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I still remember the first time I was presented with the “n” players in a knockout tournament, no ties, how many matches in the tournament question. n was set at 128; some people were doing the traditional 64+32+16+..+ thing, the rest of us had listened to the teacher. He very pointedly said “Think. How many losers?”. And we were ecstatic children, discovering for ourselves the truth that each match produced precisely one loser, and that the tournament needed 127 people to lose….