the others rotated clockwise one position This is repeated until you end up almost back at the initial position: To see that - with an even number of competitors - this algorithm realizes every possible combination of them (equivalently, that all pairs realized are pairwise different), we argue as follows. First, the algorithm obviously realizes every pair of competitors if one of them equals (the non-moving competitor). Next, for pairs of non- competitors, let their distance be the number 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 (), has distance to and to and it has distance to and to . In a round, a non-leftmost position (not including ) can only be taken by competitors of a fixed distance. In round of the example, in the second position competitor plays against , their distance is . In round , this position is held by competitors and , also having distance , etc. Similarly, the next position ( against in round , against in round , etc.) can only hold distance- competitors. For every , there are exactly pairs of distance . There are rounds and they all realize one distance- pair at the same position. Clearly, these pairs are pairwise different. The conclusion is that every distance- pair is realized. This holds for every , 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 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