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?

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).

So

First round:

681 players

681/2 = 340 mod 1

Number of matches in round: 340

Total number of matches: 340

Number of players at the end of the round: 340 + 1 = 341

Second round

341 players

341/2 = 170 mod 1

Number of matches in round: 170

Total number of matches: 340 + 170 = 510

Number of players at the end of the round: 170 + 1 = 171

Third round

171 players

171/2 = 85 mod 1

Number of matches in round: 85

Total number of matches: 510 + 85 = 595

Number of players at the end of the round: 85 + 1 = 86

Fourth round

86 players

86/2 = 43 mod 0

Number of matches in round: 43

Total number of matches: 595 + 43 = 638

Number of players at the end of the round: 43 + 0 = 43

Fifth round

43 players

43/2 = 21 mod 1

Number of matches in round: 21

Total number of matches: 638 + 21 = 659

Number of players at the end of the round: 21 + 1 = 22

Sixth round

22 players

22/2 = 11 mod 0

Number of matches in round: 11

Total number of matches: 659+11 = 670

Number of players at the end of the round: 11 + 0 = 11

Seventh round

11 players

11/2 = 5 mod 1

Number of matches in round: 5

Total number of matches: 670+5 = 675

Number of players at the end of the round: 5 + 1 = 6

Eight round

6 players

6/2 = 3 mod 0

Number of matches in round: 3

Total number of matches: 675+3 = 678

Number of players at the end of the round: 3 + 0 = 3

Ninth round

3 players

3/2 = 1 mod 1

Number of matches in round: 1

Total number of matches: 678+1 = 679

Number of players at the end of the round: 1 + 1 = 2

Tenth round

2 players

2/2 = 1 mod 0

Number of matches in round: 1

Total number of matches: 679+1 = 680

Number of players at the end of the round: 1 + 0 = 1 (the champion)

So, 680 matches are played altogether.