine in in the right order structure of research and design work. Research work 1 justification of the topic 2 setting goals and objectives 3 hypothesis 4 methodology

12 5 - own data 6 analysis and conclusions Project work 1 problem statement 2 definition of performance criteria 3 creation of a concept and forecasting consequences 4 - determination of available resources 5 implementation plan 6 implementation of the plan and adjustments 7 assessment of efficiency and effectiveness 17. The founder of the project method in education is : 1. L.N. Tolstoy

13 2. J. Dewey 3. S.T.Shatsky 4. N.K.Krupskaya 5. K.D.Ushinsky 6. J.J.Rousso 7. Y.A.Komnesky 18. Benefits for enrollment in Universities of the Russian Federation used by: 1. Winners and prize-winners of the All-Russian Olympiad for schoolchildren. 2. Winners of events included in the List of Olympiads and other intellectual and (or) creative competitions, events aimed at developing intellectual and creative abilities, abilities for physical education and sports, interest in scientific (research), creative, physical education sports activities, as well as the promotion of scientific knowledge, creative and sporting achievements Ministry of Education and Science of Russia. 3. Winners of Olympiads included in the list of Olympiads for schoolchildren of the Russian Ministry of Education and Science. 4. Laureates of the Government of the Russian Federation Prizes to support talented youth. 19. Which of the following actions of a psychologist are related to such an area of ​​work as “designing and diagnosing the effectiveness of the quality of the educational process based on the research activities of students”? 1. Diagnostics of students’ internal development ( psychological picture student) 2. Participation in the examination of the process of implementing educational activities and its productivity (result) 3. Group forms of work to support the effectiveness of student participation in the educational process 20. In order to diagnose the professional position of teachers - implementers of the educational and research approach, it is advisable to use the following methods: 1. Methodology for evaluating design and research work (FOPIR) CPS. (D.Treffinger) 2. BASE technique (A.L. Wenger and co-authors)

14 3. Questionnaire “Personal motivation of the leader of student research activities” (A.S. Obukhov, A.V. Leontovich) 4. Creativity Test (Torrance Test of Creative Thinking) 21. K psychological mechanisms, allowing students to carry out research activities include: 1. Divergent and convergent thinking 2. Search activity 3. Situation of uncertainty

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