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In a game of chess the moves of whites and blacks alternate

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In a game of chess the moves of whites and blacks alternate [#permalink] New post 23 Feb 2008, 07:08
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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

A. III only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

M14-09
[Reveal] Spoiler: OA

Last edited by Bunuel on 24 Jun 2014, 03:15, edited 2 times in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.
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Re: In a game of chess the moves of whites and blacks alternate [#permalink] New post 24 Feb 2008, 07:44
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marcodonzelli wrote:
incognito1 wrote:
marcodonzelli wrote:
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 Incorrect. At the most, blacks lost 4 games.
II. Blacks won more games than whites Possible
III. All games ended in a draw Possible. That white made more moves doesn't indicate that those were necessarily wins (could be draws as well)

 III only
 I and II only
 I and III only
 II and III only
 I, II, and III


(D)


OA is D. can you explain in detail?


assume that each game is made up of total 2 moves only.
since whites start, we have each game as WB.
so we have 2311 games which have this combination WB.
this means whites have made 2311 moves and blacks also 2311 moves.
now if whites have made 4 more moves than blacks, this means that in 4 games they have made an extra move.since the game starts with whites, this means that the game has also ended with whites. So, we have have 4 games with combination WBW, giving one extra white move per game.
Thus so far we have 2311 games with combination WB and 4 more games with combination WBW. so we have 2315 black moves and 2319 white moves as given.
Thus we could have blacks with more games won, than whites.
Also, its possible that every game ended in a draw as they say that the side that made the last move did not loose, which also means that the side that made the last move, either won or faced a draw.
good one+1

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Re: In a game of chess the moves of whites and blacks alternate [#permalink] New post 24 Feb 2008, 11:15
marcodonzelli wrote:
incognito1 wrote:
marcodonzelli wrote:
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 Incorrect. At the most, blacks lost 4 games.
II. Blacks won more games than whites Possible
III. All games ended in a draw Possible. That white made more moves doesn't indicate that those were necessarily wins (could be draws as well)

 III only
 I and II only
 I and III only
 II and III only
 I, II, and III


(D)


OA is D. can you explain in detail?


It's all about counting the number of games that end on either side. 4 games end with white being the last player, and 2315 end with black being the last player.

Let [NW] = number of whites moves, [NB] = number of blacks moves

Consider a single game of chess; since whites start, blacks can either win or draw on their turn. In that case:
[NB] = [NW]

Similarly, whites can win or draw on their turn. In that case:
[NW] = [NB] + 1

We're given the sum total of all white and black wins in a tournament. Clearly, whites have only won OR drawn 4 more games (2319 - 2315). Blacks, on the other hand have won OR drawn as many as 2315 games. Using this information, you can now solve the 3 choices.
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Re: In a game of chess the moves of whites and blacks alternate [#permalink] New post 20 Jun 2014, 00:34
Bunel,

I need your help in understanding the question.

whites: 2319 Moves
Black: 2315 Moves

what dos this means?

Is it talking about chess's black and white square boxes?
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Re: In a game of chess the moves of whites and blacks alternate [#permalink] New post 24 Jun 2014, 03:18
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honchos wrote:
Bunel,

I need your help in understanding the question.

whites: 2319 Moves
Black: 2315 Moves

what dos this means?

Is it talking about chess's black and white square boxes?


It talks about the moves players make with black or white figures.

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


A. III only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

From the stem it follows that there were only 4 games in which whites had the last move. These 4 games were responsible for the difference in the total number of moves made by whites and blacks during the tournament. We know that these 4 games were not won by blacks (but they could well have ended in a draw). All the other games could have been won by blacks or ended in a draw. Thus, scenarios II and III are possible.

Scenario I is impossible. It means that there were at least 5 games in which whites had the last move. If this were true then the difference between the total number of moves of whites and blacks should be at least 5. In fact, it's only 4.

Answer: D
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Re: In a game of chess the moves of whites and blacks alternate   [#permalink] 24 Jun 2014, 03:18
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