1. Principles

A mathematical battle is a competition of two teams, which consists in solving problems, narratoring their own solutions and finding and exposing shortcomings in the solutions of opponents. First, the participating teams receive the conditions of battle tasks. The set of tasks is the same for both teams and is not known to them in advance. For some time, the teams independently solve problems, and then gather in a common audience and start, in fact, the battle.

If one of the teams sends its representative —a speaker who will tell the board the solution of the problem, then the team of opponents, instead, will send an opponent who will try to find flaws in the solution of opponents, and, if he manages to prove that the speaker does not have a solution, then perhaps cites his own.

At the same time, the speeches of the opponent and the speaker are evaluated by the jury in points (for solving and for opposition, respectively).

Upon discussion by the participants of each decision, the jury makes a ruling: it distributes 12 points for it between the speaker and the opponent. The speaker receives 12 points if he independently and unmistakably told the correct solution (the optimal solution is not taken into account). When the solution contained errors, was incomplete or incorrect, contained a hole in the solution, the speaker receives the corresponding part of the total number of points (12), the opponent — half the cost of shortcomings (holes), which he pointed to. The rest of the 12 points the jury takes for itself. Under certain conditions, the opponent not only points out the shortcomings in solving opponents, but also tells how to get rid of these shortcomings. Then he also gets the second half of the cost of the hole. In the event that the solution is wrong at all, the opponent can show his own and earn all 12 points (see "Role Replacement"). For more information, see <a0><a1> What "Points allocation" item

The period of time during which the battle participants discuss one task (or find out that none of the teams know its solution) is called a round. The task, which has already been considered in any of the rounds (albeit unsuccessfully), cannot be re-considered.

The winner of the battle is a team whose points are higher. But if the difference in the results of teams does not exceed 3, it is believed that the fight ended in a draw. If, according to the rules of the tournament, this particular fight cannot end in a draw, then the jury informs the teams before the start of the fight and announces the procedure for determining the winner.

2. Team Structure

The standard number of participants in the team is determined by the rules of the competition. At the All-Ukrainian tournament it is usually 6 participants, and at Kyiv –12. At the same time, according to the decision of the organizing committee, in some teams their number may differ from this number. The smallest number of team members is 5. In case of force majeure conditions, the team may be admitted to battle and in smaller quantities by the decision of the organizing committee, but their number cannot be less than 3. In case of disrespect for the lack of a valid team at the beginning of the battle, it does not receive the conditions until the necessary 5 team members gather. If this number is not gathered before the start of the battle, then the team is credited with a technical defeat with the corresponding organizing notes. The team must determine who will be its captain and deputy captain. The captain makes a final ruling on each issue, has the right to take time out for the team, replace the speaker or opponent, contact the jury (other team members should contact the jury during the fight through the captain or his deputy). When the team captain reports or oppositions, his role is played by the deputy. In particular, the decision to replace the captain (as a speaker or opponent) is made by his deputy.

3. Distribution of conditions. Preparing for battle

The conditions of the tasks are simultaneously announced to both teams-participants, between which the battle is held, except in case of delay in the battle of the team without good reasons, as specified in paragraph 2. The set of tasks for both commands is the same.

During the preparation for battle, team members are prohibited from:

• use third-party assistance;
• use computers and any other office equipment, go online, etc.;
• use any literature.

From assistive technologies (jury permission) can only be used by conventional micro-accumulator (which does not have a programming function). In case of violations by one of the teams of these rules, the jury may warn, disqualify individual members of it or the entire team for this fight and it will be counted as a defeat.

At the time specified by the competition regulations, the team must come to the office where the fight will take place, and its captain should provide the jury with a list of team members, indicating the names of the captain and the deputy.

4. Captains Competition

To determine the order of challenges, at the beginning of the battle, the so-called captains competition is held. It is attended by representatives of the teams — one by one — who were determined by the captains (often these are the captains themselves, but not necessarily — hence the name of the competition). A jury member offers representatives a task that they must perform. This can be a task where you must first give the correct answer. The first is the participant who will raise his hand first. If the answer is correct, the team he represents wins. Otherwise, the competition will be won by rivals. If no participant raises their hands within a minute (or another period of time determined by the jury), the questions of the competition are changed, or the winner is determined by the draw. Sometimes the captains competition has a game form. A jury member explains to the participants the rules of a certain (usually, precisely for this fictional) game. The draw is chosen by the captain, who is invited to choose which of the players they want to be – starting or second. Acceptable, but not desirable, other forms of competition, including non-matic.

The team that won the captains' competition determines who will make the first challenge.

5. Order in which the call is reconciled

The caller has a moment to decide how it will work. The command can:

• Call opponents to any of those tasks that have not yet been discussed. In this case, the team captain clearly and loudly reports that his team is calling opponents, and the number of the task being called (for example: "We call the rival team to task number 5").
• Take a ten-minute break if the rules allow it (see Long breaks). If so, the team determines which task opponents will be called on the break, writes down its number and passes the recorded jury. The team has one more minute for all this. After the break, the call is announced by the jury.
• Refuse to call. This should also be clearly and loudly reported by the captain (see Refusal to call).

Hearing the decisions of rivals, the team being called up can:

• Accept the call. In this case, she sends a speaker, rivals – opponent.
• To check the correctness of the call is to refuse the call and find out if the opponents solved the problem they were called to. In this case, the team (if it wishes) sends to the opponent's board, and the opponents — the speaker. The captain informs loudly and legibly about accepting or checking the correctness of the call.
• If the opponents (as soon as or earlier) refused to call, the team can send its representative to report one of the tasks that have not yet been considered — the captain clearly calls the problem number, and the opponents (if they wish) expose the opponent. Before reporting, if rules allow, a team can take a ten-minute break (see Long breaks).

A team that wishes to save exits to the board may refuse to expose the opponent. Then she does not take part in this round (she can no longer change her decision).

6. Report

The main task of the speaker is to make a difference between the number of points that he will earn and the number of points that the opponent earns as much as possible (this difference directly affects the outcome of the battle).

The report should contain answers to all questions of the problem and full proof that they are correct. In particular, the speaker must prove each statement formulated by him or refer to him as well-known (in the latter case, if the jury agrees with the general know of this fact, the speaker must provide the full wording of the statement at the request of the opponent, but should not prove it). The speaker must repeat, at the request of the opponent or the jury, any part of his solution.

The time for the report is limited to 15 minutes after which the jury decides whether to allow the presenter to tell further. After 15 minutes, the opponent or jury may ask the speaker to give a full answer (if it is in this task), or ask the speaker to provide a solution plan. (For example, in the case of an analytical solution to the geometric problem, the jury has the right to ask the speaker to write the main results on the board, and then to discuss individual points.)

The presenter may have records with him and use them during the report, but the jury has the right to prohibit him from using them if he believes that the speaker reads the decision on the records. Any material that the speaker wishes to use, he must take to the board at the beginning of the round, first demonstrating to the jury. In exceptional cases, however, the jury may allow the presenter to take the necessary record after the timeout taken by his team. The report cannot refer to the results of counts carried out using a calculator that is not otherwise confirmed.

The speaker has the right to:

• Before the performance, bring to the board all the necessary information: drawings, counting, etc.
• Do not answer opponent's questions that are set before the discussion.
• Refuse to answer questions citing the fact that he does not know the answer to it, or he has already answered it (explaining when and how), or to the fact that the question is not correct or does not correspond to the topic of discussion. If the opponent does not agree with the last two arguments, the arbitrator acts as a jury.

After each question or comment from the opponent or jury, the presenter has a moment to think through the answer. If the presenter doesn't start responding in a minute, they think they can't reply. This minute does not include half-minute breaks taken by the speaker's team or opponent. If questions or comments are close in content to each other, the jury may limit the time a presenter has to ponder the answers.

The speaker must clearly record the end of his report by announcing "Report is over." Within 10 seconds after that, the speaker and his team have the right to say these words (the opponent is silent at this time). If this does not happen the word is passed on to the opponent.

7. 1990 In-Place

The main task of the opponent is to make the difference between the number of points earned by him and the number of points that the speaker will receive as much as possible. The opponent receives points for finding and exposing errors in solving opponents.

Until the report is completed, the opponent can only ask questions with the consent of the speaker, but he has the right to ask to repeat the parts of the solution and allow the speaker not to prove any trivial from his point of view of the statement. At the end of the report, the opponent has the right to ask questions to the speaker. If within a minute the opponent did not ask any questions, it is considered that he has no questions. If the opponent believes that the speaker is taking time, invents a solution near the board, or that a significant part of the report is not a statement of the solution to this problem, he has the right (but not earlier than 10 minutes after the start of the report) to ask the speaker to announce a response or plan for further considerations. If the jury considers this request not true, it can remove it.

If the opponent has a counter-application to the solution and this example is in itself a solution (for example, when the question is "Is it possible…?"), the opponent can declare it (to declare: "I do not agree with the solution – I have a counter-example."), but the counter-example does not cite (the jury has the right to ask the opponent to give an example to them in writing). The speaker has a moment to react to this statement, by which the opponent should give his example, and the speaker loses the right to earn points for correcting the shortcomings indicated by the example of the opponent (in some cases, this disadvantage may not be equal to the entire task).

Similarly, if the solution requires the analysis of several cases, the opponent may declare: "I do not agree with the solution, because not all cases have been disassembled." Again, the opponent should not immediately point to these cases (but must point to them to the jury if they ask).

8. Jury participation in the discussion

The jury constantly participates in the discussion. Initially, as an arbitrator, i.e. may ask for clarification of a question or answer, it can remove questions to which either an answer has been answered or that only delay the discussion time, etc. The jury has the right to discontinue a report or discussion if it has come to a standstill, has lost touch with the task being discussed, or cannot end in a reasonable time.

At the end of the discussion between the speaker and the opponent, the jury asks its questions (in fact, it acts as an opponent).If necessary, it can interfere in discussions earlier.

9. Correctness of the call. Verifying validity

If the command being called does not know the solution of the problem or does not want to tell it for other reasons, it can check the correctness of the call. In this case, this command sends to the opponent's board. The team that called now has the right to solve the problem, and it exposes the speaker. The round takes place according to the usual rules. The only exception is that there can be neither a complete nor partial replacement of roles during the validity check (see Replacement roles). If during the correctness check the opponent shows that the opponents solved the problem in general incorrectly, the call is recognized as incorrect and the next call (as punishment for incorrectness) is made by the same team. Otherwise, the call is recognized as correct and the order of calls does not change. If the opponent agreed with the speaker's decision, the call is correct, despite the fact that the jury can find any shortcomings in the solution.

If the team that called immediately admits that it has no solution, the team representatives do not go to the board, the team that checked the correctness is awarded 6 points, the call is recognized as incorrect and in the next round causes the same team.

10. Replacing roles

If the opponent has shown that the solution of the speaker's team is generally wrong, and the jury doubts that the central idea of the solution (if it is specific) can be used in the correct solution, the opponent may ask for a complete replacement of roles with the speaker. If the opponent wants it, he becomes a speaker, and the former speaker becomes the new opponent. At the same time, the presenter already has 6 points (for the error found in the opponent's report), and now the players play 6 points remaining. A new opponent can now earn points on opposition.

If the opponent pointed out some significant shortcomings in solving opponents, and the speaker failed to eliminate them, the opponent gets the right to partially replace the roles. In this case, he — now as a presenter — tells how to eliminate certain (and possibly all) shortcomings by first announcing what he will do. Now it is the duty of the new speaker to fulfill the announced. With that in sight, the former speaker may be against him. The new speaker receives points for proven useful statements (the corresponding part of 12 points), the opponent — half the cost of the errors he found. Only the statements that the new speaker formulated before the discussion are discussed are discussed. If there was a partial replacement, and there were flaws in the initial solution, which the former opponent did not notice, the jury should continue the discussion about them with the former speaker immediately after the announcement by the new speaker of the statements that he was going to prove. The former opponent (new speaker) does not participate in this discussion.

Twice in one round, roles are not replaced. Replacements, although partial, are not made when checking the correctness of the call or if earlier one of the commands refused to call. Before the full and partial replacement of roles (before the opponent decides whether to accept a replacement), the jury may (but is not obliged to) declare to the teams the exact or approximate distribution of points, which will be awarded if the opponent does not agree to a replacement.

11. Scoring

Each task is rated at 12 points, which at the end of the round are distributed between the speaker, the opponent of those juries. If the speaker, without relying significantly on the questions and comments of the jury and the opponent, gave a correct and complete solution, all 12 points are given to him. If holes were found in the solution, then the jury determines their cost at the end of the discussion. After that, the opponent immediately receives half the cost of the holes found by him (for example, if the hole was evaluated in half a task, that is, 6 points, then the opponent receives 3 points for finding it). If some of these holes were partially or completely closed, then half of the cost of closed holes is distributed between the opponent and the speaker in proportion to their contribution to the closing of these holes (if in the above example the hole was closed by the speaker, then the distribution of points will be as follows: 9 points will be received by the speaker, and 3 – the opponent. If this hole is closed by the opponent, after a partial change of roles, then both players will receive 6 points each). At the same time, the opponent's contribution may be not only closing holes (in the case of a complete or partial change of roles), but also helping the speaker (announcing some considerations or questions). All the other white jury leaves themselves.

If the holes in the solution were found by the speaker himself and they were not closed by him (or with the help of the team), then the opponent would get points for them as if he himself had found these shortcomings. In particular, if, having received a check of correctness, the captain of the team that made the call immediately admitted that they did not have a solution, then the rival team automatically receives 6 points. In this case, the presenter and opponent are not determined and the exits to the board are not taken into account.

If there was no complete change of roles, then the opponent can not get more than 6 points. An opponent who has proved incorrect in a call gets 6 points regardless of the cost of holes found by him (usually a hole of 6 points is enough to determine the incorrectness).

12. Refuse to call. Additional rounds

If the command that should call does not want to do this (for example, it no longer has solved problems and is afraid of checking correctness), it may refuse. Then rivals get the opportunity to tell the solutions of one or more problems not considered so far. Solutions are told sequentially (but in random order — it is determined by the command itself) in "additional" rounds. The team that refused to call puts up (if it wants) the opponent for each round. Rounds are played according to the usual rules. The opponent can earn points for error detection, but he can no longer replace (at least partially) the speaker.

13. Delayed rounds

If the report on the solution of a problem lasts too long and it is already clear which of the teams is calling on, the jury may suggest that team captains postpone the round — move the discussion to another room, and allow the teams to continue the battle. Points for the postponed task are awarded later in this case. Having consented to the postponement of the round, the captains must take into account that the representative of their team, while remotely discussing the task, will not be able to communicate with the team, and the team will not be able to expose him as a speaker or opponent.

14. Timeouts

During the battle (during rounds), each team can take half-minute breaks (timeouts) 6 times, during which it can communicate with its representative, who reports or opponents the solution. When one of the teams takes a break, opponents are also allowed to communicate for 30 seconds. This time is not limited, even if the initiators of the break complete communication in advance.

15.Replace a presenter or opponent

The team can replace its speaker or opponent with another participant at any time — at the expense of 2 timeouts. At the same time, she is allowed to use one or both breaks. In addition, the team can decide to replace only after one or two consecutive timeouts and it is at the expense of these breaks to replace its representative (if after the break the team representative managed to return to the board, this timeout cannot be used for replacement). If a team does not take one or both timeouts when replacing a team, the opponents cannot use them either.

16.Long breaks

The team that will make the call or tell the solution after the other team refuses to call has the right to ask the jury to organize a 10-minute break , provided that the previous break ended (or, if it is the first break, the fight began) no later than an hour and a half before. The team writes the number of the task to be called (or told) on the worksheet and submits the jury sheet. The team can also record a refuse call.

If necessary, the jury after one of the rounds may announce a forced break, noting its duration. Usually it also does not exceed 10 minutes. The command that makes the call (or tells the solution) must also perform the specified procedure.

17.restriction

Each team member can go to the board as a speaker or opponent in total no more than twice. If a team replaces its delegate, both the former presenter or opponent and the new one are counted. In the presence of certain force majeure circumstances (quarantine, transport collapse, etc.), if the team consists of only 3-5 participants, then each of them can go to the board three times or even more times. Both teams warn the jury before the fight. Under such conditions, rivals can also exercise this right, even if they have a sufficient number of participants.

Teams have to think about each common decision by minute. No time limits have been imposed on the jury.

18.The end of the battle. Determination of the winner

The fight is considered complete when the tasks for discussion are over or when one of the teams refused to call and the other refused to tell the solutions. The jury must suspend the fight after any of the rounds if the winner of the fight is already determined, if the score as such is unimportant and inform the teams about it. At the request of both teams, they can finish the battle and play unseeded tasks. Then the jury continues to conduct a battle protocol and score according to standard rules.

The result of the teams determines the sum of their points for all rounds. If the difference in points exceeds 3, the team that scored more points is recognized as the winner. Otherwise, it is believed that the fight ended in a draw. If the tournament takes place according to the scheme, which takes into account the total number of points, then the team receives 2 points for the victory, 1 point for a draw, no points are awarded for the defeat.

If the rules of the competition provide for a draw to determine the winning team, then a short additional competition is held – a blitz, the format of which is also determined by the regulations. Then the team that won without a blitz gets 3 points, the losing team – 0 points. If a team wins the blitz, it gets 2 points and the team that lost the blitz gets 1 point.

19.Structure and competence of the jury. Jury interaction with teams

The brigade that judges the fight usually consists of three to five members of the jury. Among them, one, the chairman, makes a final ruling on each issue and is responsible for the actions of all other members of the brigade.

Jury members must strictly adhere to the rules of battle and separate rules of competition (if any), but in controversial situations it is they who have the right to final discretion. The jury should explain each decision and publicly motivate the distribution of points awarded in the rounds. If the unveiled jury decision for the team or its leaders seems to be contrary to the rules of the mathematical battle, they can ask for a five-minute break during which they can talk to the jury (possibly on the sidelines) regarding the final decision. If no consensus is found, it is possible to invite the head of the jury or his representative for discussion, the decision of which on all issues is final. All these questions need to be clarified before the start of the new round.

After the start of the new round, the distribution of points in the previous round may change by the decision of the chairman of the jury upon reaching a certain consensus with the participants and managers. The course of the battle (order of calls) can no longer change. The result of the battle becomes final as soon as it is completed.

During the battle, team captains and their deputies can contact the jury with questions, requests or suggestions. Similarly, other interested persons – teachers, parents, etc. – can apply to the jury.

20.Battle Protocol

The jury protocols the course of the fight. The protocol states: status and name of the battle, names and compositions of teams, names of captains and their deputies; the name of the team whose representative won the "captains competition"; sequence and status of calls awarded to the distribution of points in each round; the number of timeouts taken and the number of replacements performed in each round; the result of the battle and the composition of the jury. Also in the protocol can record the progress and result of an additional competition, which determines the winner of the fight in the event of a draw (if such a competition is provided by the rules of battle or tournament).

On the board, the jury maintains an abbreviated protocol: records the names of the teams (possibly abbreviated), sequence and status of calls, the distribution of points and the number of times taken in each round, the result of the battle.

21.Fines and disqualification

The jury has the right to impose a fine on the team (to pick up a score or several in its favor) for the improper behavior of its participants. Individual participants may be removed from the audience where the fight takes place for improper behavior of the jury. If that doesn't help, the jury may well disqualify one or both teams for improper conduct, violation of the rules of battle, or dishonest play. If one team is disqualified, the other is awarded a technical victory. And if both, the defeat is awarded to both teams. In a tournament fight in this case, both teams score 0 points each. and at the cup, both teams come out of further struggle.