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Solve This Pzzlr: World Cup

Neatorama presents our collaboration with Pzzlr, a site where you can always find a riddle to exercise your brain. Can you solve this one?

A set of soccer games is to be organized in a round robin (e.g. every team plays a match against each other one time). If 45 games are played, how many teams participated?

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This is one of the precious few classes of problems that would've been more easily solved by ancient knowledgeable men than their modern counterparts: we rarely hear mention of triangular numbers, whereas the ancients held them at equal rank with the square numbers we're so comfortable with today. If I asked you what's the side of a square which holds 100 unit squares you'd offer the answer instantly; the problem above, however, seems a lot more counter-intuitive.
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After reading the answer I want to know how is this possibly a tournament? How are teams eliminated? How could one team in the tournament play nine games and another only one and have it be a fair testing of the skill of the soccer team?

It's an appalling way to run a tournament, and certainly unlike any "round robin" tournament I've played in the past.
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This only works if you are eliminated when you lose a game. The issue here is that the manner of elimination was not mentioned.

If every team plays every other team and you have ten teams that means each team plays nine games, or 90 games.

But with elimination at loss -- not mentioned -- then you get the answer at 10.
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I can't view the answer at Pzzlr at work but it sounds complicated so I'll offer my process:
A total of 45 games gives a total of 90 games/all teams (2 teams per game)
A round robin allows each team to play all other teams, so N teams each play N-1 games
90 = N(N-1), giving N=10
As someone pointed out earlier, the equation would be T = N(N-1)/2
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