Maths Battle Original Rules

Maths Battle is a competition between two teams in solving mathematical problems and presenting their solutions.

General Provisions

Maths Combat is a competition between two teams in solving mathematical problems. First of all, the teams are presented with the problems and given a certain time limit to solve them.

While solving the problems, the team may use any literature, but is not permitted to talk about the solution to these problems with anyone except with the members of the jury. At the end of the allotted time, the actual battle begins, when the teams present the solutions of the problems in accordance with the given rules.

One of the teams presents the solution, the other acts as an opponent and looks for mistakes (defects) in the presentation. The jury rates the speeches of the speaker (the team member presenting the solution) and the opponent (the team member opposing the speaker) by allocating points to each of them in accordance with the rules.

If the teams have neither solved any problems by the end of the given time nor found any mistakes while discussing the proposed solution, the jury of the battle can take some of the points available in the round (or even all of them).

The winner of the battle is the team that gets the most points in total.

If, at the end of battle, the results of the teams do not differ by more than three points, the battle is considered to have ended in a draw. If, for some reason, the battle cannot end in a draw, the jury is to declare this to the teams before the battle and announce the procedure for determining the winner.

The General Scheme of the Battle

The battle consists of several rounds.

At the beginning of each round (if there is no rejection of the challenge - see para. “The End of the Battle”), one of the teams challenges the other on one of the problems, the solution of which has not yet been presented (for example: “We challenge the opposing team on problem number 8”). Then, the challenged team announces whether it accepts the challenge, i.e. whether it agrees to present the solution to this problem (it is given no more than one minute to make a decision on the challenge). If a team accepts the challenge, it nominates a ‘speaker’, who explains the solution. The team that made the challenge nominates an ‘opponent’, whose duty is to look for errors in the presented solution. If the challenge is not accepted, then the team that made it is obliged to nominate a speaker, and the team that rejected the challenge has to nominate an opponent. In this way, one can consider that the reasonableness of the challenge is being assessed.

The Captain’s Competition

The ‘captain’s competition’ is held at the beginning of the battle.

The team that wins the captain’s competition determines which team makes a challenge first.

The captains are given a problem. The first captain to inform the jury that she/he knows the answer to the problem gets the right to explain it. If he/she presents the correct solution, he/she wins, but if it is incorrect – his rival wins. This is what is meant by “the correct solution”: simply the correct answer, the answer with an explanation, and so on - the jury specifies the nature of the correct answer before the competition of the captains begins.

The jury assigns a specific period of time for the solution of the problem for the captain’s competition. If, during this time, none of the captains express their desire to answer, the jury may replace the problem or determine the winner by draw. Instead of the problem the jury can offer to the captains to play in any game. There are other methods of carrying out the captain’s competition.

The jury determines the way the captain’s competition will be held in advance and informs the teams before the beginning of the battle.

Each team has the right to nominate any member of their team for the captain’s contest.

The Progress of the Round

Report

At the beginning of the round, the speaker explains the solution on the whiteboard. The presentation should contain answers to all the questions in the problem and proof of the accuracy and completeness of the received solutions. In particular, the speaker must prove each formulated interim statement or refer to it as already known. The speaker should try to ensure that his presentation is clear. In particular, she/he must repeat any part of her/his report at the request of the opponent or the jury. The report time is limited to 15 minutes, after which the report can be continued only with the permission of the jury. The speaker can use a paper with drawings during her/his presentation and (with special permission of the jury) calculations, but is not permitted to bring any text with him.

The speaker has the right to:
• write/draw any necessary information on the whiteboard before the presentation;
• not answer any questions raised by the opponent before the beginning of the discussion;
• ask the opponent to clarify his question (in particular, the speaker may propose his own version of the question: “Do I understand correctly, that you asked me about something and something?”);
• refuse to answer the question, by saying that
1. he has no answer to this question;
3. the question is incorrect or beyond the scope of scientific discussion for the stated problem. In case the opponent disagrees with the basis of 2) and 3) the jury arbitrates the dispute.
The speaker is not obliged to:
• explain how he/she worked out the solution if he/she can prove that it is correct and complete;
• compare his way of solving the problem with other possible methods, including in terms of brevity, beauty and aptitude for solving other problems.*

Opposing

While the report is in progress, the opponent can ask questions only with the consent of the speaker, but she/he has the right to ask for any part of the solution to be repeated. He/she can allow the speaker not to prove any obvious (from his perspective) facts. After the end of the report, the opponent has the right to ask the speaker questions. If the opponent does not ask a single question within a minute, it is considered that he/she has no questions. If the speaker does not begin to answer a question within a minute, it is considered that he/she has no answer. As a question, the opponent can:

• demand that any part of the report is repeated;
• ask for a clarification of any of the speaker’s statements, including:
1. asking for a definition of any term (“What do you mean by …”);
2. reformulate the speaker’s statement in his own words and ask for confirmation (“Do I understand correctly that you are stating the following: …”);
• ask the speaker to prove the statement posed by him, if it is not obvious or well-known (in the case of a dispute, the jury is to decide whether the question is known or apparent; in any case, as known is considering the facts, included in the general educational program in mathematics);
• express satisfaction or explain his motivated dissatisfaction with it after receiving the answer to the question.

If the opponent thinks that the speaker is stalling for time, coming up with a solution at the whiteboard, or that a significant part of the report does not contain the solution of the problem under discussion, she/he has the right to ask the speaker to submit a solution or an outline of the remaining argument (but no earlier than 10 minutes after the beginning of the report). The speaker and the opponent must:

• express themselves in a polite and proper manner,
• not ‘get personal’ while criticizing each other’s statements;
• repeat and refine their questions and answers at the request of each other or of the jury.

According to the results of the report and responses to the questions, the opponent has the right to give his assessment of the report and the discussion in one of the following ways:

1. recognise that the solution is correct;
2. recognise the solution (response) as basically correct, but having drawbacks and/or gaps which he/she has to identify;
3. recognise the solution (response) as incorrect, pointing out the mistakes in the proven key assertions of the report, or by providing a contrary example, or by specifying the significant gaps in the argument’s or solution’s construction.

If the opponent has agreed with the solution, she/he and his team are no longer involved in this round. If the opponent has a contrary example, refuting the solution of the speaker in general, and this contrary example is a solution to the problem itself (this happens, for example, in cases where the question of the problem sounds like “Can you …?”, “Is it true that …?” and so on), the opponent has the right to declare: “I do not agree with the solution, I have a contrary example”, but not to show the contrary example to the speaker yet (the jury is entitled to require submission of a contrary example in writing to ensure the correctness of the opponent’s statement). In this case, if the speaker does not change his solution within a minute, or after the break that was taken by the team, the opponent is entitled to present the contrary example mentioned to the speaker; in which case the speaker and his team no longer have the authority to change the solution or response.

Similarly, if the solution requires an executive search, the opponent has the right to say, “I do not agree with the decision, not all cases were considered”, without specifying to the speaker exactly what case was not considered. The further actions of the speaker, the jury and the opponent are the same as in the situation with a contrary example.

Participation of the jury in the discussion

After the end of the dialogue between the speaker and the opponent, the jury asks its questions. If necessary, it has the right to intervene before, and during the dialogue of the speaker and the opponent.

Speakers and the team

The speaker and the opponent can go to their captains with a request for a replacement or for a break for consultation. Other communication between the team and the speaker (opponent) is permitted only during a half a minute break, which can be taken by any team at any time (in this case the rivals can also use this time). Each team is allowed to take no more than six half a minute breaks during each battle (see para. “Number of Appearances at the whiteboard”).

Role reversal

The role reversal in the round can only happen if the challenge in this round was accepted. If the opponent has proven that the speaker has no solution (the jury decides if it is so, see para. “Scoring”) the opponent is entitled (but not obliged) to present his solution. If the opponent made the decision to present his solution, then there is a complete change of roles, that is, the former speaker becomes an opponent. If the opponent did not prove that the speaker has no solution, but found in the proposed solution some specific drawbacks, he/she is entitled (but not obliged) to dispose of all (or some) of these drawbacks (“patch holes”). The opponent gets the same right if she/he proves that the speaker has no solution, but refuses to present his own solution. If the opponent decides to “patch holes”, then a partial change of roles appears: the opponent states exactly what he/she is going to do (for example, analyse the case, which was not inquired into by the speaker; prove the statement unproven by the speaker and so on), and the former speaker opposes him. The inverse reversal of roles does not happen in any case!

The correctness of the challenge

If the challenge is accepted by one team, then the question of its correctness is not considered. If the challenge is not accepted, then there are two possibilities:

1. if the team that made the challenge also refuses to answer, then the challenge is automatically deemed invalid;
2. if the team that made the challenge nominates the speaker, then the correctness of the challenge depends on the consequent progression of the round.

The challenge is deemed incorrect if the opponent manages to prove that the problem is not solved. In the case of the opponent declaring the problem solved, the challenge is “automatically” accepted as correct!

Number of Appearances at the Blackboard

Each member of the team has the right to go to the whiteboard as a speaker or an opponent no more than two times per battle. The team is entitled to replace the speaker or an opponent no more than three times per battle. In each case, the appearances are counted for both team members. At each replacement, the time allotted to that team for breaks reduces by one minute (this minute can be used directly before the replacement, or it can be not used. In the latter case, the opponent’s team is also not allowed to use it).

The Challenge Procedure. End of the Battle

If the challenge has been recognised as invalid, the team has to repeat the challenge in the next round. In all other cases, the teams challenge each other in turn. At any time during the battle the team whose turn it is to make a challenge may refuse to do so (it usually happens when the team has no more solved problems, and it is risky to make a challenge, which may be flawed). Then, the other team gets the right (but not the obligation) to present the solution to the remaining problems. In this case the team which refused to make a challenge can nominate the opponents and earn points just for opposing, but has no right to present the solutions (i.e., after the rejection of the challenge, neither full, nor partial change of roles can occur). The battle ends when all the problems are discussed, or when one of the teams refuses the challenge and the other team refuses to present the solution of the remaining problems.

Earning Points

Each problem is worth 12 points, which are distributed between the speaker, the opponent and the jury by the end of the round. If the speaker presents a correct and complete solution, she/he gets all 12 points.

If the opponent was able to find significant errors in the solution, the jury decides whether the opponent was able to prove that the speaker does not solve the problem. If the opponent fails to do so, she/he can nevertheless get the points for opposing, depending on the gravity of the defects of the presentation and how the speaker (or an opponent if there was a partial change of roles) managed to fix them. Generally, the opponent receives half of the “cost” of the “holes” that were not “patched up” by the speaker in the solution (the principle of “half”). However, if the speaker was able to present a complete solution only after considerable errors in the solution are found by the opponent and/or the jury (“mud” in solution), then the jury can take away up to two points from the speaker and transfer it to the opponent, or leave it for themselves. If there was a change of roles, the former opponent gets extra points for proving the statements that were formulated prior, as does the former speaker - for her/his opposition (the “cost” of the concerned statements is defined by the jury, and the distribution of the points happens in the same way, as when opposing a complete solution - taking into account the principle of “half” and “dirt” in the debates). The remaining points are distributed between the speaker and the jury, and the round ends. If the opponent fails to prove that the speaker has no solution, she/he gets points for opposing it (taking into account the principle of “half”). If the challenge was accepted, the opponent gains the opportunity to present his solution (see para. “Role reversal”).

If this is accompanied by a complete or partial change of roles, then the scoring process proceeds according to the method described above.

If the errors or gaps in the report are pointed out by the speaker himself and are not taken care of by his team, the opponent receives points for them, as if she/he had found these drawbacks himself.

If the captain of the challenging team immediately admits that his team does not have a solution after his challenge had been refused by the other team, the opponent’s team receives 6 points for opposing (which in this case consists of a single sentence: “You have no solution”), and the challenge is accepted as incorrect. In this case, the speaker and the opponent are not nominated and the appearances at the whiteboard are not counted.

The Captain

During the battle, only the captain can speak to the jury and contestants on behalf of his/her team: to challenge the other team, to refuse or accept a challenge, to request a break, and so on. He/she has the right to stop the presentation or the opposition of a representative of his team at any time. If the captain is at the whiteboard, she/he nominates a deputy, who performs the duties of the captain at this time. The names of the captain and the deputy are reported to the jury before the beginning of the battle.

The primary responsibility of the captain during the solving of the problems is to coordinate the actions of the team members to solve as many problems as possible. To do so, the captain distributes the problems amongst the team members, taking their wishes into account and ensuring that every problem is being solved by someone. The captain also organises the checking of solutions, to make sure they are correct. The captain determines who the speaker and the opponent in the upcoming battle will be and determines the team tactics.

The captains of the teams have the right to ask the jury to provide 5-10 minutes breaks during the battle (approximately every half hour). The break may be provided only between rounds. The team, which is to make a challenge, does so in writing (without an announcement) immediately before the break and hands it to the jury, which announces the challenge immediately after the break.

The Jury

The jury is the supreme interpreter of the rules of the battle. In situations, which are not covered by the rules, it shall decide at its own discretion. The teams are bound by the jury’s decisions.

While the teams are solving the problems, every significant clarification of the problem that was given by the jury to one of the teams should be reported as soon as possible to all of the other teams.

The jury may withdraw the question of the opponent (for example, if it has no merit), or stop the presentation or the opposition if they are taking too long. If the jury cannot gain an understanding of the solution quickly, it may, with the approval of both captains, allocate its representative, who will continue to discuss the problem in cooperation with the speaker and the opponent in another room. During this time, the battle continues with other problems, and the points for these problems are awarded later.

The jury writes the battle protocol on the whiteboard. If one of the teams does not agree with the decision made by the jury regarding the problem, it has the right to immediately demand a break for a few minutes to discuss the situation with the participation of the chairman of the jury. After the beginning of the next round, the score of the previous round cannot be changed.

The jury keeps order. It can penalise the team for noisiness, inappropriate behaviour, or communication with their representative who is located at the whiteboard.

The jury is obliged to give reasons for all of its decisions that do not follow directly from the rules of the battle.