Games & Tournaments

At a glance 9 rules

1. 1Round-robin matches$n\ \text{teams} \Rightarrow \dfrac{n(n-1)}{2}\ \text{matches}$$\text{each team plays } n - 1;\ \text{each pair twice: } n(n-1)$
2. 2Knockout matchesEvery knockout match sends exactly one player out, so n players need n - 1 matches, byes or not. Out after two losses? Count losses instead: one per match.
3. 3Byes and roundsByes = the smallest power of 2 that is at least n, minus n, all in round 1 (21 players: 32 - 21 = 11). That power gives the rounds: $2^5 = 32$, so 5 rounds.
4. 4Groups, then knockoutCount each stage on its own and add: every group of g teams plays $\frac{g(g-1)}{2}$ matches, then q qualifiers in a knockout play q - 1 more.
5. 5The points poolWith 2 for a win and 1 each for a draw, every match gives out 2 points, so the table totals 2 × matches. With 3 for a win, total = 3 × matches - draws.
6. 6Wins, losses and drawsTotal wins = total losses. A drawn match sits in two teams' rows, so drawn matches = (sum of every team's draws) ÷ 2.
7. 7Read a score backwardsSplit each score into wins and draws over the games played: with 2-1-0 points and 4 games, 7 points can only be 3 wins and 1 draw. Odd points mean an odd number of draws.
8. 8Possible score listsThe list must add up to the pool, and any k players share at least $\frac{k(k-1)}{2}$ wins from their games with each other. So two players can never both win every game.
9. 9Guaranteed qualificationFor a sure top-k finish, find the most points k + 1 teams can all reach together (they beat everyone else and split their own games); one point more is safe.