## Thursday, September 19, 2013

### Sports League Scheduling - Using an Imaginary Bead Necklace

My friend Warren has to schedule a 13-week, 13-Team Billiards League and could not figure out how to do it, so he asked me for help. OY!

After spending an inordinate amount of time on and off over three days, at around 4AM this morning, I finally came up with an almost embarrassingly simple way to visualize the problem and solve it!

My solution is shown above using a spreadsheet format. Each column and each row represent a given Team, and each column/row intersection represents a given Team pairing, on a specified Week and at a specified Table. Each of the 13 weeks is shown in a different color. Since 13 is an ODD number of Teams, each week one Team has to take a "BYE" and not play. The main diagonal indicates that Team #1 will take a BYE on Week 1, Team #2 on Week 2, and so on.

AN IMAGINARY BEAD NECKLACE PROVIDES THE SOLUTION!

The diagram below shows how you can utilize an imaginary bead necklace to solve any Sports League Scheduling problem, including both Leagues with an ODD number of Teams and Leagues with an EVEN number of Teams.
The blue necklace represents a case where there are an ODD number of Teams, in this case 13. Imagine you are holding one of the beads (in this case bead "5") and allowing the other beads to stretch the necklace downwards. This represents one week of League play, say Week 5. The "5" bead you are holding is the Team that will take a BYE on that week. The next two beads ("3" and "4") indicate that Teams #3 and #4 will be paired on that week. Looking down the necklace, Teams #2 and #6 will also be paired on Week 5, as will Teams #1 and #7, #13 and #8, #12 and #9, and #11 and #10. Thus, we will need six tables to accommodate the six pairs of Teams.

To get the pairings for any other week, hold the corresponding numbered bead and read the pairings!

The pink necklace represents a case where there are an EVEN number of Teams, in this case 14. Imagine you are holding the STRING SEGMENT between two of the beads (in this case between bead "5" and bead "6") and allowing the other beads to stretch the necklace downwards. This represents one week of League play, say Week 6. Since there are an EVEN number of Teams, no one will have to take a BYE on that week. The top two beads ("5" and "6") indicate that Teams #5 and #6 will be paired on that week. Looking down the necklace, Teams #4 and #7 will also be paired on Week 6, as will Teams #3 and #8, #2 and #9, #1 and #10, #14 and #11, and #13 and #12. Thus, we will need seven tables to accommodate the seven pairs of Teams.

To get the pairings for any other week, hold the STRING SEGMENT that links the corresponding numbered bead to its predecessor and read the pairings!

CONCLUSION

I hope this will help anyone who needs to schedule a Sports League with any number of Teams. Please post a comment if you find this helpful (or if you have any suggestions on how I can make it clearer.)