
Hi Ed,
If I understand correctly, a game involves two teams,
and every week you would want the 5 large teams to play 3 games each
and the other 24 teams to play 2 games each. But this cannot work:
The number of games per week would then have to be (5x3 + 24x2)/2 =
31.5, which is not a whole number. (Here, 5x3 + 24x2 counts the total
number of games played by all teams, and I divide by two because each
game was counted twice: once credited to the winner and once to the
loser.)
In order to fix this and stay fair, the number of games per week of
the smaller teams would have to change some times:
Option 1: If every
smaller team plays 2 games a week for 7 weeks and 1 game for the other
week, then a weekly schedule can be arranged with 30 games per week.
Option
2: If every smaller team plays 2 games a week for 7 weeks and 3 games
for the other week, then a weekly schedule can be arranged with 33 games
per week.
I am confident that your other constraint (same teams
not wrestling twice) can be arranged because the schedule is short
enough. Beforehand I would need to know which of the two options you
prefer.
Claude
I read your reply in the web site and it seems that I was clear enough in what I was looking for. What happens each week is that now each week 3 to 4 teams get together to make up one match. These teams have roughly 45 to 70 wrestlers on them. The matches are taking to long to complete so I thought we could accommodate the 5 largest teams each week by having them wrestle 2 other teams at the same time which in hand would give the large team enough wrestler to match up with so all the kids get a match. All our matches are on a Saturday. We match the kids up by skill, weight and age. There is no team scores. Just individual scores. I figured that if we had 5 large teams out of 29 that would bring our remaining number to 24 teams. Then you take 10 smaller teams each week at the same time to match up the with the larger teams leaving 14 teams left to match up head to head. That would make up one days of matches. I think my math is right when I say that would be 12 matches on a Saturday. Then we need to figure this out so no team wrestles the same team twice on a 8 week schedule. In 8 weeks there would be a total of 96 matches.
The 5 teams that we deemed to be the large teams would always wrestle a tri me or 3 teams at the same time on a Saturday but never the same 2 all year. And the remaining teams each week would make up the dual meets and no one on the season would wrestle the same time. I have this part right I think. What I need help with is how to set the schedule under these boundaries. Is there are formula in which you can shift the numbers around. I thought maybe I work the 24 smaller teams into a 12 game per Saturday schedule first with no one wrestling twice then add the 5 remaining teams to each weeks schedule. This is where it starts getting hard. Can you help.
Ed
Hi Again Ed,
Let's call the small teams a1, a2, a3, ...., a10, a11, a12 and b1, b2, b3,..., b10, b11, b12. That way we can match up these teams using cyclic shifts, where
"Shift k" means team ai plays with team bi+k (and we subtract 12 when the
numbers are above 12).
For instance Shift 0 gives the matchup
a1b1, a2b2, a3b3, a4b4, a5b5, a6b6, a7b7, a8b8, a9b9, a10b10,
a11b11, a12b12,
and Shift 5 gives
a1b6, a2b7, a3b8, a4b9, a5b10, a6b11, a7b12, a8b1, a9b2, a10b3, a11b4, a12b5.
There are 12 shifts to choose from: Shift 0 through Shift 11, and if we take 8 different shifts on the 8 different weeks, the small teams will never meet twice. So we have some leeway, which is good because we need to work in the large teams.
Let's call V, W, X, Y, Z the large teams. I will just look at V's schedule for the moment I can decide for instance that V will play with a1 on week 1, with a2 on week 2, with a3 on week 3, ..., with a7 on week 7 and with a8 on week 8. That way, all the ateams that V plays are different, and I need to choose
the shifts so that the bteams are different as well.
Week 1: Shift 0, so the matchup is Va1b1.
Week 2: Shift 1, so the matchup is Va2b3.
Week 3: Shift 2, so the matchup is Va3b5.
Week 4: Shift 3, so the matchup is Va4b7.
Week 5: Shift 4, so the matchup is Va5b9.
Week 6: Shift 5, so the matchup is Va6b11.
Week 7: I cannot use Shift 6, because it would give the matchup is Va7b1 but V already played with b1. However Shift 7 is good. It gives Va7b2.
Week8: Shift 6 is no good: Va8b2. Shift 8 is good: Va8b4.
That way, V never plays the same team twice. Now the shifts for each are
pinned down and cannot move. I will matchup W, X, Y, Z by "rotating everything cyclically" as we say in the business:
The successive apartners of V were a1, a2, a3, a4, a5, a6, a7, a8.
The successive apartners of W will be a3, a4, a5, a6, a7, a8, a9, a10.
The successive apartners of X will be a6, a7, a8, a9, a10, a11, a12, a1.
The successive apartners of Y will be a8, a9, a10, a11, a12 a1, a2, a3.
The successive apartners of Z will be a11, a12 a1, a2, a3 a4, a5, a6.
(Here I jump up by 2, then 3, then 2, then 3 so that the number of trimatches of all the ateams will be roughly the same. I hope it will be the similar for the bteams.)
So, all in all, the schedule is the following:
Week 1: Va1b1, a2b2, Wa3b3, a4b4, a5b5, Xa6b6,
a7b7, Ya8b8, a9b9, a10b10, Za11b11, a12b12
Week 2: a1b2, Va2b3, a3b4, Wa4b5, a5b6, a6b7,
Xa7b8, a8b9. Ya9b10, a10b11, a11b12, Za12b1
Week 3: Za1b3, a2b4, Va3b5, a4b6, Wa5b7, a6b8,
a7b9, Xa8b10, a9b11, Ya10b12, a11b1, a12b2
Week 4: a1b4, Za2b5, a3b6, Va4b7, a5b8, Wa6b9,
a7b10, a8b11, Xa9b12, a10b1, Ya11b2, a12b3
Week 5: a1b5, a2b6, Za3b7, a4b8, Va5b9, a6b10,
Wa7b11, a8b12, a9b1, Xa10b2, a11b3, Ya12b4
Week 6: Ya1b6, a2b7, a3b8, Za4b9, a5b10, Va6b11,
a7b12, Wa8b1, a9b2, a10b3, Xa11b4, a12b5
Week 7: a1b8, Ya2b9, a3b10, a4b11, Za5b12, a6b1,
Va7b2, a8b3, Wa9b4, a10b5, a11b6, Xa12b7
Week 8: Xa1b9, a2b10, Ya3b11, a4b12, a5b1, Za6b2,
a7b3, Va8b4, a9b5, Wa10b6, a11b7, a12b8.
It turns out that every small team plays 3 or 4 times in trimatches, except b8 and b11 who play only 2 times, and b9 which plays 5 times. Let me know if this is a problem.
Claude

