Math Detours · Grade 7 · Around the corner · 12

Everyone plays everyone

Eight teams, seven rounds, nobody idle. Sit them round a table, pin one, and rotate the rest.

Name ____________________Date ____________

Tonight's idea

In a round robin, everyone plays everyone else once. With eight teams, each has 7 opponents, so the dream is 7 rounds with all eight playing in every round. Trial and error soon gets stuck. This method never does.

Sit team 1 in the middle of a round table and teams 2 to 8 round the edge. In round 1, team 1 plays the top seat and the others pair up straight across, along level lines. For each new round, turn the whole pattern of lines one seat clockwise.

2345678 1 Round 1 2345678 1 Round 2 → RoundGames 11–23–84–75–6 21–32–45–86–7 31–43–52–67–8 41–54–63–72–8 51–65–74–82–3 61–76–82–53–4 71–82–73–64–5
The pattern of lines turns one seat each round. Right: all seven rounds.

Why does every pair meet exactly once? Team 1's line points at each seat once. For two teams on the edge, picture the line joining their seats. Seven seats give the pattern seven different tilts, and that line fits exactly one of them. That is the round they meet.

Watch one: seven teams

With an odd number of teams, someone must sit out each round. That rest is called a bye.

Use the same table, but leave the middle seat empty: team 1 is a ghost. Whoever is drawn against the ghost rests. Rub team 1 out of the schedule above and round 1 becomes: team 2 rests, 3–8, 4–7, 5–6. In round 2, team 3 rests.

That gives 7 rounds of 3 games, so 21 games. Each of the seven teams rests once and plays the other six.

Your turn

  1. (a) The schedule above has 7 rounds of 4 games. How many games is that? Check it another way: each of the 8 teams plays 7 games, and every game has two teams in it. (b) For 10 teams, how many rounds are needed, how many games are in each round, and how many games are there in all?
  2. Make a schedule for 6 teams. Put team 1 in the middle and teams 2, 3, 4, 5, 6 clockwise round the edge, with 2 at the top. Write out all five rounds.
  3. Team 1 drops out of your tournament in question 2, and whoever was due to play team 1 rests instead. (a) Who rests in each round? (b) How many games are left? (c) A league has 9 teams. How many rounds does it need, and how many teams rest in each round?
  4. (a) Liam plans 6 teams without a method. Round 1 is 1–2, 3–4, 5–6. He starts round 2 with 1–3 and 2–4. Why is he stuck? (b) Noor thinks a clever enough plan could let 7 teams finish in 6 rounds. By counting games, show that no plan can.

Stretch: find the round without the table

At the 8-team table, team 5 sits exactly halfway between teams 3 and 7. Those two meet in round 4, the round when 5 is at the top of the pattern, playing team 1. Explain why. Then find the round when 2 and 3 meet. (Go the long way round.)

Answers

  1. (a) \(7 \times 4 = 28\). The other way: \(8 \times 7 = 56\) counts every game twice, once for each team in it, and \(56 \div 2 = 28\). (b) Each team has 9 opponents: 9 rounds. Ten teams make 5 games a round. \(9 \times 5 = 45\) games, and \(10 \times 9 \div 2 = 45\) agrees.
  2. Round 1: 1–2, 3–6, 4–5. Round 2: 1–3, 2–4, 5–6. Round 3: 1–4, 3–5, 2–6. Round 4: 1–5, 4–6, 2–3. Round 5: 1–6, 2–5, 3–4. That is 15 games, and every pair appears once. (If you turned the pattern the other way, you get the same rounds in a different order.)
  3. (a) Team 2 rests in round 1, then 3, 4, 5 and 6 in turn. (b) \(15 - 5 = 10\) games, 2 in each round. It is a full round robin for five teams. (c) Add a ghost to make 10 teams: 9 rounds. One team rests in each round, the one drawn against the ghost.
  4. (a) Only 5 and 6 are left, and they have already played each other in round 1. (b) Seven teams need \(7 \times 6 \div 2 = 21\) games. With seven teams, at most 3 games can be played at once, because one team is always left over. Six rounds give at most \(6 \times 3 = 18\) games. That is fewer than 21, so 7 rounds are needed.

Stretch. In every round the lines are level with each other, and team 1's line cuts each of them in half. So the seat at the top of the pattern is always halfway between the two teams in a pair. Team 5 is at the top in round 4. Going the long way from 3 to 2 you pass 4, 5, 6, 7, 8, and the halfway seat is 6. Team 6 plays team 1 in round 5, so 2 and 3 meet in round 5. The table agrees.

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