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 Richard Schurig. This constitutes a schedule where player 14 has a fixed position, and all other players are rotated counterclockwise 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 to the previous row, with the exception of player . When the result of the addition is greater than , then subtract from the sum. This schedule can also be represented as a (n-1, n-1) 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. Original construction of pairing tables by Richard Schurig (1886) For an even number or an odd number of competitors, Schurig builds a table with vertical rows and horizontal rows. Then he populates it starting from the top left corner by repeating