Math CentralQuandaries & Queries


Question from Jason, a parent:

Hello. We have a fantasy football league with 14 teams. The season is 13 weeks long and we would like to figure out a way to have two divisions of seven teams each with each team playing their inter-division opponents twice. Is there a way to set this up? I cannot seem to find a way to do it. Thanks!


If I understand correctly, you want each team to play each team in the other division twice. There are 7 teams in the other division, so that needs 14 weeks. The best you can do in 13 weeks is play 6 teams twice and 1 team once.

Suppose the divisions are $A$ and $B.$ In week 1, $A_1$ plays $B_1, A_2$ plays $B_2, …, A_7$ plays $B_7.$ In week 2, $A_1$ plays $B_2, A_2$ plays $B_3, …, A_6$ plays $B_7$ and $A_7$ plays $B_1.$ Just continue the pattern and keep going. when team $A_i$ plays team $B_j$ in some week, they play team $B_{j+1}$ in the next week. If $j$ is 7, then “wrap around” and take $j+1$ to be 1 (note $A_7$ plays $B_7$ in week 1, and $B_1$ in week 2).

Hope this helps.

PS If they play each team in their own division twice, then 12 weeks are needed. Each team can play every other team once over 13 weeks.

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