If n is the number of competitors, a pure round robin tournament requires \begin{matrix} \frac{n}{2} \end{matrix}(n - 1) games. If n is even, then in each of (n - 1) rounds, \begin{matrix} \frac{n}{2} \end{matrix} games can be run concurrently, provided there exist sufficient resources (e.g. courts for a
tennis tournament). If n is odd, there will be n rounds, each with \begin{matrix} \frac{n - 1}{2} \end{matrix} games, and one competitor having no game in that round.
Circle method The circle method is a simple
algorithm to create a schedule for a round-robin tournament. All competitors are assigned to numbers, and then paired in the first round: Next, one of the competitors in the first or last column of the table is fixed (number one in this example) and the others rotated clockwise one position: This is repeated until when the next iteration would lead back to the initial pairings: With an even number n of competitors this algorithm realizes every possible combination of them (equivalently, that all pairs realized are pairwise different). First, the algorithm obviously realizes every pair of competitors if one of them equals 1 (the non-moving competitor). Next, for pairs of non-1 competitors, let their distance be the number k of times the rotation has to be carried out in order that one competitor arrives at the position the other had. In the example given (n=14), 2 has distance 1 to 3 and to 14 and it has distance 6 to 8 and to 9. In a round, a non-leftmost position (not including 1) can only be taken by competitors of a fixed distance. In round 1 of the example, in the second position competitor 2 plays against 13, their distance is 2. In round 2, this position is held by competitors 14 and 12, also having distance 2, etc. Similarly, the next position (3 against 12 in round 1, 2 against 11 in round 2, etc.) can only hold distance-4 competitors. For every k, there are exactly n-1 pairs of distance k. There are n-1 rounds and they all realize one distance-k pair at the same position. Clearly, these pairs are pairwise different. The conclusion is that every distance-k pair is realized. This holds for every k, hence, every pair is realized. If there are an odd number of competitors, a dummy competitor can be added, whose scheduled opponent in a given round does not play and has a
bye. The schedule can therefore be computed as though the dummy were an ordinary player, either fixed or rotating. Instead of rotating one position, any number
relatively prime to (n-1) will generate a complete schedule. The upper and lower rows can indicate home/away in sports, white/black in
chess, etc.; to ensure fairness, this must alternate between rounds since competitor 1 is always on the first row. If, say, competitors 3 and 8 were unable to fulfil their fixture in the third round, it would need to be rescheduled outside the other rounds, since both competitors would already be facing other opponents in those rounds. More complex scheduling constraints may require more complex algorithms. This schedule is applied in chess and draughts tournaments of rapid games, where players physically move round a table. In France this is called the
Carousel-Berger system (Système Rutch-Berger). The schedule can also be used for "asynchronous" round-robin tournaments where all games take place at different times (for example, because there is only one venue). The games are played from left to right in each round, and from the first round to the last. When the number of competitors is even, this schedule performs well with respect to quality and fairness measures such as the amount of rest between games. On the other hand, when the number of competitors is odd, it does not perform so well and a different schedule is superior with respect to these measures.
Berger tables Alternatively Berger tables, named after the
Austrian chess master
Johann Berger, are widely used in the planning of tournaments. Berger published the pairing tables in his two
Schach-Jahrbücher (Chess Annals), with due reference to its inventor This constitutes a schedule where player 14 has a fixed position, and all other players are rotated counterclockwise \frac{n}{2} positions. This schedule is easily generated manually. To construct the next round, the last player, number 8 in the first round, moves to the head of the table, followed by player 9 against player 7, player 10 against 6, until player 1 against player 2. Arithmetically, this equates to adding \frac{n}{2} to the previous row, with the exception of player n. When the result of the addition is greater than (n-1), then subtract (n-1) from the sum. This schedule can also be represented as a table, expressing a round in which players meets each other. For example, player 7 plays against player 11 in round 4. If a player meets itself, then this shows a bye or a game against player n. All games in a round constitutes a diagonal in the table. The above schedule can also be represented by a graph, as shown below: Both the graph and the schedule were reported by
Édouard Lucas in as a recreational mathematics puzzle. Lucas, who describes the method as
simple and ingenious, attributes the solution to Felix Walecki, a teacher at
Lycée Condorcet. Lucas also included an alternative solution by means of a
sliding puzzle.
Mnemonic To easily remember this method, the following mnemonic can be used. Starting from the first round, venue = 1 ╭────────────────────────────────────────────────────┐ 1—ω >>> 2—13 >>> 3—12 >>> 4—11 >>> 5—10 >>> 6—9 >>> 7—8 the next round is constructed: ω—8 >>> 9—7 >>> 10—6 >>> 11—5 >>> 12—4 >>> 13—3 >>> 1—2 and then, 2—ω >>> 3—1 >>> 4—13 >>> 5—12 >>> 6—11 >>> 7—10 >>> 8—9 ω—9 >>> ... If the number of players is odd, the player in the first venue gets a bye. If the number is even, an added player (ω) becomes the opponent.
Original construction of pairing tables by Richard Schurig (1886) For an even number n or an odd number n - 1 of competitors, Schurig == See also ==