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| Math Central | Quandaries & Queries |
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Question from Paul, a parent: So I have a question. I noticed you came up with ways to create schedules and we are having a really hard time making our lineup correctly. Basically we have 6 teams that will play over a 12 week period every Monday night. The time slots are 1 hr each playing at 6, 7 and 8. We would like each team to play 3 games at each time slot over the 12 weeks. Is this something that can actually be accomplished? |
Hi Paul,
Here is a schedule.
| w1 | w2 | w3 | w4 | w5 | w6 | w7 | w8 | w9 | w10 | w11 | w12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| t1 | 6 | 6 | 7 | 7 | 8 | 8 | 6 | 6 | 7 | 7 | 8 | 8 |
| t2 | 7 | 7 | 8 | 8 | 6 | 6 | 6 | 7 | 8 | 6 | 8 | 7 |
| t3 | 8 | 8 | 6 | 6 | 7 | 7 | 8 | 6 | 8 | 7 | 6 | 7 |
| t4 | 6 | 7 | 8 | 6 | 7 | 8 | 7 | 8 | 6 | 6 | 7 | 8 |
| t5 | 7 | 8 | 6 | 7 | 8 | 6 | 7 | 8 | 7 | 8 | 6 | 6 |
| t6 | 8 | 6 | 7 | 8 | 6 | 7 | 8 | 7 | 6 | 8 | 7 | 6 |
Week 1 to week 12 (w1 to w12) by team 1 to team 6 (t1 to t6).
The games are oddly weighted (some teams will play one team 3 times before playing another team once), but the weeks can be permuted.
Scenario:
6 teams
Every Friday for 12 weeks
3 time slots per week (6:00, 7:00, 8:00)
Conditions:
Every team should play 4 times at 6:00, 4 times at 7:00 and 4 times at 8:00
Two teams must play each other no fewer than twice, no more than 3 times.
A loose explanation of my steps:
1) Cut the season in half. I figured if I could do it for the first 6 weeks, then I could do it for 12.
2) Create a 5x5 Latin Square with entries {1,2,3,4,5}
* Entries in Row i - Team i vs. entry (Team plays itself = vs. team 6)
* Entries in column j - Matchups during Week j
3) Copy and paste the 5x5 Latin Square so that it now becomes 5x10, which covers weeks 1-10. Then copy column 1,2 = column 11,12
4) Now that the Matchups are dealt with, the times need to be taken care of:
* 6x12 Matrix with entries {6,7,8} corresponding to times
* Four each of {6,7,8} in every row and Two each of {6,7,8} in every column.
* Lay Matrix over top of Latin Squares created above
5) Result
Eric
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