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vcbabu
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In a certain play, each of four players must play with each 04 Jun 2009, 11:38
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In a certain play, each of four players must play with each of other players exactly once. If no game ended in a tie, did someone win one and lose two games?
1). No one won all the games.
2). No one lost all the games.



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In a certain play, each of four players must play with each 04 Jun 2009, 14:35
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Number of games =6

1). No one won all the games.
Suppose A won all the games.
AB-A
AC-A
AD-A
BC
BD
CD
In any case the required condition “did someone win one and lose two games” will not be met.
So (1) is SUFFICIENT.

2). No one lost all the games.
EX1.
Suppose A lost all the games.
AB-B
AC-C
AD-D
BC-B
BD
CD-D
So C won 1 game and lost 2
EX2.
Suppose A lost all the games.
AB-B
AC-C
AD-D
BC-C
BD
CD-D
So C won 2 game and lost 1

So (1) is INSUFFICIENT.

IMO A
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In a certain play, each of four players must play with each 04 Jun 2009, 17:07
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vcbabu
In a certain play, each of four players must play with each of other players exactly once. If no game ended in a tie, did someone win one and lose two games?
1). No one won all the games.
2). No one lost all the games.
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There are 4x3 games, then dividing by 2 to eliminate repeats ie (player 1 w/ 2 is the same as 2 w/ 1).

So 6.

In total we have 6 wins, 6 losses, and each person gets 3.
Nobody won all the games --> everyone gets >=1 loss. So we have 2 losses left.

A LLL
B LWW
C LWW
D LWW


2

Sufficient


After playing around with it

Will post a more detailed solution
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In a certain play, each of four players must play with each 04 Jun 2009, 17:10
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waiting for Hades's explanation .
BTW whats OA?
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In a certain play, each of four players must play with each 04 Jun 2009, 23:16
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It's a network flow.

I set it up, very basic. I tried to solve it mentally but no dice, there is no configuration that works. Let me explain it another way

We're going to have 6 matches, so 6 wins & 6 losses. Let's try and make a case where nobody Wins 1 game & Loses 2 games. If nobody can have 3 losses nor 2 losses (from condition in (2), and from additional condition we impose), then someone can have at most 1 loss. But we need to distribute 6 losses total...so let's give everyone 1 loss, and we have 2 losses left. But where do we put those? If we put another loss onto 2 different people, we violate our own condition (no 2 losses). And if we try to stick the 2 losses onto one player, we violate the condition in (2), which states that no player can lose all 3. Hence there will always be at least 1 player who wins 1 but loses 2.

B
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In a certain play, each of four players must play with each 05 Jun 2009, 00:31
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More graphically:

We're going to have \(6\) wins and \(6\) losses.

Let's represent the players via \(A/B/C/D\).

\(L\) = loss
\(W\) = win

Condition in (2): No player can have 3 losses[/m].

Let's look at the question: [b]Can a player have 2 losses?
The YES case means a player CAN have 2 losses.

Let's see if we can make a NO case, that is no player has 2 losses. If it's impossible to make a NO case,
then the answer is always YES


So we have 6 losses: \(LLLLLL\)
& 4 players:

\(A ???\)
\(B ???\)
\(C ???\)
\(D ???\)

So since nobody can have 3 losses, or 2 losses, let's give everyone 1 loss:

\(A L??\)
\(B L??\)
\(C L??\)
\(D L??\)

We have 2 L's left. Where do we put them? If we put the 2 losses onto 1 player, we have 3 losses, which violates our condition
that no player has 3 losses. But our only other choice is to put one loss onto 2 players-- which violates our assumption for the NO
case that no player has 2 losses.

Hence the NO case is impossible.

Sufficient
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In a certain play, each of four players must play with each 09 Jun 2009, 02:09
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From Hade's approach, it clearly that
We have 4 people A,B,C,D and 6 games as follow:
AB
AC
AD
BC
BD
DC
See that we have total 6 Lose and 6 Win, each person play 3 games with other people.
1) If no one wills all, at most a person has 1 lose. So remains 2 lose. These 2 loses can be distributed to one A,B,C or D to make a person who has 3 all loses and each other has 1 lose 2 wins; or they can be distributed to two other people to make 2 people who have 2 loses 1 win. So insuf
2) Because no one lose all 3 games, let's distribute 6 loses into 3 people have 2 loses each; or 2 people have 2 loses each and 2 people have 1 lose each. So suf
Hence B



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