How can 8 , 10 or 12 teams rotate through 7 or 8 games without overlaps? [closed]

Question

We are scheduling a big scout event with children, and we have 7 or 8 games organized for them to rotate and play with each other.

Set up a game schedule that follows these rules: There are 3 different concepts 8, 10, or 12 teams but please help me with whatever is feasible

• Only two teams can play a game at the same time

• Each team must play each game once.

• Each team must play against another team exactly once.

Answer 1

Yes, it is possible. Simply by assigning each team to each game, and assigning the opponent who hasn't played that game before. In the table below, the teams are A, B, C, D, E, F, G, H, and the games are numbered 1-7.

$$ \begin{array}{ccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \fbox{AB}&\fbox{AC}&\fbox{AD}&\fbox{AE}&\fbox{AF}&\fbox{AG}&\fbox{AH} \\ \fbox{ }&\fbox{BH}&\fbox{BC}&\fbox{BD}&\fbox{BE}&\fbox{BF}&\fbox{BG} \\ \fbox{CG}&\fbox{ }&\fbox{ }&\fbox{CH}&\fbox{CD}&\fbox{CE}&\fbox{CF} \\ \fbox{DF}&\fbox{DG}&\fbox{ }&\fbox{ }&\fbox{ }&\fbox{DH}&\fbox{DE} \\ \fbox{EH}&\fbox{EF}&\fbox{EG}&\fbox{ }&\fbox{ }&\fbox{ }&\fbox{ } \\ \fbox{ }&\fbox{ }&\fbox{FH}&\fbox{FG}&\fbox{ }&\fbox{ }&\fbox{ } \\ \fbox{ }&\fbox{ }&\fbox{ }&\fbox{ }&\fbox{GH}&\fbox{ }&\fbox{ } \\ \end{array} $$

Each column represent one game. Each row here doesn't really correspond to anything exact, it just helped me in constructing the table. Simply find the first available game for the next two teams, then do each game afterwards, cycling back to beginning when necessary.

Note that this table doesn't show how to do 4 different games simultaneously for 7 times, but just show which teams should play which other team at which game. If you can play 4 of the same game simultaneously, then each column corresponds to that (i.e., in round 1, everyone plays game 1, with A facing B, C facing G, D facing F, and E facing H).

Edit after updated question requiring 7 sets of 4 different games each round, by manually assigning each teams-game combo above into each round:

$$ \begin{array}{r|ccccccc} \text{Game}& 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline \text{Round }1 & \fbox{AB}&\fbox{ }&\fbox{EG}&\fbox{ }&\fbox{ }&\fbox{DH}&\fbox{CF} \\ \text{Round }2 & \fbox{ }&\fbox{AC}&\fbox{ }&\fbox{ }&\fbox{GH}&\fbox{BF}&\fbox{DE} \\ \text{Round }3 & \fbox{ }&\fbox{BH}&\fbox{AD}&\fbox{FG}&\fbox{ }&\fbox{CE}&\fbox{ } \\ \text{Round }4 & \fbox{ }&\fbox{ }&\fbox{FH}&\fbox{AE}&\fbox{CD}&\fbox{ }&\fbox{BG} \\ \text{Round }5 & \fbox{EH}&\fbox{DG}&\fbox{BC}&\fbox{ }&\fbox{AF}&\fbox{ }&\fbox{ } \\ \text{Round }6 & \fbox{DF}&\fbox{ }&\fbox{ }&\fbox{CH}&\fbox{BE}&\fbox{AG}&\fbox{ } \\ \text{Round }7 & \fbox{CG}&\fbox{EF}&\fbox{ }&\fbox{BD}&\fbox{ }&\fbox{ }&\fbox{AH} \\ \end{array} $$

Answer 2

A general process for creating a schedule for an event like this (a round robin where everybody plays everyone else exactly once) is as follows:

• assign each team a letter. If there is an odd number of teams, include an extra letter as the "bye" (the team assigned to play the "bye" team gets to sit out that round).
• line up the letters in a two-row "out-and-back" format:
  ABCD
  HGFE
• play the matches as listed vertically: AH, BG, CF, DE.
• rotate all the letters except A:
  AHBC
  GFED
• play the matches listed vertically again: AG, HF, BE, CD.
• repeat until you end up with team A playing team B. That's the last round.

This process will generate every possible matchup exactly once and in an order guaranteed to schedule every team to play a match every cycle.