Chess

[crackgmat007]

In a game of chess, the moves of whites and blacks alternate with whites having the first move. During a certain chess tournament whites have made 2319 moves altogether while blacks have made 2315 moves. If in any game the side that made the last move did not lose, which of the following can be true about the tournament?

I. Blacks lost 5 games

II. Blacks won more games than whites

III. All games ended in a draw

Choices:
III only
I and II only
I and III only
II and III only
I, II, and III

[cbenk121]

We're looking for statements that CAN be true, not MUST be true.

Statement 1 could be true - If both teams won the same number of games, their total moves would be the same, since the player to move last doesn't lose. Since there's a discrepancy of 4, that means white won four more games. So as long as n, being the number of games, is equal to or greater than 6, black could lose more than 5 games.

Statement 2 can't be true - If blacks won more games, they would have a higher move count.

Statement 3 could be true - Since the problem says the person who moved last DID NOT LOSE, not WON. They could've tied. So if n = 10, and all games were a draw, then white could've made the last move for 7 of those games, and black made the last move for 3 of those games, leading to white having 4 more total moves overall.

My answer is C, what's the OA?

[crackgmat007]

OA - D

[enniguy]

Good tricky question. Even i fell for the trap. Here's my answer justifying OA.

Note that (from the question), if white wins, white will always have more moves than black.

Win: White. Then, White > Black.

If Black wins, the number of moves of both white and black will be the same.

Win: Black. Then, White (moves) = Black (moves).
Why? Because, white cannot have the last move and lose the game (from the question).

If it is a draw, either of these possibilities exists.
Win: Draw. White > Black or Black = White.

Statement 1: Black lost 5 games. This is false because anyone who plays the last move cannot lose (from the question). IF black lost 5 games, white will have a clear lead of 5 moves. It is only 4 moves from the question.

Statement 2: This is possible. Imagine that they play about 150 games. 146 could be black wins (So black moves will be equal to white). then last 4 can be White wins (White moves > Black moves).

Statement 3: Very much possible from my explanation above for each possibility.

[cbenk121]

Good tricky question. Even i fell for the trap. Here's my answer justifying OA.

Note that (from the question), if white wins, white will always have more moves than black.

Win: White. Then, White > Black.

If Black wins, the number of moves of both white and black will be the same.

Win: Black. Then, White (moves) = Black (moves).
Why? Because, white cannot have the last move and lose the game (from the question).

If it is a draw, either of these possibilities exists.
Win: Draw. White > Black or Black = White.

Statement 1: Black lost 5 games. This is false because anyone who plays the last move cannot lose (from the question). IF black lost 5 games, white will have a clear lead of 5 moves. It is only 4 moves from the question.

Statement 2: This is possible. Imagine that they play about 150 games. 146 could be black wins (So black moves will be equal to white). then last 4 can be White wins (White moves > Black moves).

Statement 3: Very much possible from my explanation above for each possibility.

Ahaha, that is quite a trap. Nice problem, and thanks for further explanation.