The mathematical carousel is a team competition. Each team has 6 or fewer players (in case of exclusion, the team may consist of more participants). Some of the team members are on the so-called starting line during the carousel, the rest of the team is on the scoring boundary. While the competition continues, some participants move from the initial boundary to the scoring, and sometimes vice versa. Participants who are at different boundaries should not communicate with each other. Therefore, usually the organizers of the carousel place the players of the initial boundary of this team in one place, and the players of the scoring boundary — in another, at some distance from the place of the initial boundary.
First, all team players (usually six people) are located at the starting line in a certain order — from participant #1 to participant #6. The order (numbering of participants) the team determines itself before starting the carousel. According to the signal of the beginning of the competition, each team is given the first task. When the command solved the problem, it writes on the back a piece of paper on which the problem was issued, the answer to it (only the answer is a complete solution and proof is not necessary). The paper team returns to the jury member who checks the answer. If the answer is wrong, nothing happens, the command just gets the next task. If the answer is correct, one of the team members — No. 1 — switches from the initial boundary to the scoring.
Next, the players at the starting and scoring boundaries work independently. And those and others are given new tasks every time. If the players of the initial turn provided the correct answer to the next task, one of them — the one who was the first in the queue — goes to the scoring line (and sits at the end of the scoring queue). If the players of the scoring boundary gave the wrong answer, the player who was the first in the scoring queue goes back to the starting line (and sits at the end of the initial turn). Thus, team players move between the starting and scoring boundaries in a way that resembles riding a carousel. If at the initial turn the task is given the wrong answer, or on the scoring – correct, all players remain in their places.
A significant difference between the starting and scoring boundaries is that for the correct answer of the problem at the scoring boundary, the team receives certain points in its scoring, if the correct answer to the task at the initial turn is provided, then the only benefit for the team is the ability to send another participant to the scoring line. Points on the scoring boundary are awarded as follows: for the first correct answer give 3 points, for the second – 4, for the third 5 points, and for each subsequent one for the score more than the previous one. The chain, however, breaks down when any of the tasks were given the wrong answer. Then for the next task, if it is answered correctly, the team gets 5 points if this one (which was answered incorrectly), cost 6 points or more; 4 points if this one cost 5; 3 points if this one cost 4 or 3 points. The price of the following tasks increases again by 1 (while the team gives the correct answers to them). Thus, after a chain of six correct answers, one wrong and three more correct answers, the team will receive 3 + 4 + 5 + 6 + 7 + 8 + 0 + 5 + 6 + 7 points, and after a series of four correct answers, two wrong answers and another correct answer to the task 3 + 4 + 5 + 6 + 0 + 0 + 4 points (note: for the last task, the team scored 4 points, because the previous one , the sixth, the task cost 5 points).
A new task is given to players at a certain turn when they have answered (correct or incorrect) to the previous task. If the command at the turn is not able to solve a particular problem, it may not write answers, but simply give a piece of paper with the task of a jury member. This will be regarded as the wrong answer.
Tasks can be issued in a manner common to all teams, or so that the same tasks are different for different teams. Participants are informed about this in advance. Priority is the first option, which seems more reasonable and fair.
The number of tasks at both frontiers is limited (for example, 20 tasks can be prepared for each turn). The tasks for the starting and for the scoring boundary are different. The game is considered
finished for the team if there are no problems left on the scoring boundary that it has not solved, or if the allotted time for the competition has expired (usually 2 hours). Sometimes it happens that the tasks have ended at the initial turn, and the scoring team does not have a single participant. Then they also believe that the game for the team is over. The result of the team is the sum of points it received for the tasks correctly solved at the scoring boundary.
While the carousel continues, its protocol is carried out: score points received by each team for each of the tasks of the scoring boundary (in particular, show zeros for incorrectly solved tasks). This is done either directly on the board or with the help of office equipment. Each team member should be able to learn about the intermediate results.