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 altogether, in all the rounds of the tournament?

Answer:680 matches are played altogether.Step-by-step explanation:A number is odd if the rest of the division of the number by 2 is one.So, for each round, we have that the number of matches played is the number of players in the start of the round divided by 2.The number of players at the end of the round is the number of matches(each match has a winner, that remains in the tournament) plus the rest of the division(the odd player that sit out the round).SoFirst round: 681 players681/2 = 340 mod 1Number of matches in round: 340Total number of matches: 340Number of players at the end of the round: 340 + 1 = 341Second round341 players341/2 = 170 mod 1Number of matches in round: 170Total number of matches: 340 + 170 = 510Number of players at the end of the round: 170 + 1 = 171Third round171 players171/2 = 85 mod 1Number of matches in round: 85Total number of matches: 510 + 85 = 595Number of players at the end of the round: 85 + 1 = 86Fourth round86 players86/2 = 43 mod 0Number of matches in round: 43Total number of matches: 595 + 43 = 638Number of players at the end of the round: 43 + 0 = 43Fifth round43 players43/2 = 21 mod 1Number of matches in round: 21Total number of matches: 638 + 21 = 659Number of players at the end of the round: 21 + 1 = 22Sixth round22 players22/2 = 11 mod 0Number of matches in round: 11Total number of matches: 659+11 = 670Number of players at the end of the round: 11 + 0 = 11Seventh round11 players11/2 = 5 mod 1Number of matches in round: 5Total number of matches: 670+5 = 675Number of players at the end of the round: 5 + 1 = 6Eight round6 players6/2 = 3 mod 0Number of matches in round: 3Total number of matches: 675+3 = 678Number of players at the end of the round: 3 + 0 = 3Ninth round3 players3/2 = 1 mod 1Number of matches in round: 1Total number of matches: 678+1 = 679Number of players at the end of the round: 1 + 1 = 2Tenth round2 players2/2 = 1 mod 0Number of matches in round: 1Total number of matches: 679+1 = 680Number of players at the end of the round: 1 + 0 = 1(the champion)So, 680 matches are played altogether.