The sum of the points on the opposite edges of the playing cube. Playing cubes (bones). Modern playing cubes
History of playing bones
Dice is enough ancient gameBut the history of her occurrence is still unknown.
Sofokl gave the palm of the championship in this case by the Greek named Palamed, who invented this game During the siege of Troy. Herodotus was confident that the bones invented the Lydians in the era of the Board of Atis. Archaeologists based on the obtained scientific data refute these hypotheses, since the bones that were found during the excavations belong to an earlier period than the period of life of Palamened and Atis. In the ancient times, the bones treated the category of magical amulets, on which the future was wondering or predicted. Nowadays, many nations have retained the tradition of divination on the bones.
Kuasta Peter. Soldiers playing bones (1643)
Experts assure that the first playing bones were performed from the sweeping joints of the wild, then and 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 the bones of plums, apricots and peach, large seeds of various plants, deer horns, smooth stones, ceramics, teeth of predatory animals and rodents. But the main material for bones was still driving wild animals. These were bulls, moose, marals, deer Caribou. Among the ancient Greeks, elephant bone, as well as bronze, agate, crystal, ceramic, gagate and gypsum products, were very popular.
The game in the bone was often accompanied by fraud. This is evidenced by entries in ancient letters. In the sixth century BC In China, they used an almost accurate copy of modern bones. They had similar marking and cubic configuration. It is the same playing items dated by the sixth century BC to our era were found by archaeologists during excavations produced in the Criminal Republic. Earlier bone drawings made on the stones, researchers discovered in Egypt. In the Indian monument of writing entitled "Mahabharat" there are also rows about the playing bones.
Thus, the game in the bone can be bolded to be called ancient gambling entertainment. Nowadays, many games are invented in which you can play with bones.
Modern playing cubes
Modern bones, often referred to as playful cubes, are usually produced by plastic, and are divided into two groups.
The first group includes the highest quality products performed by hand. These bones buy a casino to play CRPS.
The second group includes bones made on machines. They are suitable for ubiquitous use.
The top quality of the wizard is pumped up with a special tool from extruded plastic rod. Next, the tiny holes are done on the edges, the depth of which is equal to several millimeters. Paint is poured into these holes, the weight of which is equal to the weight of the remote plastic. Then the bones are polished until it turns out the perfectly smooth and smooth surface. Such products were called "Gladkotochny".
In the gambling institution, there are usually smooth-point bones made of red, transparent plastic. The kit consists of 5 bones. The traditional bones from the gambling house is equal to two centimeters. The ribs in products are two species - blade and feather. The blade ribs are very sharp. Feather - a little sharpened. All bone sets are supplied with the ignorate logo for which they were intended. In addition to the bone monogram, there are serial numbers. They are specifically encoded to prevent fraud. In the casino, in addition to traditional hexagon products, bones with four, five and eight faces of the most different design come across. Products with concave holes today are almost no found.
Moshenianity with playing bones
In the excavated burials on all continents there are playing bones made specifically for a dishonest game. They have the shape of the wrong cube. As a result, the longest edge is most often falling. Impropiness of the form is achieved by the action of one face. Another cube can be transformed into parallelepiped. These irregularities received the nickname "Doodles". It is considered an attribute of the shoe game, and, as a rule, belong to fraudsters.
Modern blank externally cannot be distinguished from ordinary bone, as it has the form of an ideal cube. But in a blank one or more faces are extra weight. Such faces and fall the cup of others.
Another trick lies in the dubbing faces - some are quite enough, others are completely absent. As a result, some numbers will fall out too often, while others are almost never. These bones are called "tops and bottoms". Such products enjoy fraudsters with extensive experience and quite clever hands. An ordinary player often does not obey that his partner leads a dishonest game.
Some scammers train a lot with normal bones. As a result, they turn out to throw the required combinations. To this end, the bones are thrown by a special way, allowing one or two products to rotate in the vertical plane and go to the desired edge.
Other crooks choose a soft surface in the form of a blanket or coat. On such a surface, the bone is rolling like the coil. In the results of the side faces almost do not fall out, which leads to unwanted combinations for the opponent.
Playing cube scan
The usual playing cube has six faces that are the same in size. The location of the dice on the cube forming numbers is not accidental.
According to the rules, the amount of points on the opposite glands of the playing bone should always be equal to seven.
Theory of the probability of playing bone
Playing bone rushes once
When the playing bones are thrown, it is not difficult to find a chance. If we assume that we have a correct playing bone, without the various tricks described above, the likelihood of each of its faces is equal to:
1 of 6.
in fractional form: 1/6
in a densulating form: 0,1666666666666667
Playing bone rushes 2 times
If you throw two playing bones to find the probability of falling out the desired combination, it is possible to change the probability of falling out the desired face on each of the bones:
1/6 × 1/6 \u003d 1/36
In other words, the probability will be equal to 1 out of 36. 36 - these are the number of options that may result in the desired number, we will reduce all these options in the table and we will calculate the amount forms the edge 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 falling out the desired amount when throwing two playing bones:
| sum | number of favorable combinations | probability, ordinary fractions | probability, decimal fractions | 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 |
- Yakovleva Tatiana Petrovna, associate Professor of the Department of Mathematics and Physics, FGBOU VPO "Kamchatka State University. Vitus Bering", Petropavlovsk-Kamchatsky, Kamchatka Territory
Sections: Mathematics , Extracurricular work
Exercises encouraging the inner energy of the brain, stimulating the game forces
"Mental Muscles" is the solution of intelligence tasks, reducingness.
Sukhomlinsky V.A.
The humanitarian orientation today expands the content of mathematical education. It not only increases interest in the subject, as is customary, but also develops a person in students, activates their natural abilities, creates conditions for self-development. And therefore, the humanitarian aspect in teaching mathematics contributes to: the admission of students to spiritual culture, creative activity; arming them with heuristic techniques and methods of scientific search; The creation of conditions encouraging the schoolchildren to active activities and ensure its participation in it. Thinking man mainly consists of setting and solving problems. Praphrasing Descartes, you can say: live - it means to put and solve problems. And while a person solves the task - he lives.
Tasks with playing bones can be considered as a means of implementing a humanitarian orientation in teaching mathematics. They contribute: the development of spatial imagination; The formation of the ability to mentally represent the various provisions of the subject and change its position, depending on different point of reference and the ability to fix this view on the image; learning to logical rationale for geometric facts; development of design abilities, modeling; Development of research skills.
Task 1. Carefully consider the figures in the top row:
What kind of figure instead of a sign "?" From the bottom row you need to put?
Answer: "b".
Task 2. On the front face of the cube is drawn 1 point, on the back - 2, on the top - 3, on the bottom - 6, on the right - 5, on the left - 4. What the largest number of points can be seen at the same time, turning this cube in the hands?
Answer: 13 points.
Task 3. On a playing cube, the total number of points on any two opposite faces is 7. Kohl glued the column of 6 such cubes and calculated the total number of points on all external edges. What is the biggest number he could get?
Answer: number 96.
Task 4. Roll into the cube, presented in the figure, for 6 moves so that it takes to the 7th square and at the same time it would be its face with 6 points. And every move you can move the cube on a quarter turn up, down, left or right, but not diagonally.
Task 5. You see in the picture, like the king of the country of the puzzle plays with a savage in the bone.
This is an unusual game. In it, one player, throwing the bone, folds the number that fell on the upper face, with any number on one of the four side faces. And his opponent folds all other numbers on three side faces. The number on the bottom face is not taken into account. This is a simple game, although mathematics disagree in opinions as to which advantage it has a throwing bone over his opponent. At the moment, the savage throws the bone, as a result of this throw, the king was ahead of it for 5 points. Tell me what number should have fallen on the bone?
Princess The mystery is accounting to win a savage. If this is the number to translate to the Buccalozoic system familiar to the savage, it will turn out to be even more. The savages from the bungalosia, as we are well known, on each hand only three fingers, so that they are accustomed to a cheerful number system. From here there is one curious task from the area of \u200b\u200belementary arithmetic: we ask our readers to translate the number 109 778 in the Burgellaz system, so that the savage learned how many gold coins he won.
Decision. The bone should fall apart. If you add 4 on the side face here, this gives an amount equal to 5. The sum of the remaining numbers on the side of the side (5, 2 and 3) is equal to 10, which gives another player an advantage of 5 points. 2204122 Number 109778 will be recorded in the sixrical system. The digit on the right represents the units, the following digit gives the number of six, the third right of the digit means the number of "thirty-sisters", the fourth digit shows the number of "servings" of 216, etc. This system is based on degrees 6 instead of degrees. 10, as is the case in a decimal number system.
Answer: 2204122.
Task 6. On the bottom face of the cube, 6 points are drawn, on the left - 4, on the back - 2. What the largest number of points can be seen at the same time, turning this cube in the hands?
Answer: 13 points.
Task 7. Here is a playing bone: a cube with glasses marked at its faces from 1 to 6.
Peter beats about the mortgage that if you throw a cube four times in a row, then for all four times the cube will certainly fall once a single point up. Vladimir also claims that a single point is either at all falls at four drops, or will fall out more than once. Which of them is more likely to win?
Decision. With four throws, the number of all possible provisions of the playing bone is 6? 6? 6? 6 \u003d 1296. Suppose that the first throwing has already been held, and a single point fell. Then, at the following three casts, the number of all possible provisions favorable for Peter, that is, the deposits of any glasses, except for a single, 5? five ? 5 \u003d 125. In the same way, it is possible 125 more favorable for Peter locations, if a single point falls out only during the second, only at the third or only with a fourth throwing. So, there are 125 + 125 + 125 + 125 \u003d 500 different capabilities in order for a single point at four clocks one, and only once. The adverse capabilities there are 1296 - 500 \u003d 796, since all other cases are unfavorable.
Answer: Vladimir has a chance to win more than Peter: 796 against 500.
Task 8. The playing bone rushes. Determine the magnitude of the likelihood that 4 points will fall.
Decision. In the dice of 6 faces, and points from 1 to 6 are noted on them. The abandoned bone washes to lie up any of these 6 faces and show any number from 1 to 6. So we have only 6 equivalence cases. The appearance of 4 points is favored only 1. Consequently, the likelihood that exactly 4 points falls, equal to 1/6. In the case of throwing one bone, the same probability, 1/6, will also for loss of all other bone shackles.
Answer: 1/6.
Task 9. How is the probability of getting 8 points, throwing 2 bones 1 time?
Decision. Calculate the number of equilibrium cases that can happen when throwing 2 bones, it is not difficult, based on such considerations: each of the bones during throwing gives 1 of 6 equal to its cases. 6 such cases for one bone are combined by all methods with 6 cases for another bone, and thus obtains only 2 bones 6? 6 \u003d 6 2 \u003d 36 equal cases. It remains to calculate the number of equal cases that conducive the appearance of the amount 8. Here it is somewhat complicated.
We must figure out that at 2 bones sum 8 can only be thrown out in the following ways (Table 1).
Table 1
Total cases that have a favorable expected event, we have 5.
Answer: The desired chance that the bones will throw out 8 points, is 5/36.
Task 10. Throw 2 bones 3 times. What is the likelihood that although doublet will fall out (i.e., on both bones there will be the same number of points)?
Decision. All equal cases will be 3b 3 \u003d 46656. Doublets at 2 bones 6: 1 and 1, 2 and 2, 3, and 3, 4 and 4, 5, and 5, b and 6, and with each blows, any of them may appear . So, out of 36 cases, with each impact 30, no dublet is given. At the same throwing: it turns out 30 3 \u003d 27,000 underwear case. Cases that conducive to the appearance of a doublet will, therefore, 36 3 - 30 3 \u003d 19 656. The desired probability is 19656: 46656 \u003d 0.421296.
Answer: 0.421 296.
Task 11. If the playing bone throw, then any of the 6 faces may be top. For the correct (i.e., not the scaling) bone, all these six outcomes are equally possible. Absorbed independently of each other two right bones. Find the probabilities that the amount of points on the upper edges:
a) less than 9; b) more than 7; c) divided by 3; d) even.
Decision. When throwing two bones there are 36 equilibrium outcomes, since there are 36 pairs, in which each element is an integer from 1 to 6. We will be the table in which the number of points on the first bone, at the top - on the second, and at the intersection of the string and column costs Their amount (Table 2).
table 2
Second bone |
|||||||||||
First bone |
|||||||||||
The direct count shows that the likelihood that the amount of points on the upper grains is less than 9, is equal to 26/36 \u003d 13/18; that this amount is greater than 7 - 15/36 \u003d 5/18; that it is divided into 3: 12/36 \u003d 1/3; Finally, it is even: 18/36 \u003d 1/2.
Answer: a) 13/18, b) 5/18, c) 1/3, d) 1/2.
Task 12. Dice Redes up before the appearance of "six". The size of the prize is equal to three rubles multiplied by the sequence number of the "six". Should take part in the game if the entrance fee is 15 rubles? What should be an entrance fee for the game to be harmless?
Decision. Consider a random value (the value that, as a result of the test, will only take one possible value) without taking into account the entrance fee. Let x \u003d (the magnitude of the win) \u003d (3, 6, 9 ...). We will make a graph of the distribution of this random variable:
We will find a mathematical expectation (average value of the expected win), using the formula:
Answer. The mathematical wait of the winnings (18 rubles) is more than the value of the entrance fee, that is, the game is favorable for the player. For the game to be harmless, the value of the entrance fee should be set to 18 rubles.
Task 13. The amount of points on the opposite edges of the cube is 7. How to roll the cube so that it turns out to be turned like in the picture:
Task 14. Casino offers a player to a premium at 100 pounds sterling, if it gets 6 from one bone throw, as in the picture:
If he does not come out, he can make another throw. How much does the player pay for this attempt?
Answer. The first: 1/6 \u003d 6/36, the second: 5/6 1/6 \u003d 5/36, 11/36 100f.st. \u003d 30.55 f.st.
Task 15. The game in the casino, the so-called "bone game", is redone from the game, in which Bernard de Mandeville called "risk" at the beginning of the XIX century, is played by two cubes (bones), as in the figure "A" and "B" :
7 or 11 won. And which losing.
Answer: 2 - 3 - 12.
Task 16. The assignment condition is shown in Figure:
What image should I replace the sign "?" ?
Answer: "A":
Task 17. With cube sweeps, from which you can make the surface of the cube, you probably met. The number of different such swells is 11. In the figure you see the image of the cube itself and its sweep:
On the edges of Cuba, numbers 1, 2, 3, 4, 5, 6 are written. But we see only the first three numbers, and how the other numbers are located, you can understand from the scan "A". If we take the scan "b" of the same cube, then there are numbers in a different order, in addition, they turn out to be inverted. After examining the sweep "A", "B", apply to the other nine sweeps five numbers so that it corresponds to the proposed Cuba:
Check your answer, cutting out and folding the corresponding sweeps.
Task 18. On the edges of Cuba, numbers 1, 2, 3, 4, 5 and 6 are written so that the amount of numbers on any two opposite faces is 7. The figure shows this cube:
Redraw the submitted scanners (A-D) and arrange missing numbers on them in the desired order.
Answer. The numbers can be arranged as shown in the picture:
Task 19. At the cube scan, its faces are numbered (a):
Write down couples numbers of the opposite faces of a cube glued from this sweep (Gd).
Answer: (6; 3), (5; 2), (4; 1).
Task 20. On the verge of Cuba are numbers 1, 2, 3, 4, 5, 6. Three positions of this cube are depicted in the figure (A, B, B):
In each case, determine which digit is on the bottom face. List the scan of this cube (g, d) and apply missing numbers on them.
Answer. In the lower grains there are numbers 1, 5, 2; The missing numbers can be applied as shown in Figure:
Task 21. Which of the three cubes can be folded from this sweep:
Answer: "B".
Task 22. The scan is glued to the table with a painted face:
Mentally turn it out. Imagine that you look at the cube from the specified one arrows. What edge you see?
Answer: 1) A - 1, B - 4, C - 5; 2) A - 3, B - 2, C - 1.
List of references
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- Extracurricular work on mathematics in 4-5 classes / ed. S.I. Schwartzurbuda. - M.: Enlightenment, 1974. - 191 p.
- Extracurricular work on mathematics in 6-8 classes / ed. S.I. Schwartzurbuda. - M.: Education, 1977. - 288 p.
- Gardner M. and Well, guess! / lane from English - M.: Mir, 1984. - 213 p.
- Gardner M. Mathematical Wonders and Secrets: Per. from English / Ed. G.E. Shilova. - 5th ed. - M.: Science, 1986. - 128 p.
- Gardner M. Mathematical leisure: per. from English / Ed. Ya.A. Smorodinsky. - M.: Mir, 1972. - 496 p.
- Gardner M. Mathematical novels: Per. from English / Ed. Ya.A. Smorodinsky. - M.: Mir, 1974. - 456 p.
- Entertaining mathematics. 5-11 classes. (How to make mathematics lessons hopeless) / Avt.-Cost. Etc. Gavrilova. - Volgograd: Teacher, 2005. - 96 p.
- Cordemsky B.A. Mathematical leaks. - M.: Publishing House Onix: Alliance-B, 2000. - 512 p.
- Mathematics: intelligent marathons, tournaments, battles: 5-11 classes. Book for teacher. - M.: Publisher "First September", 2003. - 256 p.
- Sapeller F. Fifty entertaining probabilistic tasks with solutions / lane. from English - M.: Science, 1985. - 88 p.
- Olympic tasks in mathematics. 5-8 classes. 500 non-standard tasks for contests and Olympiads: the development of the creative essence of students / Avt.-Cost. N.V. Zobolotneva. - Volgograd: Teacher, 2005. - 99 p.
- Perelman Ya.I. Entertaining tasks and experiments. - M.: Children's literature, 1972. - 464 p.
- Russell K., Carter F. Training Intellect. - M.: Eksmo, 2003. - 96 p.
- Sharygin I.F., Shevkin A.V. Mathematics: Tasks for the smelting: studies. Manual for 5-6 CB. general education. institutions. - M.: Enlightenment, 1995. - 80 s.
It may seem that the perfectly smooth playing cube makes it difficult to make it quite difficult, especially if you consider that the face of a playing cube Must be perfectly equal to each other. After all, only then the game of a cube can be considered honest and not biased. But the complexity of creating this gaming accessory is slightly exaggerated. We offer a way to make a playing cube, light and fast.
Instructions for making a playing cube, its faces.
1. Select the material from which we will make a cube.
2. We make from this material as an accurate cube with the parties of 1 cm.
3. Remove from the sides and corners of the chamfer cube to 1 mm. At the same time, we put a file for 45 degrees. Then it is desirable to polish the product.
4. We apply for each edge of the resulting dice of numbers. The numbers of numbers can be made either using a microdel, or designate the paint, or at all, first drum down, paint the deepening of the holes.
Digital designations are applied in this order:
- on the upper face, we apply six points (three points on each side);
- on the opposite, the bottom, the face is applied one point (centered);
- on the left apply four points (at the corners);
- on the right to apply three (diagonally);
- on the front applied five points (one as in the case of a unit - in the center, four more, as in the case of the fourth - in the corners);
- on the back there should be two (in opposite corners).
Check the correctness of the application of numbers. The amount of numbers on the opposite friend of the sides of the cube should be seven.
5. Covers our cube with colorless varnish, leaving one face with one faction. On this face, the playing cube will lie until the rest of the face is dried. Then we turn over and cover it.
6. It is advisable to download a virtual playing cube program. And for this we take a mobile and install the interpreter of the computer language Baysik on it. Its without any problems can be downloaded from many sites. Run the installed interpreter and enter:
- 10 A% \u003d MOD (RND (0), 4) +3
- 20 IF A% \u003d 0 THEN GOTO 10
- 30 Print A% 40 END
Now every time you start using the RUN command this program It will be generated random numbers from 1 to 6.
7. To check if it was smooth the face of a playing cubeWe get six tens of random numbers with it, and then count how many times each of them is found. If the face of the cube is smooth, then the probabilities of the fallout for each of the numbers on the cube should be almost equal.
8. Nowadays board games Not in the go. But still do not forget the order of their holding. We draw a map with ways of the game, and maybe we had a bought in the store somewhere. Then every player puts his chip in the initial field, and the game went. We throw the bones in a circle for each other. Each player has the right to move his chip exactly on so many divisions as the cubes thrown to him. Next, follow the instructions. If you got to the division of "skip the move", then the next circle rest, "repeat the move" we throw again in a row, and so on. One wins the nerves and whose chip, in the end, will come to the finish.
Rectangular parallelepiped
Answers to page 111
500. A) The edge of the cube is 5 cm. Find the surface area of \u200b\u200bthe Cuba, that is, the sum of the squares of all its faces.
b) The edge of the cube is 10 cm. Calculate the surface area of \u200b\u200bthe cube.
a) 1) 5 2 \u003d 25 (cm 2) - the area of \u200b\u200bone face
2) 25 6 \u003d 150 (cm 2) - Cuba surface area
OT V E T: Cube surface area 150 cm 2.
b) 1) 10 2 \u003d 100 (cm 2) - the area of \u200b\u200bone face
2) 100 6 \u003d 600 (cm 2) - Cuba surface area
OT V E T: Cube surface area 600 cm 2.
501. On the edges of the cube (Fig. 104), numbers 1, 2, 3, 4, 5, 6 were written so that the amount of numbers on two opposite faces is equal to seven. Next to the cube depicts its expands on which one of these numbers is indicated. Specify the remaining numbers.
502. Figure 105 shows a playing cube and its expandment. What number is depicted on:
a) lower face;
b) side face down on the left;
c) lateral edge from behind? 
a) on the bottom face number 6.
b) on the side face left number 1.
c) on the side edge behind the number 2.
503. Figure 106 shows two identical playing cubes in different positions. What numbers are depicted on the lower edges of the cubes? 
a) The number on the bottom face is the opposite of the number 5. Judging by Figure A), it cannot be numbers 6 and 3, and judging by Figure B), it cannot be numbers 1 and 4. Only 2 remains.
b) the number on the bottom face is the opposite number 1. Judging by the figure b) and the previous solution, it cannot be numbers 2, 4 and 5. Also, judging by the location of the numbers in Figure A), it may not be the number 3. remains Only number 6.
504. Masha gathered to glue cubes, and for this she painted various billets (Fig. 107). The older brother looked at her work and said that some of them were not dice sweeps. What billets are cube sweeps? 
Cube blanks are variants a), c) and d).
A playing cube, which is also called a playing bone, is a small cube, which in a fall on a flat surface occupies one of several possible positions by one face up. Playing dices are used as means of generating random numbers or points in gambling.
Description of playing cube
The traditional playing bone is a cube, on each of the six faces of which numbers from 1 to 6 are applied. These numbers can be represented as numbers or a certain number of points. The latter is most often used.
Amount of glasses on a pair of opposite faces
Under the assignment condition, the amount of points on each pair of the opposite faces is the same.
There are only 6 faces on which numbers from 1 to 6 are applied. The sum of all glasses is defined as the sum of arithmetic progression by the formula
S (n) \u003d (A (1) + A (N)) * N / 2, where
- n is the number of progression members, in this case n \u003d 6;
- a (1) - the first term of the progression of A (1) \u003d 1;
- a (N) is the last term A (6) \u003d 6.
S (6) \u003d (1 + 6) * 6/2 \u003d 7 * 3 \u003d 21.
So, the sum of all glasses on the playing cube is 21.
If 6 faces are divided into pairs, then there are 3 pairs.
Thus, 21 points are distributed to 3 pairs of faces, that is, 21/3 \u003d 7 points on each pair of the faces of the playing cube.
These may be the following options:
The solution of the problem.
1. Find how much the faces have a playing cube.
2. Calculate how many points on all the edges of the cube.
1 + 2 + 3 + 4 + 5 + 6 \u003d 21 points.
3. We define how many pairs of opposite faces of a playing cube.
6: 2 \u003d 3 pairs of opposite faces.
4. Calculate the number of points on each pair of the opposite faces of the playing cube.
21: 3 \u003d 7 points.
Answer. The amount of glasses on each pair of opposing faces of the playing cube is 7 points.