Math CentralQuandaries & Queries


Question from John, a teacher:

I have six teams, two sites, and four days. We have decided that we want to have tri-meets, where there would be three teams competing against each other at the same site each day. Here is the problem, I need to make sure that each team sees each other team at least once and each team goes to each site at least once.

Hi John,

It is possible. Instead of just giving you a schedule, I'll try to explain how to find one so you can do this yourself in the future. Suppose the teams are 1,2,3,4,5,6. Draw six points, labelled with these numbers, in the shape of a hexagon. The points will represent the teams. A game between two teams will be represented by a line segment joining them. The line segments will be both numbered (by day) and coloured (red for site 1 and blue for site 2). The three teams meeting at each site on a day correspond to two triangles that use up all of the six points. This might become more clear once we get started.

Make a triangle coloured red and numbered 1 on the points 1, 2, 3, and one coloured blue and numbered 1 on the points 4,5,6. This tells you that 1,2,3 play at site 1 on day 1, while 4,5,6 play at site 2 on day 1.

Now make a triangle coloured red and numbered 2 on the points 1, 5, 6, and one coloured blue and numbered 2 on 2, 3, 4. This tells you that 1,5,6 play at site 1 on day 2, while 2,3,4 play at site 2 on day 2. Notice that 2 and 3, and 5 and 6 are joined by 2 line segments (because they play each other twice, so far).

Let's look at what has happened so far. The pairs not joined by a line segment are 1,4; 2,5; 2,6; 3,5; and 3,6. And, looking at the colours, point 1 is incident with (touching) only red segments while point 4 is incident with only blue segments segments. Thus, so far, team 1 only visits site 1 and team 4 only visits site 2. Every other team visits both sites.

Now it is easy to get to where you want. Put a red triangle numbered three on 2,5,6 and a blue one numbered three on 1,3,4. (now we just need team 4 to visit site 1 in the last round). Complete the schedule by putting a red triangle numbered four on 1,2,4 and a blue one numbered four on 3,5,6.

This gives you:
Day 1: site one 1,2,3; site two 4,5,6
Day 2: site one 1,5,6; site two 2,3,4
Day 3: site one 2,5,6; site two 1,3,4
Day 4: site one 1,2,4; site two 3,5,6

Unfortunately, in this schedule, teams 5 and 6 play in every round. I'm sure you can use the same method but do better.

Good luck!

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