In chess, the winner receives 1 point and the loser receives 0 points. In case of a draw, each of the players receives point.
Fourteen chess players, no two of whom were the same age, took part in a competition in which everyone played against everyone else. After the competition was concluded, a ranking list was drawn up. Of two players with the same number of points, the younger one receives the better placement.
After the competition, Jan noticed that the three top-placed players together received exactly as many points as the total number of points of the last nine players. Jörg remarked on this that, in doing so, the number of games that ended in a draw was maximal. Determine the number of drawn games.
Solution
The total number of points of the last nine players is at least points (for if each of the nine were to play only against one other of the nine, then, since one point is awarded in every game, there would already be 36 points in total). The total number of points of the three top-placed players, however, is at most points, if they were to win all games against the remaining 13 players.
It follows, then, that the last nine players win none of the games against the others, and the top three win all of them, whereby among themselves they may well play to a draw.
The number of games ending in a draw (which, according to Jörg, is indeed maximal!) must therefore be .
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