Problems with dice as a means of implementing a humanitarian orientation in teaching mathematics. Problems with dice as a means of implementing a humanitarian orientation in teaching mathematics Description of a dice

History of dice

Dice is a rather ancient game, but the history of its origin is still unknown.

Sophocles gave the palm in this matter to a Greek named Palamedes, who invented this game during the siege of Troy. Herodotus was sure that bones were invented by the Lydians during the reign of Atys. Archaeologists, based on the scientific data obtained, refute these hypotheses, since the bones that were found during the excavations date back to an earlier period than the period of the life of Palamedes and Atys. In ancient times, bones belonged to the category of magical amulets, on which they guessed or predicted the future. Today, many peoples have preserved the tradition of divination on the bones.

Quast Peter. Soldiers playing dice (1643)

Experts assure that the first dice were made from the hoof joints of wild, then domestic animals, which were called "grandmothers". They were not symmetrical, and each surface had its own individual characteristics.

However, our ancestors also used other material to obtain "magic" bones. They used plum, apricot and peach pits, large seeds of various plants, deer antlers, smooth stones, ceramics, teeth of predatory animals and rodents. But the main material for the bones was still supplied by wild animals. These were bulls, elks, deer, caribou. Among the ancient Greeks, ivory was very popular, as well as bronze, agate, crystal, ceramic, jet and plaster products.

The game of dice was often accompanied by fraud. This is evidenced by records in ancient writings. In the sixth century BC, almost exact copies of modern bones were used in China. They had a similar layout and cubic configuration. It is these playing objects dating back to the sixth century BC that were found by archaeologists during excavations carried out in the Celestial Republic. Earlier drawings of bones made on stones were discovered by researchers in Egypt. In the Indian written monument called "Mahabharata" there are also lines about dice.

Thus, the game of dice can be safely called the oldest gambling entertainment. Nowadays, there are many games that can be played with dice.

Modern dice

Modern dice, often referred to as dice, are usually made of plastic and are divided into two groups.

The first group includes products of the highest quality, made by hand. These dice are purchased by casinos for playing craps.

The second group includes bones made on machines. They are suitable for general use.

Bones of the highest quality are cut by craftsmen with a special tool from an extruded plastic rod. Further, tiny holes are made on the faces, the depth of which is several millimeters. Paint is poured into these holes, the weight of which is equal to the weight of the removed plastic. Then the bones are polished until a perfectly smooth and even surface is obtained. Such products are called "smooth-point".

A gambling establishment usually has smooth-pointed dice made of red, transparent plastic. The set consists of 5 bones. For traditional dice from a gambling house, it is equal to two centimeters. The ribs of the products are of two types - blade and feather. Blade edges are very sharp. Feather - a little sharpened. All sets of dice are supplied with the logo of the gambling establishment for which they were intended. In addition to the monogram on the bones, there are serial numbers. They are specially coded to prevent fraud. In the casino, in addition to traditional hexagonal products, there are dice with four, five and eight faces of various designs. Products with concave holes are almost never found today.

Dice scam

In excavated burials on all continents, there are dice made specifically for foul play. They are in the shape of an irregular cube. As a result, the longest edge most often falls out. Irregular shape is achieved by grinding one edge. Another cube can be transformed into a parallelepiped. These irregular bones were nicknamed "blanks". It is considered an attribute of a cheating game, and, as a rule, belongs to scammers.

A modern blank cannot be distinguished externally from an ordinary bone, since it has the shape of a perfect cube. But in a blank, one or more faces have additional weight. Such faces fall out more often than others.

Another trick is to duplicate the faces - some are quite a lot, others are completely absent. As a result, some numbers will fall out too often, while others almost never. These bones are called "tops and bottoms". Such products are used by fraudsters with extensive experience and rather dexterous hands. An ordinary player often does not notice that his partner is playing a dishonest game.

Some cheaters train hard with normal bones. As a result, they manage to throw out the required combinations. For this purpose, the dice are thrown in a special way, allowing one or two products to rotate in a vertical plane and fall on the required face.

Other crooks choose a soft surface such as a blanket or coat. On such a surface, the bone rolls like a coil. As a result, the side faces almost do not fall out, which leads to combinations that are undesirable for the opponent.

Dice development

A regular dice has six faces, all of the same size. The arrangement of points on the cube, forming numbers along the faces, is not accidental.

According to the rules, the sum of the dots on opposite sides of a die must always equal seven.

Dice Probability Theory

The dice is thrown once

When dice are rolled, finding the probability is not difficult. If we assume that we have the correct dice, without the various tricks described above, then the probability of falling out of each of its faces is equal to:

1 of 6
in fractional form: 1/6
decimal: 0.1666666666666667

The dice is thrown 2 times

If two dice are thrown, you can find the probability of getting the desired combination by multiplying the probabilities of getting the desired face on each of the dice:

1/6 × 1/6 = 1/36

In other words, the probability will be equal to 1 out of 36. 36 is the number of options that can be obtained when the desired number falls out, let's summarize all these options in a table and calculate the sum in it that forms the faces of both cubes.

The probability of getting the required amount when throwing two dice:

• Yakovleva Tatyana Petrovna, Associate Professor of the Department of Mathematics and Physics, Vitus Bering Kamchatka State University, Petropavlovsk-Kamchatsky, Kamchatka Territory

Sections: Mathematics , Extracurricular work

Exercises that stimulate the internal energy of the brain, stimulate the play of forces
“mental muscles” is the solution of tasks for ingenuity, sharpness.

Sukhomlinsky V.A.

Humanitarian orientation today expands the content of mathematical education. It not only increases interest in the subject, as is commonly believed, but also develops a personality in students, activates their natural abilities, and creates conditions for self-development. Therefore, the humanitarian aspect in teaching mathematics contributes to: introducing students to spiritual culture, creative activity; arming them with heuristic techniques and methods of scientific research; creating conditions that encourage the student to be active and ensure his participation in it. Human thinking mainly consists of setting and solving problems. To paraphrase Descartes, we can say: to live means to set and solve problems. And while a person solves problems, he lives.

Problems with dice can be considered as a means of implementing a humanitarian orientation in teaching mathematics. They contribute to: the development of spatial imagination; the formation of skills to mentally represent the various positions of the object and changes in its position depending on different points of reference and the ability to fix this representation in the image; teaching the rationale of geometric facts; development of design abilities, modeling; development of research skills.

Task 1. Carefully consider the figures in the top row:

What figure instead of the sign “?” from the bottom row you need to put?

Answer: "b".

Task 2. 1 dot is drawn on the front face of the cube, 2 on the back, 3 on the top, 6 on the bottom, 5 on the right, and 4 on the left. What is the largest number of dots that can be seen simultaneously by turning this cube in your hands?

Answer: 13 points.

Problem 3. On a dice, the total number of points on any two opposite faces is 7. Kolya glued a column of 6 such dice and counted the total number of points on all outer faces. What is the biggest number he could get?

Answer: number 96.

Task 4. Roll the cube shown in the figure in 6 moves so that it gets to the 7th square and at the same time its edge with 6 points is on top. And each turn, you can move the die a quarter turn up, down, left, or right, but not diagonally.

Problem 5. You see in the picture how the king of the Land of Puzzles plays with a dice with a savage.

This is an unusual game. In it, one player, throwing a die, adds the number that has fallen on the top face with any number on one of the four side faces. And his opponent adds up all the other numbers on the three side faces. The number on the bottom face is ignored. It is a simple game, although mathematicians disagree as to exactly what advantage the thrower of the dice has over his opponent. At the moment, the savage throws a die, as a result of this throw, the king is ahead of him by 5 points. Tell me, what number should have fallen on the dice?

Princess Riddle keeps score of the savage's winnings. If this number is translated into the Bungaloz system familiar to the savage, then it will turn out to be even larger. The savages of Bungalosia, as we well know, have only three fingers on each hand, so they are accustomed to the six-digit system. From this arises a curious problem in the field of elementary arithmetic: we ask our readers to convert the number 109,778 into the Bungaloz system, so that the savage will know how many gold coins he has won.

Solution. The die should fall one up. If you add 4 on the side face here, then this gives a sum equal to 5. The sum of the remaining numbers on the side faces (5, 2 and 3) is 10, which gives the other player a 5-point advantage. In the hexadecimal system, the number 109778 is written 2204122. The number on the right represents ones, the next number gives the number of sixes, the third number from the right indicates the number of "thirty-sixes", the fourth number shows the number of "servings" of 216, etc. This system is based on powers of 6 instead of powers of 10, as is the case in the decimal number system.

Answer: 2204122.

Problem 6. There are 6 dots on the bottom side of the cube, 4 on the left, and 2 on the back. What is the largest number of dots that can be seen at the same time by turning this cube in your hands?

Answer: 13 points.

Problem 7. Here is a die: a die with points from 1 to 6 marked on its faces.

Peter bets that if you throw a die four times in a row, then for all four times the die will certainly fall once with a single point up. Vladimir, on the other hand, claims that a single point will either not fall out at all during four throws, or it will fall out more than once. Which one is more likely to win?

Solution. With four throws, the number of all possible positions of the dice is 6? 6? 6? 6 = 1296. Suppose that the first throw has already taken place, and a single point has fallen. Then, during the next three throws, the number of all possible positions favorable for Peter, that is, the loss of any points, except for a single one, is 5? 5 ? 5 = 125. In the same way, 125 favorable arrangements for Peter are possible if a single point falls only on the second, only on the third, or only on the fourth throw. So, there are 125 + 125 + 125 + 125 = 500 different possibilities for a single point to appear once, and only once, in four 6 drops. There are 1296 - 500 = 796 unfavorable possibilities, since all other cases are unfavorable.

Answer: Vladimir has more chances to win than Peter: 796 against 500.

Problem 8. A dice is thrown. Determine the probability of getting 4.

Solution. There are 6 faces in a die, and points from 1 to 6 are marked on them. A tossed die can lie up any of these 6 faces and show any number from 1 to 6. So, we have only 6 equally possible cases. The appearance of 4 points is favored only by 1. Therefore, the probability that exactly 4 points will fall is 1/6. In the case of throwing one die, the same probability, 1/6, will be for the fall of all other shackles of the die.

Answer: 1/6.

Problem 9. What is the probability of getting 8 points by throwing 2 dice 1 time?

Solution. It is not difficult to calculate the number of all equally possible cases that can occur when throwing 2 dice, based on the following considerations: each of the dice, when thrown, gives 1 out of 6 equally possible cases for it. 6 such cases for one bone are combined in all ways with the same 6 cases for another bone, and thus it turns out for 2 bones in total 6? 6 = 6 2 = 36 equally likely cases. It remains to calculate the number of all equally probable cases favoring the appearance of the sum 8. Here the matter becomes somewhat more complicated.

We must realize that with 2 dice, the sum 8 can be thrown out only in the following ways (Table 1).

Table 1

In total, we have 5 cases favorable to the expected event.

Answer: The desired probability that the dice will roll in the amount of 8 points is 5/36.

Problem 10. Throw 2 dice 3 times. What is the probability that at least once a doublet will fall out (i.e., both dice will have the same number of points)?

Solution. All equally possible cases will be 3b 3 = 46656. Doublets with 2 bones 6: 1 and 1, 2 and 2, 3 and 3, 4 and 4, 5 and 5, b and 6, and with each hit, any of them may appear . So, out of 36 cases, 30 never give a doublet on each hit. With three throws: it turns out 30 3 \u003d 27,000 non-doubling cases. There will be, therefore, 36 3 - 30 3 = 19 656 cases favorable for the appearance of a doublet. The desired probability is 19656: 46656 = 0.421296.

Answer: 0.421296.

Problem 11. If a dice is thrown, then any of the 6 faces can be the top one. For a correct (i.e., not cheating) die, all these six outcomes are equally likely. Two regular dice are thrown independently of each other. Find the probabilities that the sum of the points on the upper faces:

a) less than 9; b) more than 7; c) is divisible by 3; d) even.

Solution. When throwing two dice, there are 36 equally possible outcomes, since there are 36 pairs in which each element is an integer from 1 to 6. Let's make a table in which the number of points on the first bone is on the left, on the second on the top, and at the intersection of the row and column is their sum (Table 2).

table 2

Direct calculation shows that the probability that the sum of points on the upper edges is less than 9 is 26/36 = 13/18; that this sum is greater than 7 - 15/36 = 5/18; that it is divisible by 3: 12/36 = 1/3; finally, that it is even: 18/36 = 1/2.

Answer: a) 13/18, b) 5/18, c) 1/3, d) 1/2.

Problem 12. A die is tossed until a “six” appears. The size of the prize is equal to three rubles multiplied by the ordinal number of the “six”. Should I take part in the game if the entry fee is 15 rubles? What should be the entry fee for the game to be harmless?

Solution. Consider a random variable (a variable that, as a result of the test, will take only one possible value) without taking into account the entrance fee. Let X = (gain value) = (3, 6, 9...). Let's make a distribution graph of this random variable:

Let's find the mathematical expectation from the graph (the average value of the expected payoff) using the formula:

Answer. The mathematical expectation of winning (18 rubles) is greater than the entry fee, that is, the game is favorable for the player. To make the game harmless, you need to set the entry fee equal to 18 rubles.

Problem 13. The sum of the points on the opposite sides of the cube is 7. How should the cube be rolled so that it turns out to be rotated as in the figure:

Problem 14. The casino offers the player a bonus of 100 pounds sterling if he gets 6 from one throw of the dice, as in the figure:

If he fails, he can make another throw. How much does the player have to pay for this attempt?

Answer. First: 1/6=6/36, second: 5/6 1/6=5/36, 11/36 £100=£30.55

Problem 15. A game in a casino, the so-called “dice game”, remade from the game that Bernard de Mandeville called “risk” at the beginning of the 19th century, is played with two dice (dice), as in the figure “a” and “b” :

7 or 11 win. And which ones lose.

Answer: 2 - 3 - 12.

Task 16. The task condition is shown in the figure:

What image should replace the “?” ?

Answer: "a":

Problem 17. You have probably met with the developments of a cube, from which the surface of a cube can be made. The number of different such sweeps is 11. In the figure you see an image of the cube itself and its sweep:

The numbers 1, 2, 3, 4, 5, 6 are written on the faces of the cube. But we see only the first three numbers, and how the rest of the numbers are located can be understood from the “a” scan. If we take the scan “b” of the same cube, then the numbers there are in a different order, in addition, they turn out to be inverted. Having studied the sweeps “a”, “b”, apply five numbers to the remaining nine sweeps so that it corresponds to the proposed cube:

Check your answer by cutting and stacking the corresponding nets.

Problem 18. The numbers 1, 2, 3, 4, 5 and 6 are written on the faces of a cube so that the sum of the numbers on any two opposite faces is 7. The figure shows this cube:

Redraw the presented sweeps (a-d) and place the missing numbers on them in the correct order.

Answer. Numbers can be arranged as shown in the figure:

Problem 19. On the development of the cube, its faces are numbered (a):

Write down in pairs the numbers of the opposite faces of the cube glued from this development (b-d).

Answer: (6; 3), (5; 2), (4; 1).

Problem 20. Numbers 1, 2, 3, 4, 5, 6 are marked on the faces of the cube. Three positions of this cube are shown in the figure (a, b, c):

In each case, determine which digit is on the bottom face. Redraw the scans of this cube (d, e) and put the missing numbers on them.

Answer. On the lower faces are the numbers 1, 5, 2; the missing numbers can be applied as shown in the figure:

Problem 21. Which of the three cubes can be added from the given sweep:

Answer: "B".

Problem 22. The development is glued to the table with a painted edge:

Roll it up mentally. Imagine that you are looking at the cube from the side indicated by one of the arrows. What edge do you see?

Answer: 1) A - 1, B - 4, C - 5; 2) A - 3, B - 2, C - 1.

References

1. Bizam D., Herceg J. Game and Logic. 85 logical problems / transl. from Hung. Yu.A. Danilova. – M.: Mir, 1975. – 358 p.
2. Extracurricular work in mathematics in grades 4-5 / ed. S.I. Schwarzburd. – M.: Enlightenment, 1974. – 191 p.
3. Extracurricular work in mathematics in grades 6-8 / ed. S.I. Schwarzburd. - M.: Enlightenment, 1977. - 288 p.
4. Gardner M. Come on, guess! / per. from English. – M.: Mir, 1984. – 213 p.
5. Gardner M. Mathematical miracles and secrets: per. from English. / ed. G.E. Shilov. – 5th ed. – M.: Nauka, 1986. – 128 p.
6. Gardner M. Mathematical leisure: per. from English. / ed. Ya.A. Smorodinsky. – M.: Mir, 1972. – 496 p.
7. Gardner M. Mathematical short stories: per. from English. / ed. Ya.A. Smorodinsky. – M.: Mir, 1974. – 456 p.
8. Entertaining mathematics. 5-11 grades. (How to make math lessons not boring) / ed. T.D. Gavrilov. - Volgograd: Teacher, 2005. - 96 p.
9. Kordemsky B.A. Mathematical tricks. - M .: Publishing House ONIKS: Alliance-V, 2000. - 512 p.
10. Mathematics: Intellectual marathons, tournaments, fights: grades 5-11. The book for the teacher. - M .: Publishing house "First of September", 2003. - 256 p.
11. Mosteller F. Fifty entertaining probabilistic problems with solutions / per. from English. – M.: Nauka, 1985. – 88 p.
12. Olympiad problems in mathematics. 5-8 grades. 500 non-standard tasks for competitions and olympiads: the development of the creative essence of students / ed. N.V. Zobolotnev. - Volgograd: Teacher, 2005. - 99 p.
13. Perelman Ya.I. Entertaining tasks and experiences. - M.: Children's literature, 1972. - 464 p.
14. Russell K., Carter F. Intelligence training. – M.: Eksmo, 2003. – 96 p.
15. Sharygin I.F., Shevkin A.V. Mathematics: tasks for ingenuity: textbook. allowance for 5-6 cells. general education institutions. - M.: Enlightenment, 1995. - 80 p.

cuboid

Answers to page 111

500. a) The edge of a cube is 5 cm. Find the surface area of ​​the cube, that is, the sum of the areas of all its faces.
b) The edge of a cube is 10 cm. Calculate the surface area of ​​the cube.

a) 1) 5 2 \u003d 25 (cm 2) - the area of ​​one face
2) 25 6 \u003d 150 (cm 2) - cube surface area
Answer: the surface area of ​​the cube is 150 cm 2.

b) 1) 10 2 = 100 (cm 2) − area of ​​one face
2) 100 6 \u003d 600 (cm 2) - cube surface area
Answer: the surface area of ​​the cube is 600 cm 2.

501. On the faces of a cube (Fig. 104) the numbers 1, 2, 3, 4, 5, 6 were written so that the sum of the numbers on two opposite faces is equal to seven. Next to the cube are its scans, which indicate one of these numbers. Enter the rest of the numbers.

502. Figure 105 shows a dice and its development. What number is shown on:
a) the bottom edge;
b) side face on the left;
c) side face behind?

a) Number 6 on the bottom face.
b) Number 1 on the left side.
c) Number 2 on the side face at the back.

503. Figure 106 shows two identical dice in different positions. What numbers are shown on the bottom faces of the dice?

a) The number on the lower face is the opposite of the number 5. Judging by the figure a), these cannot be the numbers 6 and 3, and judging by the figure b), these cannot be the numbers 1 and 4. Only the number 2 remains.

b) The number on the lower face is the opposite of number 1. Judging by figure b) and the previous solution, these cannot be numbers 2, 4 and 5. Also, judging by the arrangement of numbers in figure a), this cannot be number 3. It remains only the number 6.

504. Masha was going to glue the cubes, and for this she drew various blanks (Fig. 107). The older brother looked at her work and said that some of them are not cube scans. What blanks are the development of a cube?

Cube blanks are options a), c) and d).

A dice, also called a dice, is a small cube that, when dropped on a flat surface, occupies one of several possible positions with one face up. Dice are used as a means of generating random numbers or points in gambling.

Description of the dice

The traditional dice is a dice with six faces marked with numbers from 1 to 6. These numbers can be represented as numbers or a certain number of dots. The latter is used most often.

Sum of points on a pair of opposite faces

By the condition of the task, the sum of points on each pair of opposite faces is the same.

There are 6 faces in total, on which numbers from 1 to 6 are applied. The sum of all points is determined as the sum of an arithmetic progression according to the formula

S(n) = (a(1) + a(n)) * n/2, where

• n is the number of members of the progression, in this case n = 6;
• a(1) - the first member of the progression a(1) = 1;
• a(n) - last term a(6) = 6.

S(6) = (1 + 6) * 6/2 = 7 * 3 = 21.

So the sum of all the points on the dice is 21.

If 6 faces are divided into pairs, then 3 pairs will be obtained.

Thus, 21 points are distributed over 3 pairs of faces, that is, 21 / 3 = 7 points on each pair of faces of the dice.

These may be the following options:

The solution of the problem.

1. Find how many faces a dice has.

2. Calculate how many points there are on all sides of the cube.

1 + 2 + 3 + 4 + 5 + 6 = 21 points.

3. Determine how many pairs of opposite faces the dice has.

6: 2 = 3 pairs of opposite faces.

4. Calculate the number of points on each pair of opposite faces of the dice.

21:3 = 7 points.

Answer. The sum of points on each pair of opposite faces of the dice is 7 points.

It may seem that making a perfectly even dice with your own hands is quite difficult, especially when you consider that sides of a dice should be perfectly equal. After all, only then the game with a cube can be considered truly honest and not biased. But the complexity of creating this play accessory is slightly exaggerated. We offer a way to make a dice, easy and fast.

Instructions for making a dice, its faces.

1. We choose the material from which we will make a cube.

2. We make from this material, if possible, an exact cube with sides of 1 cm.

3. We remove chamfers up to 1 mm from the sides and corners of the cube. In this case, we put the file at 45 degrees. Then it is desirable to polish the product.

4. We put numbers on each face of the resulting cube. The points of the numbers can be made either with a microdrill, or marked with paint, or even, having first drilled holes, paint the recesses of the holes with paint.

Numbers are applied in the following order:

• put six points on the upper face (three points on each side);
• on the opposite, which has become the bottom, we apply one point (in the center);
• on the left we put four points (in the corners);
• on the right we apply three (diagonally);
• on the front we put five points (one, as in the case of a unit, in the center, four more, as in the case of a four, in the corners);
• on the back there should be two (at opposite corners).

Check if the numbers are correct. The sum of the numbers on opposite sides of the die must equal seven.

5. We cover our cube with colorless varnish, while leaving one face untouched. The dice will lie on this face until the rest of the faces dry. Then turn over and cover it too.

6. It is advisable to download the virtual dice program. And for this we take a mobile phone and install the BASIC computer language interpreter on it. It can be easily downloaded from many sites. Run the installed interpreter and enter:

• 10 A%=MOD (RND(0),4)+3
• 20 IF A%=0 THEN GOTO 10
• 30 PRINT A%40 END

Now, every time you run it with the RUN command, this program will generate random numbers from 1 to 6.

7. To check if they turned out even sides of a dice, we use it to get six dozen random numbers, and then count how many times each of them occurs. If the faces of the dice are even, then the probabilities for each of the numbers on the dice should be almost equal.

8. Nowadays, board games are not in use. But still, do not forget the order of their conduct. We draw a map with the paths of the game, or maybe we have a store-bought one lying around somewhere. Then each player puts his chip in the initial field, and the game started. We roll the dice in a circle one after another. Each player has the right to move his chip exactly as many spaces as the die he threw showed him. Next, follow the instructions. If you hit the “skip move” division, then we rest for the next round, “repeat the move” we throw again in a row, and so on. The winner is the one whose nerves do not give up and whose chip, in the end, will be the first to reach the finish line.