This problem is taken from the delightful book "problems for mathematicians, young and old" by paul r. halmos. suppose that 681 tennis players want to play an elimination tournament. that means: they pair up, at random, for each round; if the number of players before the round begins is odd, one of them, chosen at random, sits out that round. the winners of each round, and the odd one who sat it out (if there was an odd one), play in the next round, till, finally, there is only one winner, the champion. what is the total number of matches to be played together, in all the rounds of the tournament