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256 teams play in a state soccer tournament. A team is elimi [#permalink]

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28 Apr 2013, 06:07

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256 teams play in a state soccer tournament. A team is eliminated from the tournament after one loss. In the first round, all 256 teams play one game. If a team wins, it advances to the next round, where it plays another winning team. This process repeats itself until only one team is left, having advanced through each round without losing. How many games are played in the tournament?

Re: 256 teams play in a state soccer tournament. A team is elimi [#permalink]

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28 Apr 2013, 06:33

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In order to one team to win, all other teams should lose. So we can calculate the number of games as the number of team-losers. There are 255 such teams. The answer should be A.

P.S. If course you can calculate as sum of 128 games in first round, 64 in second, 32 in third, 16 in fourth, 8 in fifth, 4 in sixth, 2 in seventh, and 1 in eights. Still you get the same answer A. But not so fast.
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Re: 256 teams play in a state soccer tournament. A team is elimi [#permalink]

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28 Apr 2013, 06:42

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The most beautiful thing in this problem that you don't need to do any calculations. The answer will be always the number if teams-1.
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Re: 256 teams play in a state soccer tournament. A team is elimi [#permalink]

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28 Apr 2013, 07:01

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retailingvnsupernova wrote:

smyarga wrote:

The most beautiful thing in this problem that you don't need to do any calculations. The answer will be always the number if teams-1.

Yes, after solving this problem as I posted above, I think that if you have \(2^n\) teams, there's will always be \(2^n-1\) games played smyarga, please explain to me more clearly why there's 255 losers so we will have 255 games played. I think your solution is much faster and more clever than mine.

With pleasure. Every game has exactly one loser. To calculate the number of games is the same as calculate the number of losers. Every team except the winner loses only one game. So the number of games is the number of teams except the winner. Hope this helps.
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Re: 256 teams play in a state soccer tournament. A team is elimi [#permalink]

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28 Apr 2013, 06:35

Hi my friends, this is my solution to this problem in round 1, there's \(2^8\) teams with \(2^7\) games are played. and then in round 2, there's \(2^7\) teams left wiht \(2^6\) games are played in the last round, there's \(2\) teams left with 1 game are played so the games that is played is \(A=2^7+2^6+...+2+1\) we must find the value of A we have \(2.A=2^8+2^7+...^2^2+2\) and then \(2.A-A=(2^8+2^7+...+2^2+2)-(2^7+2^6+...+2+1)=2^8-1\) so we have \(A=2^8-1=255\) The answer is A My friends, I think if you want to solve this problem fast, you should know that \(256=2^8\)
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Last edited by retailingvnsupernova on 28 Apr 2013, 06:36, edited 1 time in total.

Re: 256 teams play in a state soccer tournament. A team is elimi [#permalink]

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28 Apr 2013, 06:49

smyarga wrote:

The most beautiful thing in this problem that you don't need to do any calculations. The answer will be always the number if teams-1.

Yes, after solving this problem as I posted above, I think that if you have \(2^n\) teams, there's will always be \(2^n-1\) games played smyarga, please explain to me more clearly why there's 255 losers so we will have 255 games played. I think your solution is much faster and more clever than mine.
_________________

Life is not easy I knew that and now I don't even expect life to be easy

Re: 256 teams play in a state soccer tournament. A team is elimi [#permalink]

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28 Apr 2013, 07:08

smyarga wrote:

Every game has exactly one loser. To calculate the number of games is the same as calculate the number of losers. Every team except the winner loses only one game. So the number of games is the number of teams except the winner. Hope this helps.

oh my god, my brain is too lazy Thank you ^^
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Re: 256 teams play in a state soccer tournament. A team is elimi [#permalink]

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28 Apr 2013, 10:05

smyarga wrote:

In order to one team to win, all other teams should lose. So we can calculate the number of games as the number of team-losers. There are 255 such teams. The answer should be A.

P.S. If course you can calculate as sum of 128 games in first round, 64 in second, 32 in third, 16 in fourth, 8 in fifth, 4 in sixth, 2 in seventh, and 1 in eights. Still you get the same answer A. But not so fast.

i was over thinking this problem, thanks for explaining a quick and easy method..
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Re: 256 teams play in a state soccer tournament. A team is elimi [#permalink]

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07 May 2013, 23:30

We have simple set of geometrical progression with first element 128 (128 pairs = 256). We have 8 rounds to final game, so A number of games (formula of sum for n elements of geometrical progression ) = (128* (q^8-1))/0,5-1 = (128*(1-0,0625)(1+0,0625))/0,5 = First Impression that answer is 256, but you should look for slightly different answer such as 255 Where q- step of geometrical progression = 0,5!

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