We shall prove the following identity:

,

with perhaps the most beautiful proof I have ever found.

Assume that we play a tournament with players, in which every game is a knock-out game between players. So in the first round there are games to be played, one for every people. Then in the second round only half of the players remains (i.e. ), so there are another games to be played, and so forth. Right up to the semi final and final, which are just and games, respectively. So the total amount of games played in the tournament is: . Another way to see how many games total there are played is the following: in every game, exactly player gets eliminated. There is only one winner, so the other players must get elimated somewhere. So the total amount of games played must equal . Kablam.

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