Berger table for a mixed system of 27 teams. Creation of tournament schedule. This tournament system is called round robin.
Let's digress from higher mathematics and pay attention to sports. The employees I work with sometimes organize tennis tournaments and often have the task of writing a schedule of "fights" so that every athlete plays with everyone.
Such a tournament system is called a round robin.
Each participant takes turns playing with everyone else. The winner is the participant who wins the largest number of matches. At the same time, they reveal (according to the number of meetings won) the subsequent places occupied by other participants.
Obvious plus: With this method, the winner is most objectively revealed.
Disadvantage: The tournament can be very stretched in time. Especially if there are two or more circles.
On the Internet, the most commonly used so-called tBerger tables, but this is not our method.
So, in order to draw up a correct schedule of competitions, according to such a scheme, there are special formulas that will be discussed.
So the initial data: N - the number of participants.
In order for each of them to play once with the rest, we need to conduct N-1 rounds.
If the number of participants is odd, then another virtual participant is added, and if the participant has to play with such a "virtual" in the schedule, then he simply does nothing. Resting and watching others.
Analysis of the resulting table is as follows:
Half of the competitors are shown horizontally in bold type.
Each row of the table is a separate round.
The number limit of this bot.. Let's start from the number of participants no more than 50. But mathematically there are no limits, you can do two hundred.
Consider a group of 7 participants:
We write seven and get a table
| 1 | 2 | 3 | 4 |
| 7 | 6 | 5 | 8 |
| 8 | 7 | 6 | 5 |
| 2 | 1 | 7 | 6 |
| 3 | 8 | 1 | 7 |
| 4 | 3 | 2 | 1 |
| 5 | 4 | 8 | 2 |
| 6 | 5 | 4 | 3 |
There are only 7 participants, and the athlete number 8 appears in the table. This is a virtual participant
There will also be seven rounds to determine the winner (this is the number of lines)
Now what do we see:
in the first round, contestant number 1 plays contestant number 7, 2 versus 6, 3 versus 5, and contestant 4 with a "virtual" eighth. That is, the 4th participant does not participate in the first round.
In the second round, 1 athlete does not participate, the second plays with the seventh, the third with the sixth, and the fourth with the fifth.
And so in every round.
If we look at each column, we make sure that each participant plays once with everyone except himself.
This table is easy to copy into Excel, replace the numbers with the names of the participants or teams.
The sequence of rounds (lines) can also be changed as you like.
And ultimately make a beautiful poster, tournament or competition.
I hope this calculator helps someone.
Yesterday in the Premier League held a meeting of sports directors of clubs, which discussed the calendar of the second stage of the championship. "Soviet sport" knows some details.
WHY DID YOU REJECT THE MANUAL OPTION?
As Sovetsky Sport previously reported, after the blind draw procedure was abandoned (due to the inability to take into account extremely important factors: the climate, the participation of teams in European competitions), it was decided to work out the option of manually compiling the calendar. With this option, the above factors could be taken into account, but in reality it turned out that it would be difficult to “include” all the wishes of the clubs in the schedule.
In general, early planning of the calendar, especially in a situation where the first eight have decided with a high degree of probability, is an absolutely normal phenomenon. There were questions on the ethical side - they say, how can you take into account the matches with the participation of Anji, if Krasnodar did not lose the chance to break into the top eight? But, firstly, they worked on the calendar, when the gap between 8th place and 9th was about 10 points, and secondly, it would be somehow strange if the league took care of the schedule the morning after the 30th round.
Yesterday, RFPL President Sergei Pryadkin told a Sovetsky Sport correspondent Sergei EGOROV the following: "The calendar will be drawn up according to the sports principle."
What does it mean? According to our data, we are talking about the scheduling of the tournament according to the so-called Berger tables.
WHAT IS THE BERGER TABLE?
The table, named after the famous Austrian chess player and chess theorist Johann Nepomuk Berger, is a way of making a calendar.
Clubs are assigned a number corresponding to their place in the Russian Championship. Each club, except for the one that received the first number, consecutively plays with rivals in ascending numbers. That is, the team that takes eighth place plays in the first round with the first, in the second round - with the second, in the third round - with the third and so on until the seventh round. The eighth round of the opponents coincides with the second, the ninth with the third, and so on. The last round will repeat the first, only the opponents change fields.
If the position of the teams after the 30th round remains the same as after the 28th, the top eight clubs will receive the following numbers: 1. Zenit, 2. CSKA, 3. Lokomotiv, 4. Dynamo, 5. Spartak, 6. Rubin, 7. Kuban, 8. Anji.
With this system, the sports principle is preserved - the most strong team starts the tournament with the weakest. And so that our readers can follow a possible calendar online, we publish both the Berger table and an approximate calendar as of the 28th round. And you can update the schedule after each of the two remaining tours in the first stage.
The final version (unless, of course, it is decided to abandon the Berger table) we will find out on the evening of November 6, when the last match of the 30th round ends.
WHAT WILL HAPPEN TO THE SECOND EIGHT?
The calendar of the second G-8 will be determined by a blind draw on November 7 during a solemn ceremony. As Sergey Pryadkin already said in an interview with Sovetsky Sport, the matches of the teams of the first and second eights will be held on the same dates, the days of the tours will coincide.
Sonneborn-Berger system- a method for determining the best result (coefficient) if several participants in the tournament scored the same number of points. The coefficient of the participants is equal to the sum of the points of the opponents they won against and half the points of the opponents with whom they drew.
In fact, the Sonneborn-Berger odds system gives an advantage to a player who won against strong players and lost against weak ones over a “normal” player who lost against strong ones and won against weak ones. The Sonneborn-Berger odds are widely used, especially in round robin tournaments.
The Sonneborn-Berger system is not objective, therefore, in important cases (definition of a champion, admission to the next stage of a major competition), it is customary to hold an additional competition. The mixed method is also used (in case of equality of points in the additional competition, the Sonneborn-Berger coefficient decides).
Along with the Sonneborn-Berger coefficient system, other methods are used to identify advantages in case of equality of points: by the number of wins, by the result of a meeting between themselves, etc.
Berger coefficient- a way to determine places in competitions among participants who have scored an equal number of points. The method of determining the place by the Berger coefficient was originally developed for round-robin (everyone plays with everyone) chess tournaments. Later, this method was applied to other competitions, such as shogi and go.
Order of Evaluation
In round-robin tournaments, where a certain constant number of points is awarded for a victory, a draw and a defeat (for example, in chess 1 point is given for a victory, 0.5 points for a draw, 0 points for a defeat; less often - 3 - for a victory and 1 for draw, for example, in London Chess Classic 2010), it often happens that two or more participants score the same number of points. To determine which of these participants ranked higher, the Berger coefficients of the participants are calculated.
The Berger coefficient of a certain participant is the sum of all the points of opponents against whom this participant has won, plus half the sum of the points of opponents with whom this participant has drawn. The idea on which the coefficient is based: of two participants equal in number of points, the one who won against stronger opponents, that is, those who scored more points, is stronger. Therefore, a participant with a higher Berger coefficient is awarded a higher final place in the tournament.
The Berger coefficient was invented for round-robin tournaments, but can, if necessary, be used in other drawing schemes, where the players whose places need to be distributed play an equal number of games. It can also be used in tournaments according to the Swiss system, although the Buchholz coefficient is traditionally used there. In round-robin tournaments since 1985, the “simplified Berger” (proposed by M. Dvoretsky) has also been used: the points of all opponents against whom the chess player won are taken with a plus sign, and all those to whom he lost - with a minus sign, by the sum and is considered the best result. This allows you to reduce the calculations and not pre-divide in half most of the results.
Example
Summary table of the hypothetical round robin:
| № | Members | 1 | 2 | 3 | 4 | 5 | 6 | 7 | + | − | = | Glasses | Place | KB | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | Ivanov | ½ | ½ | 1 | 1 | 1 | 1 | 4 | 0 | 2 | 5 | I | 11,75 | ||
| 2 | Petrov | ½ | ½ | ½ | 1 | 1 | 1 | 3 | 0 | 3 | 4½ | II | 10 | ||
| 3 | Sidorov | ½ | ½ | ½ | ½ | 1 | 1 | 2 | 0 | 4 | 4 | III | 9 | ||
| 4 | Kuznetsov | 0 | ½ | ½ | 1 | 1 | 1 | 3 | 1 | 2 | 4 | IV | 7,75 | ||
| 5 | Smirnov | 0 | 0 | ½ | 0 | 1 | 1 | 2 | 3 | 1 | 2½ | V | 3 | ||
| 6 | Vasiliev | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 5 | 0 | 1 | VI | 0 | ||
| 7 | Nikolaev | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | VII | 0 |
Designations: 1 - victory, ½ - draw, 0 - defeat, KB - Berger coefficient.
Participants Sidorov and Kuznetsov scored the same number of points, 4 points each. Which of them will take third place is decided by the Berger coefficient.
Sidorov's Berger coefficient is: 2.5 (half of Ivanov's points) + 2.25 (half of Petrov's points) + 2 (half of Kuznetsov's points) + 1.25 (half of Smirnov's points) + 1 (all of Vasiliev's points) + 0 ( all points of Nikolaev) = 9.
Kuznetsov's Berger coefficient is as follows: 0 (for the loss to Ivanov) + 2.25 (half of Petrov's points) + 2 (half of Sidorov's points) + 2.5 (all Smirnov's points) + 1 (all Vasiliev's points) + 0 (all points Nikolaev) = 7.75.
Thus, participant Sidorov has a higher Berger coefficient than participant Kuznetsov (9 versus 7.75), so the third place is awarded to Sidorov. The Berger coefficient is higher for someone who wins or ties with stronger players (players who score more points). In the example above, winning against a participant with zero points does not contribute to the Berger coefficient.
Berger coefficient- a way to determine places in competitions among participants who have scored an equal number of points. The method of determining the place by the Berger coefficient was originally developed for round-robin (everyone plays with everyone) chess tournaments. Later, this method was applied to other competitions, such as shogi and go.
Order of Evaluation
In round robin tournaments where a certain constant number of points are awarded for a win, a draw and a loss (for example, in chess 1 point is awarded for a win, 0.5 points for a draw, 0 points for a loss), it often happens that two or more participants score the same number of points. To determine which of these participants ranked higher, the Berger coefficients of the participants are calculated.
The Berger coefficient of a certain participant is the sum of all the points of opponents against whom this participant has won, plus half the sum of the points of opponents with whom this participant has drawn. The idea on which the coefficient is based: of two participants equal in number of points, the one who won against stronger opponents, that is, those who scored more points, is stronger. Therefore, a participant with a higher Berger coefficient is awarded a higher final place in the tournament.
The Berger coefficient was invented for round-robin tournaments, but can, if necessary, be used in other drawing schemes, where the players whose places need to be distributed play an equal number of games. It can also be used in tournaments according to the Swiss system, although the Buchholz coefficient is traditionally used there. In round-robin tournaments since 1985, the “simplified Berger” (proposed by M. Dvoretsky) has also been used: the points of all opponents against whom the chess player won are taken with a plus sign, and all those to whom he lost - with a minus sign, by the sum and is considered the best result. This allows you to reduce the calculations and not pre-divide in half most of the results.
Example
The final table of a hypothetical round-robin tournament:
| № | Members | 1 | 2 | 3 | 4 | 5 | 6 | 7 | + | − | = | Glasses | Place | KB | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | Ivanov | ½ | ½ | 1 | 1 | 1 | 1 | 4 | 0 | 2 | 5 | I | 11,75 | ||
| 2 | Petrov | ½ | ½ | ½ | 1 | 1 | 1 | 3 | 0 | 3 | 4½ | II | 10 | ||
| 3 | Sidorov | ½ | ½ | ½ | ½ | 1 | 1 | 2 | 0 | 4 | 4 | III | 9 | ||
| 4 | Kuznetsov | 0 | ½ | ½ | 1 | 1 | 1 | 3 | 1 | 2 | 4 | IV | 7,75 | ||
| 5 | Smirnov | 0 | 0 | ½ | 0 | 1 | 1 | 2 | 3 | 1 | 2½ | V | 3 | ||
| 6 | Vasiliev | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 5 | 0 | 1 | VI | 0 | ||
| 7 | Nikolaev | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | VII | 0 |
Designations: 1 - victory, ½ - draw, 0 - defeat, KB - Berger coefficient.
Participants Sidorov and Kuznetsov scored the same number of points, 4 points each. Which of them will take third place is decided by the Berger coefficient.
Sidorov's Berger coefficient is: 2.5 (half of Ivanov's points) + 2.25 (half of Petrov's points) + 2 (half of Kuznetsov's points) + 1.25 (half of Smirnov's points) + 1 (all of Vasiliev's points) + 0 ( all points of Nikolaev) = 9.
Kuznetsov's Berger coefficient is as follows: 0 (for the loss to Ivanov) + 2.25 (half of Petrov's points) + 2 (half of Sidorov's points) + 2.5 (all Smirnov's points) + 1 (all Vasiliev's points) + 0 (all points Nikolaev) = 7.75.
Thus, participant Sidorov has a higher Berger coefficient than participant Kuznetsov (9 versus 7.75), so the third place is awarded to Sidorov. The Berger coefficient is higher for someone who wins or ties with stronger players (players who score more points). In the example above, winning against a participant with zero points does not contribute to the Berger coefficient.
Story
The Czech chess master Oscar Gelbfus was the first to propose such a scoring system in August 1873. In practice, for the first time such a distribution system was used by William Sonneborn (-) and Johann Berger at a tournament in Liverpool in 1882. In 1886, scoring by the Berger coefficient was put into practice.
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Literature
- Chess: encyclopedic dictionary / ch. ed. A. E. Karpov. - M .: Soviet Encyclopedia, 1990. - S. 357-358. - 624 p. - 100,000 copies. - ISBN 5-85270-005-3.
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An excerpt characterizing the Berger Coefficient
The officers got up. Prince Andrei went out with them outside the shed, giving his last orders to the adjutant. When the officers left, Pierre went up to Prince Andrei and just wanted to start a conversation, when the hooves of three horses clattered along the road not far from the barn, and, looking in this direction, Prince Andrei recognized Wolzogen and Clausewitz, accompanied by a Cossack. They drove close, continuing to talk, and Pierre and Andrei involuntarily heard the following phrases:– Der Krieg muss im Raum verlegt werden. Der Ansicht kann ich nicht genug Preis geben, [The war must be transferred into space. This view I cannot praise enough (German)] - said one.
“O ja,” said another voice, “da der Zweck ist nur den Feind zu schwachen, so kann man gewiss nicht den Verlust der Privatpersonen in Achtung nehmen.” [Oh yes, since the goal is to weaken the enemy, then private casualties cannot be taken into account (German)]
- O ja, [Oh yes (German)] - confirmed the first voice.
- Yes, im Raum verlegen, [transfer to space (German)] - Prince Andrei repeated, angrily snorting his nose, when they drove by. - Im Raum then [In space (German)] I left a father, and a son, and a sister in the Bald Mountains. He doesn't care. This is what I told you - these gentlemen Germans will not win the battle tomorrow, but will only tell how much their strength will be, because in his German head there are only arguments that are not worth a damn, and in his heart there is nothing that alone and you need it for tomorrow - what is in Timokhin. They gave all of Europe to him and came to teach us - glorious teachers! his voice screamed again.
"So you think tomorrow's battle will be won?" Pierre said.
“Yes, yes,” Prince Andrei said absently. “One thing I would do if I had the power,” he began again, “I would not take prisoners. What are prisoners? This is chivalry. The French have ruined my house and are going to ruin Moscow, and have insulted and insult me every second. They are my enemies, they are all criminals, according to my concepts. And Timokhin and the whole army think the same way. They must be executed. If they are my enemies, they cannot be friends, no matter how they talk in Tilsit.
“Yes, yes,” Pierre said, looking at Prince Andrei with shining eyes, “I completely, completely agree with you!”
The question that had been troubling Pierre from Mozhaisk Mountain all that day now seemed to him completely clear and completely resolved. He now understood the whole meaning and significance of this war and the forthcoming battle. Everything that he saw that day, all the significant, stern expressions of faces that he caught a glimpse of, lit up for him with a new light. He understood that latent (latente), as they say in physics, warmth of patriotism, which was in all those people whom he saw, and which explained to him why all these people calmly and, as it were, thoughtlessly prepared for death.
“Do not take prisoners,” continued Prince Andrei. “That alone would change the whole war and make it less brutal. And then we played war - that's what's bad, we are magnanimous and the like. This generosity and sensitivity is like the generosity and sensitivity of a lady, with whom she becomes dizzy when she sees a calf being killed; she is so kind that she cannot see the blood, but she eats this calf with sauce with gusto. They talk to us about the rights of war, about chivalry, about parliamentary work, to spare the unfortunate, and so on. All nonsense. In 1805 I saw chivalry, parliamentarianism: they cheated us, we cheated. They rob other people's houses, let out fake banknotes, and worst of all, they kill my children, my father and talk about the rules of war and generosity towards enemies. Do not take prisoners, but kill and go to your death! Who has come to this the way I did, by the same suffering...
Prince Andrey, who thought that it was all the same to him whether Moscow was taken or not taken in the same way as Smolensk was taken, suddenly stopped in his speech from an unexpected convulsion that seized him by the throat. He walked several times in silence, but his body shone feverishly, and his lip trembled when he began to speak again:
- If there was no generosity in the war, then we would go only when it is worth it to go to certain death, as now. Then there would be no war because Pavel Ivanovich offended Mikhail Ivanovich. And if the war is like now, then the war. And then the intensity of the troops would not be the same as now. Then all these Westphalians and Hessians led by Napoleon would not have followed him to Russia, and we would not have gone to fight in Austria and Prussia, without knowing why. War is not a courtesy, but the most disgusting thing in life, and one must understand this and not play war. This terrible necessity must be taken strictly and seriously. It's all about this: put aside lies, and war is war, not a toy. Otherwise, war is the favorite pastime of idle and frivolous people ... The military estate is the most honorable. And what is war, what is needed for success in military affairs, what are the morals of a military society? The purpose of the war is murder, the weapons of war are espionage, treason and encouragement, the ruin of the inhabitants, robbing them or stealing for the food of the army; deceit and lies, called stratagems; morals of the military class - lack of freedom, that is, discipline, idleness, ignorance, cruelty, debauchery, drunkenness. And despite that - this is the highest class, revered by all. All kings, except for the Chinese, wear a military uniform, and the one who killed the most people is given a big reward ... They will converge, like tomorrow, to kill each other, they will kill, maim tens of thousands of people, and then they will serve thanksgiving prayers for beating there are many people (of which the number is still being added), and they proclaim victory, believing that the more people are beaten, the greater the merit. How God watches and listens to them from there! - Prince Andrei shouted in a thin, squeaky voice. “Ah, my soul, lately it has become hard for me to live. I see that I began to understand too much. And it’s not good for a person to eat from the tree of the knowledge of good and evil ... Well, not for long! he added. “However, you are sleeping, and I have a pen, go to Gorki,” Prince Andrei suddenly said.
