Elementary mathematical modeling problem

Scheduling issues

problem ： Five football teams play football on the same field . Play a single round robin , In other words, every two teams of these five teams will play in pairs once in this game . A total of ten games , Play once a day for ten consecutive days . How to arrange the schedule of this game is fair to all teams ？

Problem analysis ：1. What do you want -> Schedule 2. What is the goal ？-> fair

The schedule of any one of the five teams is as follows ：

1.AB 2.BC 3.AD 4.DE 5.BD 6.AE 7.CD 8.BE 9.AC 10.CE

The number of games between each team's two games

A:1,2,2; B:0,2,2; C:4,1,0; D:0,0,1; E:1,1,1

Obviously, this schedule is right for A,E Advantageous , Yes D Unfair .

What is fairness ？-> How to use mathematical language to express fairness ？

Fair definition ： The interval between the two games of each team is the same . It can't be fair ！

for example , There are only three teams , Three matches .

1.AB 2.BC 3.AC

Try to be fair

Fair definition ： Each team reaches the maximum possible number of games at the minimum interval of its adjacent games .

Mathematical problems ： Find the maximum in the minimum interval of the race schedule .

consider n A schedule of a team's single round robin . Note that the minimum interval is r. therefore , Here r Three teams in the two games before and after the game do not participate in this r game , And there are 2r There are two different teams participating in this r game , therefore 2r<=n-3

The biggest possibility is r=[(n-3)/2].

Model ：n One team round robin , A fair schedule is one in which teams play at intervals between their adjacent matches =[(n-3)/2].

Conclusion ： arrange 5 The minimum number of games between teams in their adjacent games is 1 Our schedule is fair to all teams .