Question 36
Scheduling a tournament in which all $n$ teams meet over $n-1$ rounds (for example, each half of the Brasileirão) is a fairly elaborate mathematical problem with several solution methods. Here we explore one approach.
Consider a tournament with 6 teams, labeled $1,2,3,4,5$ and $\infty$ (infinity). The first round is $1\times\infty$, $2\times5$, $3\times4$. To construct round $i$, add $i-1$ to each number in the first-round matches, with these conventions:
- if the sum exceeds 5, subtract 5;
- $\infty$ plus any positive integer is $\infty$.
For example, the second round is:
$$\begin{aligned}(1+1)\times(\infty+1)&:\quad 2\times\infty,\\(2+1)\times(5+1)&:\quad 3\times1,\\(3+1)\times(4+1)&:\quad 4\times5.\end{aligned}$$
a. Determine the three remaining rounds by this method.
b. Using this procedure, display the seven rounds of a tournament with eight teams.
Consider a tournament with 6 teams, labeled $1,2,3,4,5$ and $\infty$ (infinity). The first round is $1\times\infty$, $2\times5$, $3\times4$. To construct round $i$, add $i-1$ to each number in the first-round matches, with these conventions:
- if the sum exceeds 5, subtract 5;
- $\infty$ plus any positive integer is $\infty$.
For example, the second round is:
$$\begin{aligned}(1+1)\times(\infty+1)&:\quad 2\times\infty,\\(2+1)\times(5+1)&:\quad 3\times1,\\(3+1)\times(4+1)&:\quad 4\times5.\end{aligned}$$
a. Determine the three remaining rounds by this method.
b. Using this procedure, display the seven rounds of a tournament with eight teams.