Problems with dice as a means of implementing a humanitarian orientation in teaching mathematics. On a dice, the sum of points on each pair of opposite faces is the same. What is this sum? On the opposite faces of a dice there are

It may seem that it is quite difficult to make a perfectly even dice with your own hands, especially when you consider that dice faces must be perfectly equal to each other. After all, only then can dice play be considered truly fair and unbiased. But the difficulty of creating this gaming accessory is slightly exaggerated. We offer a method for making dice that is easy and fast.

Instructions for making a dice and its faces.

1. Select the material from which we will make the cube.

2. We make from this material the most accurate cube with sides of 1 cm.

3. We chamfer up to 1 mm from the sides and corners of the cube. At the same time, set the file at 45 degrees. Then it is advisable to polish the product.

4. We put numbers on each face of the resulting cube. The number points can be made either using a microdrill, or marked with paint, or even by first drilling holes and painting the recesses of the holes with paint.

Numerical designations are applied in the following order:

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

We check the correctness of the numbers. The sum of the numbers on opposite sides of the cube must equal seven.

5. Cover our cube with colorless varnish, leaving one side untouched. The dice will lie on this face until the other faces dry. Then we turn it over and cover it too.

6. It is advisable to download the virtual dice program. And to do this, we take a mobile phone and install the BASIC computer language interpreter on it. It can be downloaded from many sites without any problems. Launch 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 using the RUN command, this program will generate random numbers from 1 to 6.

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

8. Nowadays Board games not in use. But still, do not forget the order in which they are carried out. We draw a map with the paths of the game, or maybe we have a store-bought one lying around somewhere. Then each player places his chip in the starting field, and the game begins. We throw the dice in a circle, one after another. Each player has the right to move his piece exactly as many spaces as the dice he threw showed him. Next we follow the instructions. If you hit the “skip move” space, then rest for the next round, throw “repeat move” again in a row, and so on. The winner is the one who doesn't lose his nerve and whose chip, in the end, reaches the finish line first.

History of dice

Bones are enough 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 excavations date back to an earlier period than the period of life of Palamedes and Atis. In ancient times, bones were classified as magical amulets, which were used to tell fortunes or predict the future. Nowadays, many peoples have preserved the tradition of divination with bones.

Kuast Peter. Soldiers playing dice (1643)

Experts claim that the first dice were made from the claw joints of wild, and then domestic, animals, which were called “grandmas.” 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, and teeth of predatory animals and rodents. But the main material for bones still came from wild animals. These were bulls, moose, deer, and caribou. Ivory, as well as bronze, agate, crystal, ceramic, jet and plaster products, were extremely popular among the ancient Greeks.

Dice games were often accompanied by fraud. This is evidenced by records in ancient writings. In the sixth century BC, China used an almost exact copy of modern bones. They had similar layouts and cubic configurations. It is precisely these playing objects dating back to the sixth century BC that were found by archaeologists during excavations carried out in the Celestial Republic. Researchers discovered earlier drawings of bones made on stones in Egypt. The Indian scripture called the Mahabharata also contains lines about dice.

Thus, playing dice can be safely called the oldest gambling entertainment. Nowadays, many games have been invented that can be played with dice.

Modern dice

Modern dice, more often called dice, are usually made of plastic and are divided into two groups.

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

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

Craftsmen cut bones of the highest quality with a special tool from an extruded plastic rod. Next, tiny holes are made on the edges, 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. The bones are then polished until a perfectly smooth and even surface is obtained. Such products are called “smooth-pointed”.

A gambling establishment usually has smooth-dot dice made of red, transparent plastic. The set consists of 5 bones. For traditional gambling house dice it is two centimeters. There are two types of ribs on products – blade and feather. The blade ribs are very sharp. Feathers are slightly sharpened. All sets of dice are equipped with the logo of the gambling establishment for which they were intended. In addition to the monogram, the bones have serial numbers. They are specially coded to prevent fraud. In casinos, in addition to traditional six-sided products, there are dice with four, five and eight sides of a wide variety of designs. Products with concave holes are almost never found today.

Dice scam

In excavated burials on all continents, dice are found that were made specifically for dishonest play. They have the shape of an irregular cube. As a result, the longest edge falls out most often. Irregularity of shape is achieved by grinding down one edge. Another cube can be transformed into a parallelepiped. These irregular bones are nicknamed "dummy bones." It is considered an attribute cheating game, and, as a rule, belong 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 edges fall out more often than others.

Another trick is to duplicate the edges - some are quite numerous, others are completely absent. As a result, some numbers will appear too often, while others will appear almost never. These bones are called “tops and bottoms.” Such products are used by scammers great experience and quite dexterous hands. An ordinary player will often not notice that his partner is playing unfairly.

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

Other scammers choose a soft surface in the form of a blanket or coat. On such a surface the bone rolls like a reel. As a result, the side edges almost never fall out, which leads to combinations that are undesirable for the opponent.

Development of a dice

A regular dice has six sides of equal size. The location of the dots on the cube, forming numbers along the faces, is not random.

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

Dice probability theory

The dice are rolled once

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

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

The dice are rolled 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 side 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 is rolled out. Let’s put all these options in a table and calculate in it the sum that forms the sides of both cubes.

combination number	combination	sum
1		2
2		3
3		4
4		5
5		6
6		7
7		3
8		4
9		5
10		6
11		7
12		8
13		4
14		5
15		6
16		7
17		8
18		9
19		5
20		6
21		7
22		8
23		9
24		10
25		6
26		7
27		8
28		9
29		10
30		11
31		7
32		8
33		9
34		10
35		11
36		12

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

sum	number of favorable combinations	probability, fractions	probability, decimals	probability, %
2	1	1/36	0,0278	2,78
3	2	2/36	0,0556	5,56
4	3	3/36	0,0833	8,33
5	4	4/36	0,1111	11,11
6	5	5/36	0,1389	13,89
7	6	6/36	0,1667	16,67
8	5	5/36	0,1389	13,89
9	4	4/36	0,1111	11,11
10	3	3/36	0,0833	8,33
11	2	2/36	0,0556	5,56
12	1	1/36	0,0278	2,78

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

Description of the dice

A traditional die is a die with numbers from 1 to 6 printed on each of its six faces. These numbers can be represented as numbers or a specific number of dots. The latter is used most often.

Sum of points on a pair of opposite faces

According to the conditions of the task, the sum of points on each pair of opposite faces is the same.

There are only 6 faces, on which numbers from 1 to 6 are printed. 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 terms of the progression, in this case n = 6;
• a(1) - the first term of the progression a(1) = 1;
• a(n) is the last term of a(6) = 6.

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

So, the sum of all points on dice equals 21.

If 6 faces are divided into pairs, you get 3 pairs.

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

These may be the following options:

The solution of the problem.

1. Let's find how many faces a die has.

2. Let's 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 die 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 sides of the die is 7 points.

Rectangular parallelepiped

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 the cube is 10 cm. Calculate the surface area of ​​the cube.

a) 1) 5 2 = 25 (cm 2) - area of ​​one face
2) 25 6 = 150 (cm 2) - surface area of ​​the cube
Answer: the surface area of ​​the cube is 150 cm2.

b) 1) 10 2 = 100 (cm 2) - area of ​​one face
2) 100 6 = 600 (cm 2) - surface area of ​​the cube
Answer: the surface area of ​​the cube is 600 cm2.

501. On the faces of the cube (Fig. 104) they wrote the numbers 1, 2, 3, 4, 5, 6 so that the sum of the numbers on two opposite faces is seven. Next to the cube there are its scans, on which one of these numbers is indicated. Enter the remaining numbers.

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

a) On the bottom edge is the number 6.
b) On the side face on the left is the number 1.
c) On the side face at the back there is the number 2.

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

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

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

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

The cube blanks are options a), c) and d).

• Yakovleva Tatyana Petrovna, Associate Professor, Department of Mathematics and Physics, Federal State Budgetary Educational Institution of Higher Professional Education "Kamchatsky" State University them. Vitus Bering", Petropavlovsk-Kamchatsky, Kamchatka Territory

Sections: Mathematics , Extracurricular activities

Exercises that stimulate the internal energy of the brain, stimulating the play of forces
“mental muscles” is solving problems using quick wits and ingenuity.

Sukhomlinsky V.A.

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

Tasks with dice can be considered as a means of implementing a humanitarian orientation in teaching mathematics. They contribute to: the development of spatial imagination; developing the ability to mentally imagine different positions of an object and changes in its position depending on different reference points and the ability to fix this idea in the image; teaching logical justifications of geometric facts; development of design abilities, modeling; development of research skills.

Task 1. Carefully look at the figures in the top row:

What figure instead of the “?” from the bottom row must be placed?

Answer: “b”.

Problem 2. There is 1 dot 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 dots on any two opposite faces is 7. Kolya glued a column of 6 such cubes and counted the total number of dots on all outer faces. What's the biggest number he could get?

Answer: number 96.

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

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

This unusual game. In it, one player, having tossed a die, adds the number dropped on the top side 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 edge is not taken into account. It is a simple game, although mathematicians disagree as to exactly how much advantage the thrower of the die has over his opponent. At the moment, the savage is throwing 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 Bungalozo system familiar to the savage, it will turn out to be even greater. The savages of Bungalosia, as we well know, have only three fingers on each hand, so they are accustomed to the six-digit number system. This raises a curious problem in the realm of elementary arithmetic: we ask our readers to convert the number 109,778 into the Bungalow system, so that the savage will know how many gold coins he has won.

Solution. The die should land one up. If you add the 4 on the side edge here, it gives a total of 5. The sum of the remaining numbers on the side edges (5, 2 and 3) is 10, which gives the other player an advantage of 5 points. In the sixfold system, the number 109778 would be written 2204122. The digit on the right represents the ones, the next digit gives the number of sixes, the third digit from the right represents the number of “thirty-sixes”, the fourth digit shows the number of “portions” of 216, etc. This system is based on powers of 6 instead of powers 10, as is the case in the decimal number system.

Answer: 2204122.

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

Answer: 13 points.

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

Peter bets that if you throw the dice four times in a row, then in all four times the dice will certainly land once with a single point up. Vladimir claims that a single point will either not come up at all after four throws, or it will come up 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. Let’s assume that the first throwing has already taken place, and the result is a single point. Then, during the next three tosses, the number of all possible positions favorable for Peter, that is, the number of any points except one, is 5? 5 ? 5 = 125. In the same way, 125 positions favorable for Peter are possible if a single point occurs 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, on four 6drops. There are 1296 – 500 = 796 unfavorable possibilities, since all other cases are unfavorable.

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

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

Solution. There are 6 sides of a die, and they are marked with points from 1 to 6. A tossed die can land on any of these 6 sides and show any number from 1 to 6. So, we have a total of 6 equally possible cases. The appearance of 4 points is favored only by 1. Therefore, the probability that exactly 4 points will appear is 1/6. In the case of throwing one die, the same probability, 1/6, will be for all other bones falling out.

Answer: 1/6.

Problem 9. How likely is it to get 8 points by rolling 2 dice once?

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 dice, when thrown, gives 1 out of 6 equally possible cases for its case. 6 such cases for one bone are combined in all ways with 6 cases for another bone, and thus it turns out for a total of 2 bones 6? 6 = 6 2 = 36 equally possible cases. It remains to count the number of all equally possible cases favorable to the appearance of the sum 8. Here the matter becomes somewhat more complicated.

We must realize that with 2 dice, the sum of 8 can only be rolled 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 a total of 8 points is 5/36.

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

Solution. There will be 3b 3 = 46656 of all equally possible cases. There are 6 doublets with 2 dice: 1 and 1, 2 and 2, 3 and 3, 4 and 4, 5 and 5, b and 6, and with each hit one of them is possible . So, out of 36 cases with each blow, 30 in no case give a doublet. With three tosses: it turns out 30 3 = 27,000 non-doublet cases. The cases favorable for the appearance of a doublet will therefore be 36 3 – 30 3 = 19 656. The desired probability is 19656: 46656 = 0.421296.

Answer: 0.421 296.

Problem 11. If you throw a die, then any of the 6 faces can be the top. For a correct (i.e., non-cheating) die, all six of these outcomes are equally possible. Two fair dice are thrown independently of each other. Find the probability that the sum of points on the upper faces:

a) less than 9; b) more than 7; c) 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 create a table in which the number of points on the first dice is on the left, on the second at the top, and at the intersection of the row and column is their sum (Table 2).

table 2

		Second bone
First bone

Direct calculation shows that the probability that the sum of points on the upper faces is less than 9 is 26/36 = 13/18; that this amount 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. The die is tossed until a “six” appears. The prize size is equal to three rubles multiplied by the serial number of the “six” rolled. Should I take part in the game if the entry fee is 15 rubles? What should the entry fee be for the game to be harmless?

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

Using the graph, we find the mathematical expectation (the average value of the expected win) using the formula:

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

Problem 13. The sum of points on opposite sides of the cube is 7. How to roll the cube so that it turns out as in the picture:

Problem 14. A casino offers a player a bonus of £100 if he gets a 6 with one throw of the dice, as in the picture:

If he doesn't succeed, he can take another shot. How much should the player 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 casino game, the so-called “dice” game, converted from a game that Bernard de Mandeville called “risk” at the beginning of the 19th century, is played with two dice (dice), as in figure “a” and “b” :

7 or 11 win. And which ones lose.

Answer: 2 – 3 – 12.

Problem 16. The condition of the task is shown in the figure:

What image should replace the “?” ?

Answer: “a”:

Problem 17. You have probably come across cube developments, from which you can make up the surface of a cube. The number of different such developments is 11. In the figure you see an image of the cube itself and its development:

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 remaining numbers are located can be understood from the “a” scan. If we take the “b” scan of the same cube, then the numbers there are arranged in a different order, in addition, they turn out to be upside down. Having studied the scans “a”, “b”, put five numbers on the remaining nine scans so that it corresponds to the proposed cube:

Check your answer by cutting out and folding the corresponding unfolds.

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 scans (a-d) and place the missing numbers on them in the required order.

Answer. The numbers can be arranged as shown in the figure:

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

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

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

Problem 20. On the edge of the cube there are numbers 1, 2, 3, 4, 5, 6. Three positions of this cube are shown in the figure (a, b, c):

In each case, determine which number is on the bottom edge. 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 entered as shown in the figure:

Problem 21. Which of the three cubes can be folded from this development:

Answer: “B”.

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

Mentally roll it up. Imagine that you are looking at the cube from the side indicated by one arrow. What edge do you see?

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

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