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

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.

• Your description is a little unclear. If each team must play all games once and against all other teams once, doesn't that mean that the number of games is equal to the number of other teams, i.e. the number of teams is one more than the number of games? Jul 30, 2022 at 8:23
• Sorry for that, yes I get what you mean. Unfortunately only 7 games have been designed at the moment. It makes no sense if it's 7 games now that I think of it. So, let's make it 8 games, not 7. Should I change the title ? Jul 30, 2022 at 8:35
• 7 games with 8 teams would make sense as each team plays 7 others, 1 for each game. Jul 30, 2022 at 9:29

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:

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

• An easier scheme is to number the teams 0 to 6, and number the games 0 to 6, and when team A plays against team B they play game A+B mod 7. Jul 30, 2022 at 12:08
• Hello, thank you for your answer. Very descriptive. I didn't explain well my assignment though. My question includes how to do 4 different games simultaneously for 7 times as you said. For example, in round 1, the 8 teams play against each other in 4 different games. In the next round, they should play against another team in a different game than they played in the past until they played all games. I am will upload an excel picture if that's alright, me trying to do the same with 12 teams. Please let me know whether it is possible to do the above I described. Jul 30, 2022 at 17:16
• With 12 teams and 7 games it's not possible as mentioned in the comment. From the teams side each team needs to play 11 games, but from the game side each team needs to play 7 times. Jul 30, 2022 at 18:02
• @BillPapadodemas Updated to show configuration as per your updated requirements. Jul 30, 2022 at 18:32
• Thank you so much for the solution, I am curious to know, but don't tell me if I put you in much trouble, which other combination of teams/games makes sense? I am trying to configure the best way possible way since we don't know yet how teams are we going to separate them. If it's gonna be 8-10 or 12 teams. Is there a better solution than 8 teams, and 7 games? like 6 games 8,10,12 member teams or 8 games with 8,10,12 member teams. Sorry for bothering you so much... Jul 30, 2022 at 21:47

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.

• Thank you for the comment, nice processing steps. Appreciate the help. Jul 31, 2022 at 19:56
• @Hellion, nice tips. any nice tips for also assigning those teams to different games each time? And after that, to ensure each team plays each game once? Aug 8, 2022 at 5:41